Structural Properties and Convergence Approach for Chance-Constrained Optimization of Boundary-Value Elliptic Partial Differential Equation Systems

Abstract

This work studies the structural properties and convergence approach of chance-constrained optimization of boundary-value elliptic partial differential equation systems (CCPDEs). The boundary conditions are random input functions deliberated from the boundary of the partial differential equation (PDE) system and in the infinite-dimensional reflexive and separable Banach space. The structural properties of the chance constraints studied in this paper are continuity, closedness, compactness, convexity, and smoothness of probabilistic uniform or pointwise state constrained functions and their parametric approximations. These are open issues even in the finite-dimensional Banach space. Thus, it needs finite-dimensional and smooth parametric approximation representations. We propose a convex approximation approach to nonconvex CCPDE problems. When the approximation parameter goes to zero from the right, the solutions of the relaxation and compression approximations converge asymptotically to the optimal solution of the original CCPDE. Due to the convexity of the problem, a global solution exists for the proposed approximations. Numerical results are provided to demonstrate the plausibility and applicability of the proposed approach.

1. Introduction

Partial differential equations (PDEs) are widely used to describe the spatial variations of physical, biological, and social systems as well as processes in mechanical engineering, thermodynamic, chemical engineering, medicine, industrial manufacturing, etc. [1, 2]. Moreover, practical PDE models involve uncertainties arising from imprecise model parameters and the system’s operational environment. In real-life applications, external influences have a non-negligible impact and seriously affect system behaviors [2, 3, 4, 5, 6]. For example, ambient temperature, wind, and pressure are uncertain external influences that seriously impact system performances.

External input uncertainties will cause output uncertainties in system state variables [4, 7, 8, 9, 10, 11]. Such random inputs usually affect the boundary of the system and thus should be compensated by distributed boundary control. Hence, we consider in this study the randomness of the boundary condition of elliptic PDE systems and solve the chance-constrained optimization problems of such systems [12].

This study is an extension of the previous works in [13, 14] in which the randomness from the model parameters of a PDE system was considered but without considering boundary-valued control. In the present study, we consider the randomness from a nonhomogeneous boundary condition of a PDE system, which implies that the required state solution of the chance-constrained optimization of boundary-value elliptic partial differential equation (CCPDE) is a random field [15]. The control input is applied deterministically at the boundary function to compensate for the random disturbances. As a result, the study addresses the issue of chance-constrained optimization of a randomly boundary-valued PDE system.

Mathematically, in this work, we consider a random parameter $\xi \in \mathrm{\Omega }$ coupled with a spatial variable $x\in D$ at the boundary condition of the PDE system. We assume that the uncertainty is under a given probabilistic measure $\mathit{Pr}$ of the complete probability space $\left(\mathrm{\Omega },\mathrm{\Sigma },\mathit{Pr}\right)$ where $\mathrm{\Sigma }$ is a sigma-Algebra in the Borel set $\mathrm{\Omega }$. This study analyzes the properties of infinite-dimensional optimization problems in the reflexive and separable Bochner space with the elliptic PDE system as equality constraint and its probabilistic state constraints as inequality constraints. In general, for CCPDE problems, significant difficulties arise from chance constraints. Specifically, the main structural properties such as continuity, compactness, convexity, and differentiability of the probabilistic state-constraint functions are difficult to analyze. In addition, solving chance-constraints problems is generally not a trivial task.

Therefore, our investigation first focuses on the theoretical analysis of the main structural properties of the probability pointwise state-constrained functions in the CCPDE. The presence of uncertainties on the nonhomogeneous and nonlinear Dirichlet boundary conditions impacts the required state solutions. It is necessary to investigate the optimality conditions to the existence and uniqueness of the solution to the CCPDE problem. Subsequently, since such CCPDE problems are generally difficult to solve directly and also potentially nonsmooth [6, 16], this work proposes smoothing approximation methods to address this difficulty [2, 13].

The numerical computation for solving the CCPDE problem needs a finite-dimensional representation of the infinite-dimensional space through a discretization coupled with an appropriate sampling of the random variables by the multilevel Mont-Carlo method (ML-MCM). Since the resulting finite-dimensional chance-constrained optimization problem is generally nonsmooth, nonconvex, and difficult to solve directly, we use the recently proposed inner-outer approximation approach [6] for the solution of the CCPDE problem. Several structural properties of the inner-outer approximation-based CCPDE are also analyzed in this study. In the previous work, [13] the convexity of the outer approximation was investigated. In this study, we address the convexity issue of the inner approximations to guarantee the optimal global solution of the CCPDE.

2. Problem definition

We consider chance-constrained optimization of a boundary-value elliptic PDE system (CCPDE),

$$\begin{array}{l}\mathit{\text{CCPDE}}:\underset{u}{min}E\left[J\left(y,u,\xi \right)\right]≔\left\{E\left[{‖y-y_d‖}_{H_g^1\left(D\right)}^2\right]+\frac{\rho }{2}{\left[\parallel u{\parallel }^2\right]}_{L^2\left(D\right)}\right\}\\ \text{subject\hspace{0.17em}to}:\end{array}$$E1

$$-\nabla .\left(\kappa \left(x\right)\nabla y\right)=f\left(x\right)\text{in}D\times \mathrm{\Omega },$$E2

$${\left.y\right|}_{\partial D}=g\left(x,u,\xi \right),\xi \in \mathrm{\Omega },x\in \partial D,u\in U$$E3

$$\mathit{Pr}\left\{y_{\mathit{min}}\left(x\right)\le y\left(x,u,\xi \right)\le y_{\mathit{max}}\left(x\right)\right\}\ge \alpha ,x\in D,$$E4

$$u_{\mathit{min}}\le u\left(x\right)\le u_{\mathit{max}},u\in U$$E5

where $D\subset {\mathbb{R}}^n$ is a given bounded convex open spatial domain with Lipschitz boundary $\partial D$ and $n\ge 2$, $\rho$ is a given regularization parameter, $D_c$ is a given compact subset of the closure $\overline{D}$ of $D$. $\nabla .$ and $\nabla$ represent the divergence and gradient operator w.r.t. $x$ in the weak sense of Sobolev spaces, respectively. The state function $y\left(x,u,\xi \right):\overline{D}\times U\times \mathrm{\Omega }\mapsto \mathbb{R}$ is a random continuous function in $H_g^1\left(D\right)=\left\{\mathrm{\Gamma }\in W^{1,2}\left(D\right)|\mathrm{\Gamma }\left(x\right)=g\left(x\right),x\in \partial D\right\}$ with $H_g^1\left(D\right)$ being a closed subspace of the Sobolev space $H^1\left(D\right)=W^{1,2}\left(D\right)$ for any $\xi \in \mathrm{\Omega }$. $y_d\in H^1\left(D\right)$ is a given function describing the desired profile of the state and is assumed twice differentiable w.r.t. $x\in D$.¹ With $<h,g{>}_{H_g^1\left(D\right)}$ and $<h,g{>}_{H^1\left(D\right)}$_, we denote the related standard scalar product (see [1, 2, 3, 4, 5, 7, 8, 17, 18] for more details on Sobolev spaces).

The triple $\left(\mathrm{\Omega },\mathrm{\Sigma },\mathit{Pr}\right)$ represents a complete probability space, with a set of all possible outcomes $\mathrm{\Omega }\subset {\mathbb{R}}^p$, with $\sigma$-Algebra $\mathrm{\Sigma }\subset 2^{\mathrm{\Omega }}$ and probability measure $\mathit{Pr}\left(\cdot \right):\mathrm{\Sigma }\mapsto \left[0,1\right]$ and $\mathit{Pr}\left(\mathrm{\Omega }\right)=1$. The parameter $\xi$ represents uncorrelated input random vector variables distributed homogeneously acting on the system through the boundary $\partial D$. Such disturbances are position-dependent random parameters and distributed inside or outside of the boundary of the spatial domain $\partial D$. In general, such infinite-dimensional random parameters can be treated by a dimensional-reduction method using the Karhunen-Loeve (KL) expansion (see [19]) or a finite-dimensional representation using a discretization method [3, 4].

The input data $g\left(x,u,\xi \right)$ might vary randomly from one point of the boundary domain $\partial D$ to another point and thus their uncertainty should be described in terms of random fields, which can be dealt with a sampled covariance from multilevel Monte Carlo method (MLMCM) [2, 9, 20]. The expected value $E\left[\cdot \right]$ is taken with respect to the probability space and the probability measure possesses the Radon-Nikodym derivative $\varphi$ w.r.t. the Lebesgue measure $\mu$, i.e., $\mathit{dPr}\left(\xi \right)=\varphi \left(\xi \right)\mathit{d\mu }\left(\xi \right)$. Moreover, we suppress the measure $\mu$ and write simply $\mathit{dPr}\left(\xi \right)=\varphi \left(\xi \right)\mathit{d\xi }$. The random variable $\xi$ is assumed to have a continuous probability density function $\varphi \left(\xi \right)$ with $\mathrm{\Omega }$ being its sample space of the support set. For each $u\in U$, due to the random variable $\xi$, the solution $y$ of the boundary values (2)–(3) is a stochastic linear boundary value state function indicated by $y\left(u,\xi ;x\right)$.

After the solution of the PDE system, Eq. (4) expressed by $P\left(.,x\right)=\mathit{Pr}\left\{y\left(x,u,\xi \right)\le y_{\mathit{max}}\left(x\right)\right\}\ge \alpha ,\forall x\in D$ defines a single pointwise probability state constraint, for each $x\in D$, to be satisfied with a given reliability level $\alpha$, where $\alpha \in \left(0,1\right]$. It should be noted that chance constraints for the PDE system considered in this study can be expressed in the following two forms:

1. Single chance constraints

   $$\mathit{Pr}\left\{y_{\mathit{min}}\left(x\right)\le y\left(x,u,\xi \right)\le y_{\mathit{max}}\left(x\right)\right\}\ge \alpha ,\forall x\in D,$$E6

2. Joint chance constraint

   $$\mathit{Pr}\left\{y_{\mathit{min}}\le y\left(x_i,u,\xi \right)\le y_{\mathit{max}},\forall x_i\in D\right\}\ge \alpha ,$$E7

The first one describes the chance constraints imposed on individual points in D (i.e., pointwise chance constraints), while the second one requires the satisfaction of the constraints at all points with a probability level. In this study, only the form of single pointwise constraints is considered. A joint CCPDE is mathematically complex and needs further studies.

The right-hand side of Eq. (2) is assumed to be a function in $L^2\left(D\right)$. Due to (3), $y$ depends on $u$ and $\xi$, and therefore, we need to analyze the existence and uniqueness of a weak solution. In Section 3, we will prove these by verifying the continuous-bilinear and coercivity form in the Bochner space with the associated expectation of the norm $E\left(\parallel \cdot {\parallel }_{H^2\left(D\right)}\right)$ based on the Lax-Milgram theorem.

Specifically in this study, (2) and (3) lead to a boundary value of an elliptic PDE system with nonhomogeneous Dirichlet boundary condition ${\left.y\right|}_{\partial D}=g\left(x,u,\xi \right),\xi \in \mathrm{\Omega }$ and $u\in U$_. $u$ is a control variable bounded by $u_{\mathit{min}}$ and $u_{\mathit{max}}$ by (5). Since the output $y$ is constrained, we have to find an optimal control profile w.r.t. $x\in D,$ in the admissible set $\left(U_{\mathit{adm}}\right)$ by variational analysis.

Now, we define a separable and reflexive Bochner space by mapping $\mathcal{W}≔L\left(\mathrm{\Omega };W\left(D\right)\right)$ from the Borel space to the Sobolev space $W\left(D\right)$

$$\begin{array}{ccc}H\left(\mathrm{\Omega };W\left(D\right)\right)& =& \left\{v:\mathrm{\Omega }\to W\left(D\right):v\text{is\hspace{0.17em}measurable},\parallel v{\parallel }_{\mathcal{W}}\right.\\ & =& \left.{\int }_{\mathrm{\Omega }},\parallel ,v,(,\cdot ,,,\xi ){\parallel }_{W\left(D\right)}^2\varphi \left(\xi \right)\mathit{d\xi }<+\infty \right\}\end{array}$$E8

From the PDE system defined in (2)–(3), the related function spaces of individual inputs are defined as follows:

$$\mathcal{H}≔L^2\left(\mathrm{\Omega };H_g^1\left(D\right)\right),\mathcal{L}≔L^2\left(D\right),\mathcal{G}≔L^2\left(\mathrm{\Omega },H^2\left(D\right)\right),\mathcal{B}≔L^2\left(\mathrm{\Omega },H^{1/2}\left(D\right)\right),\mathcal{K}≔L^{\infty }\left(D\right).$$E9

Since the spaces defined above are separable, the weak measurability for the random PDE system is equivalent to the strong measurability (see [[21], Section 3.5 Cor. 2]).

In addition, we define scalar products in $\mathcal{L}$ and $\mathcal{H}$ spaces, respectively,

$${⟨a,b⟩}_{\mathcal{L}}={\int }_{\mathrm{\Omega }}{\int }_{D}a\left(x,\xi \right)b\left(x,\xi \right)\mathit{dx\varphi }\left(\xi \right)\mathit{d\xi },\parallel a{\parallel }_{\mathcal{L}}^2=<a,a{>}_{\mathcal{L}},$$E10

$${⟨a,b⟩}_{\mathcal{H}}={\int }_{\mathrm{\Omega }}{\int }_{D}a\left(x,\xi \right)b\left(x,\xi \right)\mathit{dx\varphi }\left(\xi \right)\mathit{d\xi }+{\int }_{\mathrm{\Omega }}{\int }_{\partial D}{\left(\nabla a\left(x,\xi \right)\right)}^t\nabla b\left(x,\xi \right)\mathit{dx\varphi }\left(\xi \right)\mathit{d\xi },$$E11

$$\parallel a{\parallel }_{\mathcal{H}}^2=<a,a{>}_{\mathcal{H}}=<a,a{>}_{\mathcal{L}}+<\nabla a,\nabla a{>}_{\mathcal{L}},$$E12

for $a,b\in \mathcal{L}$ and $a,b\in \mathcal{H}$, respectively. In addition, in (9),

$$\begin{array}{l}\mathcal{B}=\left\{v:\mathrm{\Omega }\to W\left(\partial D\right):v\text{is\hspace{0.17em}measurable,}{L}_2\left(\mathrm{\Omega }\times H^{-1/2}\left(\partial D\right)\right)\to \mathbb{R}\right\},\\ \begin{array}{rcl}{‖v‖}_{\mathcal{B}}& =& {\left({\int }_{\mathrm{\Omega }}{‖v‖}_{H^{-1/2}\left(\partial D\right)}^2\varphi \left(\xi \right)\mathit{d\xi }\right)}^{1/2}\\ & =& {\left[{‖v‖}_{L_2\left(\mathrm{\Omega }\times \partial D\right)}^2+{\sum }_{\left|m\right|=\left[s\right]}{\int }_{\mathrm{\Omega }}{\int }_{\partial D}\frac{{\left|{\partial }^{m}v\left(x_1\right)-{\partial }^{m}v\left(x_2\right)\right|}^2}{{\left|x_1-x_2\right|}^{n+2+2\left(s-\left[s\right]\right)}}\mathit{d\sigma \varphi }\left(\xi \right)\mathit{d\xi }\right]}^{1/2},\end{array}\\ {‖v‖}_{\mathcal{B}}^2=\left[{‖v‖}_{L_2\left(\mathrm{\Omega }\times \partial D\right))}^2+{\int }_{\mathrm{\Omega }}{\int }_{\partial D}\frac{{\left(|v\left(x_1\right)-v\left(x_2\right)\right)}^2|}{{\left|x_1-x_2\right|}^{\frac{s+1}{2}}}\mathit{d\sigma \varphi }\left(\xi \right)\mathit{d\xi }\right]<\infty ,s=1/2,\end{array}$$E13

is a norm of trace function in the boundary space $\mathcal{H}$ of the boundary value with a compact embedding from $\mathcal{B}$ where $\mathit{d\sigma }$ is the surface measure at the boundary of $D$ [22]. Finally, the space of the model parametric coefficient $\mathcal{K}=\left\{v:L^{\infty }\left(D\right)\to \mathbb{R}:\mathrm{v}\text{is measurable}:\parallel v{\parallel }_{L^{\infty }\left(D\right)}={\mathit{\text{maxsup}}}_{n\to \infty }{\left|v_n\right|}^2<\infty \right\}$. Thus, $v\in \mathcal{H}$, it implied that $v\left(\cdot ,\xi \right)\in H_g^1\left(D\right)$ and $E\left[\parallel v\left(\cdot ,\xi \right){\parallel }_{H_g^1\left(D\right)}^2\right]<+\infty$.

The probability density $\varphi$ is assumed to be Lebesgue measurable and almost everywhere positive on $\mathrm{\Omega }$. Hence, the spaces $\mathcal{L},\mathcal{G},\text{and ℋ}$ are a reflexive Bochner space, e.g., Hilbert spaces using the standard equivalence classes. Note also that $\mathcal{G},\mathcal{H},\mathcal{K},\mathcal{B}$ are dense subspaces of $\mathcal{L}$ in the topology of $\mathcal{L}$.

The variable $u\in L^2\left(D\right)$ is a decision variable that belongs to the set of admissible decisions

$$U_{\mathit{adm}}≔\left\{u\left(x\right)\in L^2\left(D\right)|u_a\le u\left(x\right)\le u_b\right\},\forall x\in D,\text{for}{u}_a\ge u_{\mathit{min}}\text{and}{u}_b\le u_{\mathit{max}},$$E14

where $u_a,u_b\in L^2\left(D\right)$ are given functions with $u_a\le u_b$. Observe that equalities and inequalities of functions in the Lebesgue space $L^2\left(D\right)$ and corresponding Sobolev spaces are valid only almost everywhere on $D.$ The term almost everywhere (a.e.) will be suppressed in this study assuming almost surely (a.s.) without any confusions arising. Note that $U_{\mathit{adm}}$ is a nonempty, convex, closed, and bounded subset of $L^2\left(D\right)$.

In the elliptic PDE system (2)–(3), the random parameters in the boundary condition in $\mathcal{B}$² represent the effect of external and internal disturbances such as ambient temperature, pressure, and wind; also, there is a factor of imprecise model parameters, while those in the forcing term $f$ are nonrandom input function. For the sake of simplicity of presentation, the coefficient and forcing term ($\kappa ,f$) respectively are considered as nonrandom input functions in this study. Moreover, the forcing term is continuous w.r.t. $x\in D$ a.s. and $f\left(x\right)\in \mathcal{L}.$³.

As a result, pointwise probabilistic state constraint $P\left(.,x\right)=\mathit{Pr}\left\{y_{\mathit{min}}\le y\le y_{\mathit{max}}\right\}\ge \alpha ,$ overall the spatial region $x\in \mathit{Da}.s.,$ is conservative (worse-case) if, its reliability level $\alpha =1,$ with no chance of constraints violation. The internal function $y_{\mathit{min}}\le y\le y_{\mathit{max}},\forall x\in D$ cannot be computed deterministically. Hence, the expression in (4) defines a chance (probabilistic) constraint by stipulating the satisfaction of the inequality constraint on $y\in \mathcal{H}$ with a given probability value of a reliability level $\alpha \in \left[\mathrm{0.95,1}\right)$. Moreover, (4) represents a pointwise chance constraint, i.e., the constraint on the state variable is required to hold with the same reliability level $\alpha$ at each individual point of $x\in D$.

The required random state solution $y\left(x,u,\xi \right)$ is a function in the infinite-dimensional space $\mathcal{H},$ so that, the infinite number of probabilistic constraints make sense whenever in the equivalence class $y$ is w.r.t. $x$ a continuous element, which is ensured by Sobolev embedding theorems in $L^2(\mathrm{\Omega },H^2\left(D\right)\cap H_g^1\left(D\right)$), excellent properties of the inhomogeneity term $f$, and the convexity of $D$. From the embedding theory of Sobolev space, one can use a more general setting in ${\mathcal{H}}^p=W^{1,p}$ with $p>d$ and sufficient regular $D$. Thus, the convexity of $D$ can be relaxed. We give here only one opportunity, where it works. It is essential for our approximation approach (see, e.g., theorem 3.5 continuity of $y$, that space for $y$ can be continuously embedded in the space of continuous functions).

For instance, at some critical spatial locations $x\in D_c\subset D$, the reliable level $\alpha \in \left[\mathrm{0.95,1}\right)$ can be chosen, as a result, this study focuses on the solution of CCPDE with pointwise constraints but considers reliable level independent of $x$ for simplicity of representation, and it is not trivial to directly extend our inner and outer approximation concept in [6] to joint and uniform chance constraints for the infinite number of $x\in D$. Therefore, solving the CCPDE problem is not a trivial task since there is no simple equivalent deterministic representation. Also, there is no closed-loop analytic representation for the probabilistic state constraint in the equation expressed in (4). The structural properties of (4) are not yet analyzed properly, generally unknown, nondifferentiable, and nonconvex.

3. Existence of the solution of the PDE system

In this paper, we need to solve the weak variational form of the random PDE system with nonhomogeneous Dirichlet boundary value of the elliptic PDE system as we defined in equations expressed (2) and (3), the control is applied on the boundary of polygonal spatial domain $D$ and the control $u\in L^2\left(D\right),$

$$-\nabla \cdot \left(\kappa \left(x\right)\nabla y(x,\xi \right)=f\left(x\right),\mathrm{on}D\times \mathrm{\Omega }\mathrm{a}.\mathrm{s}.,$$E15

$${\left.y\left(x,u,\xi \right)\right|}_{x\in \partial D}=g\left(x,u,\xi \right),\xi \in \mathrm{\Omega }\mathrm{a}.\mathrm{s}.$$E16

For every test function $v\in \mathcal{H}$, we can apply integration by part,

$$E\left[{\int }_D\kappa \left(x,\cdot \right)\nabla y(x,\cdot )\nabla v(x,\cdot )\mathit{dx}\right]=E\left[{\int }_{D}f\left(x\right)v(x,\cdot )\mathit{dx}\right]+E\left[{\int }_{\partial D}g\left(u,x,\cdot \right)v(x,\cdot )\mathit{ds}\right],\forall v\in \mathcal{H},$$E17

which is the weak form of the PDE system (15)–(16) with $f\left(x\right)$ in the Hilbert space $\mathcal{L}$ and solution $y\left(.\right)\in \mathcal{H}$. The relevant functions of the inputs are in separable and reflexive Bochner space. Since $f\left(x\right)\in \mathcal{L}$, $\kappa \in \mathcal{K}$_, and $g\in \mathcal{B}$_, the spatial domain D is convex, the well-known shift statements (see, for instance, [[23], Th. 3.30]; i.e., higher regularity of $f$ is shifted to higher regularity of $y$) imply that $x\mapsto y\left(u,x,\xi \right)\in C\left(\overline{D}\right)$. Since the continuity of $x\mapsto y\left(u,x,\xi \right)$ is required only on the subset $D$, the convexity of $D$ is not necessary whenever the critical spatial domain $D_c\subset \mathit{\text{intD}}$, $\mathit{\text{intD}}$ is the interior of space $D$. However, to guarantee the well-posedness of the weak form, our investigation is based on the following standard assumptions.

Assumption 3.1. (A1.1) The domain $D$ is convex, the set $D_c\subset \overline{D}$ is compact and $y_c\in C\left(D\right)\cap H_g^1\left(D\right),y_d\in H_g^1\left(D\right)\cap H^2\left(D\right)$.

(A1.2) The coefficient $\kappa \left(\cdot \right)\in \mathcal{K}$ is positive and bounded such that

$$0<{\kappa }_{\mathit{min}}\le \kappa \left(x\right)\le {\kappa }_{\mathit{max}},x\in D\mathrm{a}.\mathrm{s}.,$$E18

where ${\kappa }_{\mathit{min}},{\kappa }_{\mathit{max}}$ are finite constants.

(A2.1) For each $u\in L^2\left(D\right)$, the random forcing term $u\mapsto f\left(u,\cdot \right):L^2\left(D\right)\to \mathcal{L}$ is continuous.

(A2.2) For each $u\in L^2\left(D\right)$, the random forcing term $u\mapsto f\left(u,\cdot \right):L^2\left(D\right)\to \mathcal{L}$ is continuously Fréchet differentiable.

(A3) The forcing term has a Taylor expansion form m $f\left(x,u\right)=u\left(x\right)+{\sum }_{n=0}^{\infty }{f}^n\left(x_0\right)/\left(n!\right))\ast {\left(x-x_0\right)}^n$, where $u\in L^2\left(D\right)$ and $f_0\in \mathcal{L}$.

(A4) For each $u\in L^2\left(\partial D\right)$, the random forcing term $u\mapsto g\left(u,\cdot ,\cdot \right):L^2\left(\partial D\right)\to \mathcal{L}$ is continuous. For each $u\in L^2\left(\partial D\right)$, the random forcing term $u\mapsto g\left(u,\cdot ,\cdot \right):L^2\left(\partial D\right)\to \mathcal{L}$ is continuously Fréchet differentiable. and $g$ is linear w.r.t. $u.$

(A5) The random variables ${\xi }^{{\rm T}}=\left({\xi }_1,\dots ,{\xi }_p\right)$ are independently, identically distributed with a continuous joint multivariate probability density function $\varphi \left(\xi \right)={\mathrm{\Pi }}_{i=1}^p{\varphi }_i\left({\xi }_i\right)$ and the set $\mathrm{\Omega }={\mathrm{\Pi }}_{i=1}^p{\mathrm{\Omega }}_i$, where ${\mathrm{\Omega }}_i\subset \mathbb{R},i=1,\dots ,p$, such that

$$f\left(x,u\right)=u\left(x\right)+\sum _{n=0}^{\infty }{f}^{\left(n\right)}\left(x_0\right)/\left(n!\right)\ast {\left(x-x_0\right)}^n$$E19

$$g=u\left(x\right)+g_0\left(\xi \right)\left(\frac{\partial y_0\left(x\right)}{\partial x_0}\right)+\sum _{n=1}^n{g}_k\left(\xi \right)\frac{\partial y\left(x\right)}{\partial x_k}=u\left(x\right)+\sum _{k=1}^N{g}_k\left(\xi \right)\frac{\partial y_k}{\partial x_k}$$E20

with $u,a_k\in L^2\left(D\right)$, $k=\mathrm{0,1,2},\dots ,n.$.

In the assumption A5, $f$ and $g$ are commonly given as a series, which is called finite-dimensional noise representation (20) (see [4, 5, 20]) for the boundary condition $g$. In fact, for numerical computations, it is essential to reduce the dimension of the uncertainties in $g$ from KL dimension reduction method.

3.1 Solution of the random PDE with nonhomogeneous boundary control

From the equations expressed in (15) and (16), we have to show that the Lax-Milgram theorem of the continuous-bilinear and coercivity property for every test function in $v\in \mathcal{H},$

$$-\left(\nabla \cdot \left({\kappa }_0\nabla y\right)+y_{\partial D}\right)v\left(x,\xi \right)=\mathit{fv}\left(x,\xi \right)+\mathit{gv}\left(x,\xi \right).$$E21

This implies for $v\in \mathcal{H}$,

$$\begin{array}{l}-{\int }_{\mathrm{\Omega }}{\int }_D(\nabla \cdot \left({\kappa }_0\nabla y+y_{\partial D})\right)v\left(x,\xi \right)\mathit{dx\varphi }\left(\xi \right)\mathit{d\xi }\\ ={\int }_{\mathrm{\Omega }}{\int }_D\mathit{fv}\left(x,\xi \right)\mathit{dx\varphi }\left(\xi \right)\mathit{d\xi }+{\int }_{\mathrm{\Omega }}{\int }_{\partial D}\mathit{gv}\left(x,\xi \right)\mathit{d\sigma \varphi }\left(\xi \right)\mathit{d\xi }.\end{array}$$E22

It implies the following integration functions of the expectation,

$$-E\left[{\int }_D\nabla \cdot \left({\kappa }_0\nabla y+y_{\partial D})\right)v\left(x,\xi \right)\mathit{dx}\right]=E\left[{\int }_D\mathit{fv}\left(x,\xi \right)\mathit{dx}\right]+E\left[{\int }_{\partial D}\mathit{gv}\left(x,\xi \right)\mathit{d\sigma }\right],\forall v\in \mathcal{H}.$$E23

The Sobolev space plays several roles in the study of stochastic PDE system [7]. The space $L^2\left(D\right)=H^0\left(D\right)$ equivalence class of real-valued Lebesgue measure and square integrable function defined on the spatial domain D. Let $H^1=H^1\left(D\right)$ denote the vector subspace of $H$ defined by $H^1$=$\left\{v\in H^1:\nabla v_{\mathit{xi}}\in H^1\right\}$ $i=\mathrm{1,2,3},\dots ,n$ the equipped with the norm

$$\parallel v{\parallel }_{H^1}^2=\parallel v{\parallel }_{L^2\left(D\right)}^2+\parallel {\nabla }_{x_i}v{\parallel }_{L^2\left(\partial D\right)}^2.$$E24

Space $H^1\left(D\right)$ is Hilbert space, and it is known as the Sobolev space of order 1. Let $\mathit{\text{Domain}}\left(D\right)$ denote the $C^{\infty }\left(D\right)$ with a compact support. The closure of $\mathit{\text{Domain}}\left(D\right)$ is norm topology of $H^1\left(D\right)$, the required random solution is in $H_g^1\left(D\right)$ is subspace of $H^1\left(D\right).$ The dual of $H_g^1$ is $H^{-1}$, is the dual space of continuous linear function on $H_g^1$, both of Sobolev space $H_g^1$ and $H^{-1}$ are space of functional distributions in the sense of Schwartz and have nonunique representations, and the functional at boundary has a polynomial approximation in Eq. (20), with $g_k\left(\xi \right)$ and $\frac{\partial y}{\partial x_k}\left(x\right)$ for $k=\mathrm{1,2,3}.\dots ,n.$ Where the derivative of $y_k\left(x\right)$ is understood in the sense of distribution. Generally, the input random boundary conditions,

$$g\in L^2\left(\mathrm{\Omega };H^{1/2}\left(\partial D\right)\right)\equiv \mathcal{B},$$E25

that defined by above expansion of orthonormal function the adjoint is in $L^2(\mathrm{\Omega };H^{-1/2}\left(\partial D\right)$ for the defined $\xi \in \mathrm{\Omega }$. Since there is nonzero and nonlinear random function $g\ne 0$ at the Dirichlet boundary condition. The boundary g is approximated by Fourier transform, and one can define Sobolev space one can define Sobolev space $H^s$ for all real numbers s, $s<0$ these are genuine distributions as characteristic function, for $s=0$, we have $H^0=L^2\left(D\right)$ for $s>0$ these are the regular function spaces contained in $H^1$, for example, y belongs to $H_g^1\left(D\right)$ see [21, 24]. There is the trace operator $\mathrm{\Gamma }:L^2(\mathrm{\Omega };H_g^1\left(D\right))\mapsto L^2(\mathrm{\Omega };H^{1/2}\left(\partial D\right),\mathrm{\Gamma }\left(y\right)=y=g\text{for}x\in \partial D$. The space $H^m\subset H^{m-k}$ for $k>0$ and the injection is compact. The trace function loses its interior smoothness and may be a distribution on the boundary as $\frac{\partial y}{\partial n}$ belongs to $H^{-1/2}\left(\partial D\right))$, n is a normal to the test function $v,$ i.e., $\frac{\partial y}{\partial v}\in H^{-1/2}\left(\partial D\right))$. We shall define the inner product space (.,.) or $\prec .,.\succ$ by the double integral function on the set $D$ and $\mathrm{\Omega }$, there is the duality pairing between $H_1$ and $H_1^{\ast }$ at the trace operator. In each case of boundary-value condition of bilinear-continuity form $a\left(y,v\right)$ and coercive (elliptic) form, we can formulate the following equation:

$$a\left(y,v\right)-\left(f,v\right)-\left(g,v\right)={\int }_{\mathrm{\Omega }}{\int }_{D}v\left(\mathit{Ly}-f\right)\mathit{dx\varphi }\left(\xi \right)\mathit{d\xi }+\sum \left[a_{\mathit{ik}}{\partial }_{k}y-g\right]\mathit{vds}$$E26

that vanishes the right-hand side of second integral, and $L$ and $a_{\mathit{ik}}$ are the linear and differential operators, respectively [7]. The weak PDE system gives

$$-E\left[{\int }_D\nabla \cdot \left({\kappa }_0\nabla y\right)v\left(x,\xi \right)\mathit{dx}\right]=E\left[{\int }_D\mathit{fv}\left(x,\xi \right)\mathit{dx}\right]+E\left[{\int }_{\partial D}\left(y-g\right)v\left(x,\xi \right)\mathit{d\sigma }\right],v\in \mathcal{H}.$$E27

Definition 3.2. The system of elliptic PDE in Eqs. (2) and (3) has a weak solution $y\in L^2(\mathrm{\Omega };H_g^1\left(D\right)),$ if there exists a measurable random variable $y$ w.r.t. $\xi$ defined in $\mathrm{\Omega }$ such that $E\left[a\left(y,\vartheta \right)\right]=E\left[l\left(y,v\right)\right]=E[\left(f,\vartheta \right)+E\left[\left(g,\vartheta \right)\right]$ for all $\vartheta \in \mathcal{H}=L^2(\mathrm{\Omega };H_g^1\left(D\right)).$ The operator $a$ satisfies the continuous bilinear and coercive form.

Definition 3.3. The system of elliptic PDE said to be stable in $L^2(\mathrm{\Omega };H_g^1\left(D\right))$, if it has a weak solution $y\in L^2(\mathrm{\Omega };H_g^1\left(D\right))$ and the forcing term $f\in \mathcal{L}$ expressed in Eq. (9) and the boundary input $g\in L_2(\mathrm{\Omega },H^{1/2}\left(\partial D\right)$. The solution $y$ is continuous depending on the random parameter, i.e., $E\left[y\right]=E\left[y\left(f,g\right)\right]$. The subspace of all function form $L^2(\mathrm{\Omega },H^l\left(D\right)$ whose generalized derivatives up to order $l$ exist and belong to $L^2(\mathrm{\Omega },H^l\left(D\right)$. The space $H^l\left(D\right)={W^l}_2\left(D\right)$ is called Sobolev space order $l.$.

Theorem 3.4. Lax-Milgram Theorem: Let $\kappa \left(.\right)\in L^{\infty }\left(D\right)$ be a functional, and there exists a constant ${\kappa }_{\mathit{min}}>0$ and ${\kappa }_{\mathit{max}}>\kappa \left(x\right)>{\kappa }_{\mathit{min}}$ almost surly and the test function $v\in \mathcal{H}=L^2\left(\mathrm{\Omega }\times H_g^1\left(D\right)\right),g\in L_2\left(\mathrm{\Omega }\times H^{1/2}\left(\partial D\right)\right)$ and $f\left(x\right)\in {\left(H^1\left(D\right)\right)}^{\ast }$. The operators $a$ and $l$ defined in Eq. (25) hold continuous-bilinearity and coercivity. Thus, the variational problem defined in Eq. (1) and (2) has unique solution $y\in L_2(\mathrm{\Omega },H_g^1\left(D\right))$ for all $\xi \in \mathrm{\Omega }$.

Proof. The elliptic PDEs in Eqs. (15) and (16) have weak solution if there exists a test function $v\in \mathcal{H},$ this boundary condition $g$ moves to left-hand side

$$\begin{array}{l}-\nabla \left(\kappa \left(.\right)\nabla y\right)v=f\left(x\right)v\text{in}D;\\ \left(y-g\left(.,\xi ,u\right)\right)v=0\mathrm{on}U\times \partial D\times \mathrm{\Omega };\\ \Rightarrow {\int }_{\mathrm{\Omega }}{\int }_D-\nabla \kappa \left(.\right)\nabla y.\mathit{vdx}\mathrm{\Phi }\left(\xi \right)\mathit{d\xi }\\ ={\int }_{\mathrm{\Omega }}{\int }_D\mathit{\text{fvdx}}\mathrm{\Phi }\left(\xi \right)\mathit{d\xi }+{\int }_{\mathrm{\Omega }}{\int }_{\partial \mathit{tD}}\left(g-y\right)\left(.,\xi ,u\right).\mathit{vd\sigma }\mathrm{\Phi }\left(\xi \right)\mathit{d\xi }.\end{array}$$E28

Where $\mathit{d\sigma }=\overleftarrow{n}.\overleftarrow{\mathit{ds}}$ is the surface measure at the boundary of D from integration by part,

$$\begin{array}{l}\Rightarrow {\int }_{\mathrm{\Omega }}{\int }_D\kappa \left(.\right)\nabla y.\nabla \mathit{vdx}\mathrm{\Phi }\left(\xi \right)\mathit{d\xi }-{\int }_{\mathrm{\Omega }}{\int }_{\partial D}\frac{\partial y}{\partial n}\left(\xi ,x\right)v\left(\xi ,x\right).\mathit{d\sigma }\mathrm{\Phi }\left(\xi \right)\mathit{d\xi }\\ ={\int }_{\mathrm{\Omega }}{\int }_D\mathit{\text{fvdx}}\mathrm{\Phi }\left(\xi \right)\mathit{d\xi }+{\int }_{\mathrm{\Omega }}{\int }_{\partial D}\left(g-y\right).\mathit{vd\sigma }\mathrm{\Phi }\left(\xi \right)\mathit{d\xi }\end{array}$$E29

Since $y.v\left(\xi ,x\right)=0$ for $x\in \partial D$ because the unit normal vector is perpendicular to the boundary $\partial D$. Let

$$\begin{array}{c}a\left(y,v\right)={\int }_{\mathrm{\Omega }}{\int }_D\kappa \left(.\right)\nabla y.\nabla \mathit{vdx}\mathrm{\Phi }\left(\xi \right)\mathit{d\xi };\\ l\left(y,v\right)={\int }_{\mathrm{\Omega }}{\int }_D\mathit{\text{fvdx}}\mathrm{\Phi }\left(\xi \right)\mathit{d\xi }+{\int }_{\mathrm{\Omega }}{\int }_{\partial D}g\left(.,\xi ,u\right).\mathit{vd\sigma }\mathrm{\Phi }\left(\xi \right)\mathit{d\xi };\\ \Rightarrow a\left(y,v\right)=l\left(y,v\right).\end{array}$$E30

We need to show that continuous-bilinear and coercivity form. These two properties are sufficient for the existence and uniqueness of weak solution [7, 8]. The input random functions are in a reflexive and separable Bochner space. Thus, weak solution is the same as the classical solution, the weak measurability is also similar to the strong measurability seen in [24].

$$\begin{array}{r}\Rightarrow \mid a\left(y,v\right)\mid =\mid {\int }_{\mathrm{\Omega }}{\int }_D\kappa \left(x\right)\nabla y.\nabla \mathit{vdx}\mathrm{\Phi }\left(\xi \right)\mathit{d\xi }\mid \le \mid k\left(x\right)\mid {\int }_{\mathrm{\Omega }}{\int }_D\mid \nabla y.\nabla \mathit{vdx}\mathrm{\Phi }\left(\xi \right)\mathit{d\xi }\mid \\ \le \parallel k\left(.\right){\parallel }_{L^{\infty }(D}\parallel \nabla y{\parallel }_{L^2\left(\mathrm{\Omega }\times D\right)}\parallel \nabla v{\parallel }_{L^2\left(\mathrm{\Omega }\times D\right)}\\ \le \parallel \kappa \left(.\right){\parallel }_{L^{\infty }(D}\parallel y{\parallel }_{L^2\mathrm{\Omega }\times {H^1}_g\left(D\right)}\parallel v{\parallel }_{L^2\left(\mathrm{\Omega }\times D\right)}.\end{array}$$E31

Hence, the operator $a$ is continuous bilinear. The same case for $l\left(y,v\right)$ this is integral of duality product between a mapping in Hölder theorem

$$\begin{array}{r}\mid l\left(y,v\right)\mid =\left|{\int }_{\mathrm{\Omega }}{\int }_D\mathit{\text{fvdx}}\mathrm{\Phi }\left(\xi \right)\mathit{d\xi }+{\int }_{\mathrm{\Omega }}{\int }_{\partial D}g.\mathit{vd\sigma }\mathrm{\Phi }\left(\xi \right)\mathit{d\xi }\right|\\ \le \left|{\int }_{\mathrm{\Omega }}{\int }_D\mathit{\text{fvdx}}\mathrm{\Phi }\left(\xi \right)\mathit{d\xi }\right|+\left|{\int }_{\mathrm{\Omega }}{\int }_{\partial D}g.\mathit{vd\sigma }\mathrm{\Phi }\left(\xi \right)\mathit{d\xi }\right|\\ \le \parallel f{\parallel }_{(L^2\left(D\right)}\parallel v{\parallel }_{(L_2\left(\mathrm{\Omega }\times D\right)}+\parallel g{\parallel }_{(L^2\left(\mathrm{\Omega }\times \partial D\right)}\parallel v{\parallel }_{(L_2\left(\mathrm{\Omega }\times \partial D\right)}\\ \le \parallel f{\parallel }_{(H^1\left(D\right)}\parallel v{\parallel }_{(L_2\left(\mathrm{\Omega }\times D\right)}+\parallel g{\parallel }_{{\left(L_2\left(\mathrm{\Omega },H^{1/2}\left(\partial D\right)\right)\right)}^{\ast }}\parallel v{\parallel }_{(}{L}_2\left(\mathrm{\Omega },\partial D\right).\end{array}$$E32

Therefore, both $a$ and $l$ are continuous bilinear form. To see coercivity, $\forall v\in V$ Fubini theorem implies that $\mid a\left(v,v\right)\mid ={\int }_{\mathrm{\Omega }}{\int }_D\mid \kappa \left(x\right)\nabla v.\nabla \mathit{vdx}\mathrm{\Phi }\left(\xi \right)\mathit{d\xi }\mid \ge {\kappa }_{\mathit{min}}{\int }_{\mathrm{\Omega }}{\int }_D\mid \nabla v.\nabla \mathit{vdx}\mathrm{\Phi }\left(\xi \right)\mathit{d\xi }\mid$ because ${\kappa }_{\mathit{min}}$ is bounded from below a.s., and independent of random $\xi$, ${\kappa }_{\mathit{min}}\left(x\right)$ is lower bounded as well. Thus, $\mid a\left(v,v\right)\mid \ge {\kappa }_{\mathit{min}}{\int }_{\mathrm{\Omega }}{\int }_D\parallel \nabla v{\parallel }_V^2\mathit{dx}\mathrm{\Phi }\left(\xi \right)\mathit{d\xi }\ge k_{\mathit{min}}/C\parallel v{\parallel }_V^2$ Where $C=c^2$ by Poincare-Friedrichs inequality $\mid a\left(v,v\right)\mid \ge {\kappa }_{\mathit{min}}/c^2\parallel v{\parallel }_{\mathcal{H}}^2\ge {\kappa }_{\mathit{min}}/c^2\parallel v{{\parallel }^2}_V$ the same for $l$. The required numerical solution is obtained from stochastic finite difference method (SFDM) or finite element method (SFEM), for the indexed $x\in D$ the numerical solution

$$y\left(x,u,\xi \right)=A^{-1}(f_{\mathit{ij}}/\left(\kappa \right)\left(u\right)+g_{\mathit{ij}}\left(u,\xi \right)\forall x\in \mathit{Da}.s.$$E33

The $y\left(x,u,\xi \right)$ have a linear property w.r.t. $u,\xi$ jointly. This case the matrix $A$ obtained from the discretization operator $a$ is positive definite [22]. Therefore, the weak solution $y\in L^2\left(\mathrm{\Omega }\times {H^1}_g\left(D\right)\right)$ exists and is unique. We will analyze the continuously dependent of the solution $y\in L^2\left(\mathrm{\Omega }\times {H^1}_g\left(D\right)\right)$ on $f,g,u$ and $k,\forall \xi \in \mathrm{\Omega }.$ □

Theorem 3.5. Suppose the coefficient operator $\kappa ={\sum }_{\mathit{ij}}{\kappa }_{\mathit{ij}}{x}^i\left(\omega \right)x^j\left(\omega \right)$ and ${\kappa }_{\mathit{ij}}\left(x\right)\in L^{\infty }\left(D\right)$ is nonrandom, is independent of $\xi$ seen in [7] and deterministic coefficient function of $\kappa \left(.\right)$ is indexed by $x\in D$. There exists a lower bound number ${\kappa }_{\mathit{min}}>0$ such that ${\kappa }_{\mathit{min}}\le \kappa \left(x\right)\le {\kappa }_{\mathit{max}}$ a.s. and ${\sum }_{\mathit{ij}}{\kappa }_{\mathit{ij}}{x}^i{x}^j\ge {\kappa }_{\mathit{min}}{x}^2$ for $x\in D$ subset of ${\mathbb{R}}^n$. Thus elliptic PDE system is $L^2$ stable in the sense of distribution in Definition (3.2) and Definition (3.3). There exist $c>0$, $c$ is independent of f, g for all fixed $x\in D$, and $c$ is dependent only $\kappa$, such that

$$E{\left(\parallel y\left(f,g\right){\parallel }^2\right)}_{H^1\left(D\right)}\le c\left\{E{\left(\parallel f{\parallel }^2\right)}_{H^1\left(D\right)}+E{\left(\parallel g{\parallel }^2\right)}_{H^{-1/2}\left(\partial D\right))}\right\}$$E34

$\forall f\in H^1\left(D\right)$ and $g\in L^2\left(\mathrm{\Omega },H^{\frac{-1}{2}}\left(\partial D\right)\right),E\left(\parallel f{\parallel }^2\right)={\int }_{\mathrm{\Omega }}\parallel f{\parallel }^2\mathrm{\Phi }\left(\xi \right)\mathit{d\xi }=\parallel f{\parallel }^2{\int }_{\mathrm{\Omega }}\mathrm{\Phi }\left(\xi \right)\mathit{d\xi }=\parallel f{\parallel }^2.$ This theorem is proved and extended in the work [25].

Remark: For dimension $n\le 3$_, the map of the solution is continuous embedding from $L^2\left(\mathrm{\Omega }\times H^1\left(D\right)\right)\to L^2(\mathrm{\Omega }\times {\left(H^1\left(D\right)\right)}^{\ast }\cap L^2\left(\mathrm{\Omega }\times H^{-1/2}\left(\partial D\right)\right)$is fulfilled and the mapping is continuous and linear $E\left[\parallel y{\parallel }_{{H^1}_g\left(D\right)}\right]\le c\{\left[\parallel f{\parallel }_{H^1\left(D\right)}^{\ast }\right]+E\left[\parallel g{\parallel }_{H^{1/2}\left(\partial D\right)}\right]$} for each $\xi$ a.s., see the prove in [7].

Theorem 3.6. Let U and V be Hilbert spaces. Then, the linear mapping $L:U\to V$ is an isomorphic mapping if and only if the associated form $a:U\times V\mapsto \mathbb{R}$ satisfies

1. Continuity, there exists $C>0$ such that $\mid a\left(u,v\right)\mid \ge C\parallel u{\parallel }_U\parallel v{\parallel }_V$ for all $u\in U$

2. Inf-sup condition, i.e., there exists $c>0$ such that $\mathit{sup}\left(\frac{a\left(u,v\right)}{\parallel v{\parallel }_V}\right)\ge c\parallel u{\parallel }_U,\forall u\in U.$

3. For every $v\in V$, there exists $u\in U$ with $a\left(u,v\right)\ne 0$ and if we assume continuity and inf- sup conditions above, then $L:U\to \left\{v\in V:a\left(u,v\right)=0,\forall u\in U\right\}\subset V$ is an isomorphism. Thus, the equation $\mathit{sup}\left(\frac{a\left(u,v\right)}{\parallel v{\parallel }_V}\right)\ge c\parallel u{\parallel }_U$ is equivalent to $\parallel \mathit{Lu}{\parallel }_V\ge c\parallel u{\parallel }_U,\forall u\in U.$ It follows the equivalent formulation ${\mathit{inf}}_{u\in U}\left\{{\mathit{sup}}_{v\in V}\left(\frac{a\left(u,v\right)}{\parallel v{\parallel }_V\parallel U{\parallel }_U}\right)\right\}\ge c>0$.

Proof. We need to show that injective and surjective map $L:U\to V$. The equivalence of continuity of $L:U\to V,$ $L\left(u_1\right)={\mathit{Lu}}_1=L\left(u_2\right)={\mathit{Lu}}_2,u_1,u_2\in U$

$$\Rightarrow a\left(u_1,v\right)=a\left(u_2,v\right),\forall v\in V$$E35

$\Rightarrow a\left(u_1-u_2,v\right)=0$ implies $u_1-u_2=0.$ For the surjective $\forall f\in L\left(u\right),L\left(u\right)$ from the image of the pre image $u$. There exists a unique $u=L^{-1}\left(f\right).$ Thus, $c\parallel u{\parallel }_U\le \mathit{sup}\left(\frac{a\left(u,v\right)}{\parallel v{\parallel }_V}\right)=\mathit{sup}(\frac{a\left(f,v\right)}{\parallel v{\parallel }_V}=\parallel f^{\ast }\parallel$.□

3.2 Reduced optimization of PDE with random data

The problems of CCPDEs are not properly studied. Moreover, the pointwise or uniform probabilistic state-constrained function is a nonsmooth, nonconvex, intractable, and infinite-dimensional state constraints. The following assumptions are needed for reducing dimension and variability for analyzing the structural properties of CCPDE.

Assumption 3.7. Assume that the functionals

1. The $\kappa \left(x\right)\in \mathcal{K}$ of Eq. (9), the $g\left(x,.,.\right)\in \mathcal{B}$, the $f\left(.\right)\in H^{\ast }$, the $H^{\ast }$ is the dual in Sobolev space of $\mathcal{H}$ defined by Eq. (9) and $D_c$ is subset of convex set D have Lipschitz smooth boundary $\partial D$.

2. The control functional $u\left(.\right)\in \mathcal{L}$ is compact and convex, the $\mathcal{L}$ is subset of the admissible set $U_{\mathit{adm}}$.

3. The objective functional E[J(y(u))] is mapping from $L_2\left(\mathrm{\Omega },H^2\left(D\right)\right))\times L_2\left(\mathrm{\Omega },H^{1/2}\left(\partial D\right)\right))\to \mathbb{R}$ is weakly sequentially lower semicontinuous (wsls) and bounded below by $\frac{\rho }{2}\parallel u{\parallel }_{L^2\left(D\right)}^2$.

4. The $E\left(J\left(y\left(u\right)\right)\right)$ is a convex w.r.t. u.

5. The f w.s.l.s w.r.t. $x\in D$ and g are w.s.l.s and convex w.r.t $\left(u,\xi \right)$ simultaneously.

The optimization problem of

$$\underset{u\left(.\right)\in \mathcal{L}}{min}E\left[J(y\left(x,\xi \right),u\left(.\right))\right]$$E36

subject to:

$$P\left(u;x\right)=p_x\left\{u\right\}=\mathit{Pr}\left\{y\left(x,u,\xi \right)\le y_{\mathit{max}}\right\}\ge \alpha ,x\in D$$E37

the random variable $\xi \in \mathrm{\Omega }$ have a log-concave density function $\varphi \left(\xi \right)$. If the objective function in Eq. (36) is convex, then the chance-constrained programming is a convex optimization problem and has a globally optimal solution.

Proof. The functional J is a convex function from the convexity property of norm. The expectation $E$ is the integral of the convex function, is convex. Moreover, the solution of the PDE system $y$ is a linear w.r.t $\left(x,\xi \right)$. The internal functional $\gamma =y-y_{\mathit{max}}$ is a quasiconcave from proposition (3.4), then $P_x\left(u\right)$ is a convex w.r.t. u. Suppose the constrained $P=\left\{u\in U/P\left(u;x\right)>\alpha \right\}$ is a convex set [26]. Hence, the composition functional $E\left(J\left(y\left(u\right)\right)\right)$ is the composition of convex and lower-continuous function. Therefore, $E\left(J\left(y\left(u\right)\right)\right)$ is a convex function and the set $\mathcal{B}=L^2(\mathrm{\Omega },H^{1/2}\left(\partial D\right)\subset P\cap U_{\mathit{adm}}$, convex intersection of convex set is convex.□

The optimization problem of CCPDE reduced to the following programming form and the solution $y$ is continuous dependent on the parameters $f$ and $g$ in theorem (3.5),

$$\underset{u\left(.\right)\in P\cap U_{\mathit{adm}}}{min}q\left(u\left(.\right)\right)=E\left[J\left(y\right(f\left(x\right),\left(g\right(x,\xi ),u\left(.\right))\right],$$E38

where the probability function in Eq. (37) expressed $P\left(.,x\right)=\mathit{Pr}\left\{\gamma =y\left(f\left(x\right),g\left(\xi \right),u\left(.\right)\right)-y_{\mathit{max}}\le 0,\forall x\in D\right\}$, the set $P=\left\{u\in U/P\left(x;u\right)>\alpha \right\}$ this optimization problem admits unique optimal solution. The random variable $\xi \in \mathrm{\Omega }$ has a log-concave density function $\varphi \left(\xi \right)$. If the objective function in Eq. (47) is a convex, then the chance-constrained programming is a convex optimization problem and has a global optimal solution.

3.3 The structural property of probabilistic constrained function

Most of the recent research works on the chance-constrained optimization problems do not include a probabilistic state constraints. In this study, we have deliberated a pointwise probabilistic state constraints, which are expressed in Eq. (41). The structural properties of the probabilistic state-constrained function are not properly analyzed. The continuity, differentiability, compactness, and convexity properties of the state constraints are important for guaranteeing the optimality criteria. The function of state constrained looks like,

$$P\left(u;x\right)≔p_x\left\{u\right\}≔\mathit{Pr}\left\{y\left(\left(x,\xi \right),u\right)\right)\le y_{\mathit{max}},\forall x\in D\}\ge \alpha .$$E39

The internal part of the probability function is

$$\gamma \left(u,x,\xi \right)=y\left(\left(x,\xi \right),u\right))-y_{\mathit{max}}\le 0,\forall x\in D,$$E40

which is a continuous differentiable function from the equation expressed (33), it does not imply the continuity and differentiability of the probability function in Eq. (41). The following propositions guaranteed the structural properties for the probabilistic uniform constrained functions, and these are a strong form of a pointwise state constrained,

$$P\left(u;x\right)≔p_x\left\{u\right\}≔\mathit{Pr}\left\{y\left(\left(x,\xi \right),u\right)\right)=y_{\mathit{max}},,\forall x\in D\}=0.$$E41

it holds the measure zero property

$$P\left(u;x\right)≔p_x\left\{u\right\}≔\mathit{Pr}\left\{y\left(\left(x,\xi \right),u\right)\right)=y_{\mathit{max}},,\forall x\in D\}=0.$$E42

Proposition 3.8. Assume that $\gamma \left(x,.,\xi \right)$ are Borel measurable w.r.t $\xi \in \mathrm{\Omega }$ for all $u\in U$ and for all $x\in D$ and $\gamma \left(x,.,\xi \right)$ are weakly sequentially lower semicontinuous (wsls) or weakly sequentially upper semicontinuous (wsus) vice versa for $x\in D$ and $\xi \in \mathrm{\Omega }$. Then, $P\left(u;x\right)=\mathit{Pr}\left(\gamma \left(x,u,\xi \right)\le 0\right)$ is wsls or wsus vice versa and $\gamma \left(x,u,\xi \right)=y-y_{\mathit{max}}\le 0,\forall x\in D$. The same case, for fixed $x\in D$ the function $\mathcal{P}\left(u\right)=\left\{u\in U_{\mathit{adm}}:P\left(.,x\right)\ge \alpha \right\}$ is wsls or wsus vice versa.

Proof. From a given assumption, $\mathcal{P}$ is well defined by Borel measurability of $\gamma \left(x,u,.\right)$ w.r.t. $\xi \in \mathrm{\Omega }$ in the second argument. Fix an arbitrary $\widehat{u}$ and let $u_n\in U_{\mathit{adm}},$ $u_n\to \widehat{u},$ arbitrary weakly convergent sequences in $U_{\mathit{adm}}$ and any arbitrary fixed $x\in D$ and let $x_n\in D,$ $x_n\to \widehat{x}$. Denote by $u_{n_l}$ a subsequence such that ${\mathit{liminf}}_{n\to \infty }\mathcal{P}\left(u_n\right)={\mathit{lim}}_{l\to \infty }\mathcal{P}\left(u_{n_l}\right).$ The variable of the decision $u=u\left(x\right)$ linearly continuous dependent on $x\in D.$.

Define the sets $P=\left\{\xi \in \mathrm{\Omega }:\gamma \left(x,u,\xi \right)\le 0,\forall \xi \in \mathrm{\Omega }\right\}$ and $P_n=\left\{\xi \in \mathrm{\Omega }:\gamma \left(x,u_n,\xi \right)\le 0\forall \xi \in \mathrm{\Omega }\right\},n\ge n_o\in \mathcal{N}.$ Since, by $\gamma$ being wsls in the first argument, we have

$${\mathit{liminf}}_{n\to \infty }\gamma \left(x,u_n,\xi \right)\ge {\mathit{lim}}_{l\to \infty }\gamma \left(x,u_{n_l},\xi \right)\ge \gamma \left(x,\widehat{u},\xi \right),\forall \xi \in \mathrm{\Omega }/P.$$E43

Consequently, $\gamma \left(x,u_n,\xi \right)<0,\forall \xi \in \mathrm{\Omega }/P$ and all $n\ge n_0,\xi \in \mathrm{\Omega }$.

Denoting by

$$h\left(\xi \right)=\left\{\begin{array}{cc}1& \text{if}\xi \in P\\ 0& \text{if}\xi \notin P\end{array}\right.$$E44

the characteristic function of a set $U_{\mathit{adm}}$, this entails that $h\left(P_n\left(\xi \right)\right)\to 0$ as $n\to \infty ,\forall \xi \in \mathrm{\Omega }/P.$ By the Lebesgue dominance convergence theorem, ${\int }_{\mathrm{\Omega }}h[\gamma \left(x,u_n,\xi \right)\varphi \left(\xi \right)\mathit{d\xi }\to 0$ for all $\xi \in \mathrm{\Omega }/P$.

On the other way $E[h\left[\gamma \left(x,u_n,\xi \right)\right]\le E[h\left[\gamma \left(x,u,\xi \right)\right]=1,\forall \xi \in P$.

Therefore,

$$\mathit{lim}\mathcal{P}\left(u_{n_l}\right)={\mathit{lim}}_{l\to \infty }\mathit{Pr}\left[\gamma \left(x,u_{n_l},\xi \right)\le 0\right]={\mathit{lim}}_{l\to \infty }{\int }_{\mathrm{\Omega }}h\left(\gamma \left(x,u_{n_l},\xi \right)\right)\varphi \left(\xi \right)\mathit{d\xi }$$

$$={\mathit{lim}}_{l\to \infty }{\int }_{\mathrm{\Omega }/P}h[\gamma \left(x,u_n,\xi \right)\varphi \left[\xi \right]\mathit{d\xi }+{\mathit{lim}}_{l\to \infty }{\int }_{P}h[\gamma \left(x,u_n,\xi \right)\varphi \left(\xi \right)\mathit{d\xi }$$

$$\ge {\mathit{lim}}_{l\to \infty }\mathit{inf}{\int }_{\mathrm{\Omega }/P}h[\gamma \left(x,u_n,\xi \right)\varphi \left[\xi \right]\mathit{d\xi }+{\mathit{lim}}_{l\to \infty }\mathit{inf}{\int }_{P}h[\gamma \left(x,u_n,\xi \right)\varphi \left(\xi \right)\mathit{d\xi }$$

$$\ge {\mathit{lim}}_{l\to \infty }\mathit{inf}{\int }_{P}h[\gamma \left(x,u_n,\xi \right)\varphi \left(\xi \right)\mathit{d\xi }$$E45

$\ge {\mathit{lim}}_{l\to \infty }\mathit{inf}{\int }_P\varphi \left(\xi \right)\mathit{d\xi }=\mathit{Pr}\left[\xi \in P\right]=\mathit{Pr}\left[\xi \in P:\gamma \left(x,.,\xi \right)\le 0\right]=\mathcal{P}\left(\widehat{u}\right)$. Thus, the function $\mathcal{P}$ is wsls from the equation in above ${\mathit{liminf}}_{n\to \infty }\mathcal{P}\left(u_n\right)={\mathit{lim}}_{l\to \infty }\mathcal{P}\left(u_{n_l}\right)\ge \mathcal{P}\left(\widehat{u}\right).$ For wsusc property, related propositions also proved in work of [11, 16]. □

Proposition 3.9. Assume that D is compact subset of ${\mathbb{R}}^n$, if $\gamma$ is wsus then $P\left(u,x\right)$ is wsus at $\forall u\in U$ satisfying $\mathit{pr}\left({\gamma }^{\ast }\left(u,\xi \right)=0\right)=0$, this is said to be measure zero property where ${\gamma }^{\ast }=\mathit{inf}\left(\gamma \left(x,u,\xi \right)\right)$ defined in proposition (3.8).

The prove is the similar to the proof of (3.8), has been proved in [16]. Let $x\in D$ be fixed. We need to show the convexity property of probability function $u\to p\left(u,\xi \right)$ needs convex property with respect to $\left(u,\xi \right)$ along the continuous probability density function $\varphi \left(\xi \right)$. From the solution of PDE system, we have a continuous $y$ in Bochner space and $y\left(x,\xi ,u\right)=\left(\left(A^{-1}\right)/\left(k\right)\right)\left(j_H\left(f\left(x\right)+\mathit{kg}\left(u,.,.\right)\right)\left(x.\xi \right)\right)\forall x\in D,\forall \xi \in \mathrm{\Omega }$. Thus, $\gamma \left(x,.,.\right)=\left(\left(A^{-1}\right)/\left(k\right)\right)(j_H\left(f\left(x\right)+k\left(g\left(u,.,.\right)\right)\left(x,u,\xi \right)-y_{\mathit{max}}\right)\le 0$ and $\mathit{Pr}\left\{\left(\left(A^{-1}\right)\left(j_H\left(f\left(x\right)+\left(k\right)g\left(u,.,.\right)\right)\left(x.\xi \right)-\left(k\right)y_{\mathit{max}}\right)\right)\le 0\right\}\ge \alpha$ is a convex w.r.t. $\left(u,\xi \right)$ jointly from the linearity of the internal function for any fixed $x\in D.$ Therefore, it is a sufficient in the finite-dimensional optimization, continuity, and linearity of $y$ guarantee for continuity and convexity of $P\left(u;x\right)$ [26, 27]. Our aim is to extend it to infinite-dimensional case.

Remark: Assume that $\gamma \left(u,x,\xi \right)=\left(A^{-1}/\left(k\right)\right)\left(j_{H}f\left(x\right)\right)+g\left(x,.,.\right)-y_{\mathit{max}}$ is a concave w.r.t.$\xi \in \mathrm{\Omega }$, $\forall u\in U$ and $\forall x\in D$, for each u there exists random vector $\xi \in \mathrm{\Omega }$ such that $\gamma \le 0,\forall x\in D$ then, $\xi$ has a density $\varphi \left(\xi \right)$ distributed continuously and

$$\mathit{Pr}\left\{{\gamma }^{\ast }\left(u,\xi \right)=0\right\}=0=\mathit{Pr}\left\{\left(A^{-1}/\left(k\right)\right)\left(j_{H}f\left(x\right)\right)+g\left(u,.,.\right)\left(x,\xi \right)=y_{\mathit{max}}\right\}=0,\forall x\in D,$$E46

where, ${\gamma }^{\ast }=\mathit{inf}\left(\gamma \left(x,u,\xi \right)\right)={\mathit{inf}}_u\left\{\left(A^{-1}/\left(k\right)\right)\left(j_{H}f\left(x\right)\right)+g\left(u,.,.\right)\left(x,\xi \right)-y_{\mathit{max}},\forall x\in D\right\}.$

Proposition 3.10. Let $U$ be Banach space and D be arbitrary index set. Let the n-dimensional random vector $\xi \in \mathrm{\Omega }$ have a log-concave density, i.e., density of its logarithm is possibly extended valued concave function. Assume that the function $\gamma \left(x,u,\xi \right)=\left(A^{-1}/\left(k\right)\right)\left(j_H\left(f\left(x\right)\right)+g\left(u,.,.\right)\left(x,\xi \right)\right)-y_{\mathit{max}}$ is quasi-concave for all x in D, then the set $M=\left\{u\in U/P\left(u;x\right)>\alpha \right\}$ is a convex set.

Proof. From structural property of wslsc on the proposition (3.8) pick up ${\gamma }^{\ast }={\mathit{inf}}_u\left(\gamma \left(x,u,\xi \right)\right)$ and we defined $P\left(u;x\right)≔\mathit{Pr}\left({\gamma }^{\ast }\left(x,u,\xi \right)\le 0\right)$ for all u in Banach space $U_{\mathit{adm}}=M\subset U$ to fix arbitrary elements in $U$, $u_1,u_2\in U$ and quasi concavity of ${\gamma }^{\ast }$ such that ${\xi }_1$ and ${\xi }_2$ in $\mathrm{\Omega }$ and $\left(u_1,{\xi }_1\right),\left(u_2,{\xi }_2\right)\in U\times \mathrm{\Omega }$ and $\lambda \in \left[0,1\right]$ there exists indexed $x\in D$ such that ${\gamma }^{\ast }\left(\lambda \left(u_1,{\xi }_1\right)+\left(1-\lambda \right)\left(u_2,{\xi }_2\right)\right)\ge \mathit{min}\left\{\right({\gamma }^{\ast }\left(u_1,{\xi }_1\right),{\gamma }^{\ast }\left(u_2,{\xi }_2\right)\}$ and the density $\xi$ having log-concavity density shown in Prekopa ([11, 28] Theorem [4.2.1]) for the given log-concave distribution. The $\mathit{pr}\left(\xi \in \mathit{\lambda P}+\left(1-\lambda \right)q\right)\ge \mathit{pr}{\left(\xi \in p\right)}^{\lambda }\mathit{Pr}{\left(\xi \in q\right)}^{1-\lambda })$ for p and q are convex subset of $\mathrm{\Omega }$. We need to show the convexity of set M in $u_1$ and $u_2$ in M and $\lambda \in \left(0,1\right)$ we define a map $H:U\to \mathbb{R}$ u in U observe that $H\left(u_1\right)$ and $H\left(u_2\right)$ are the convex set an immediate consequence of the quasi concave on ${\gamma }^{\ast }$. The sets $\mathit{\lambda H}\left(u_1+\left(1-\lambda \right)H\left(u_2\right)\right)\subset H\left(\lambda u_1+\left(1-\lambda \right)u_2\right)$. Thus, $H$ is a concave set ($\gamma$ is a quasi-concave). In other words, let ${\xi }_1\in H\left(u_1\right)$ and ${\xi }_2\in H\left(u_2\right)$, $\xi \in P\left(\lambda u_1+\left(1-\lambda \right)u_2\right)=\mathit{pr}\left[\xi \in H\left(\lambda u_1+\left(1-\lambda \right)u_2\right)\right]\ge \mathit{pr}[\xi \in H(u_1]{}^{\lambda \ast }\mathit{pr}{\left[\xi \in H\left(u_2\right)\right]}^{\left(1-\lambda \right)}={\alpha }^{\lambda }{\alpha }^{\left(1-\lambda \right)}=\alpha ,$ for reliable level $\alpha$. The $\xi ={\mathit{\lambda \xi }}_1+\left(1-\lambda \right){\xi }_2\in \mathrm{\Omega },$ ${\gamma }^{\ast }\left(u_1,{\xi }_2\right),{\gamma }^{\ast }\left(u_2,{\xi }_2\right)\le 0$ obtaining from the quasi concavity of ${\gamma }^{\ast }$ we have ${\gamma }^{\ast }\left(\lambda u_1+\left(1-\lambda \right)u_2,\xi \right)={\gamma }^{\ast }\left(\lambda u_1,{\xi }_1\right)+\left(1-\lambda \right)\left(u_2,\xi 2\right))\ge \mathit{min}\left({\gamma }^{\ast }\left(u_1,\xi 1\right),{\gamma }^{\ast }\left(u_2,\xi 2\right)\right)\ge 0$. Finally, the result is proved the convexity of chance-constrained, $\lambda u_1+\left(1-\lambda \right)u_2\in M$. Hence, M is a convex set.□

Proposition 3.11. Let $M=U$ be separable Banach space defined in proposition of the convexity (3.10) and $M^{\ast }$ is a dual space of $M$, assume that the $\gamma \left(x,.,.\right)\le 0$ is a weakly sequentially upper continuous (w.s.u.s) for each $u\in M$ and $x\in D$. Then, it satisfies the following three conditions and the three statements are equivalent [11, 16].

1. The $\gamma \left(x,u,.\right)$ is a Borel measurable function w.r.t. $\xi \in \mathrm{\Omega }$ $\forall u\in M$ and $x\in D$.

2. The set $M=\left\{u\in M:\mathcal{P}\left(u;x\right)\ge \alpha \right\}$ is weakly closed.

3. The $\mathcal{P}\left(u;x\right)$ is weakly sequentially upper continuous (wsus).

Proof. From the assumption, the function $\gamma \left(x,u,.\right):{\mathbb{R}}^n\to \mathbb{R}$ upper semicontinuous for each $u\in U$ and each $x\in D$. Consequently, the sets $\left\{\xi \in \mathrm{\Omega }:\gamma \left(x,u,\xi \right)\le 0,\forall x\in D\right\}$ are closed, which implies the Borel measurability w.r.t. $\xi$ (i). The proposition (3.8) explains the continuity property with measure zero property in (42), justifies to talk about probability of events as in (i) and (iii). It is an immediate consequences of (ii). Hence, in order to (i) prove (ii) let $\mathcal{P}\left(u\right)$ in $\mathbb{R}$ be arbitrary and consider weakly convergent sequences $u_n\to u$ has a convergent subsequence $u_{n_l}\in M$ with $u_n\in M$ for all n, it is from Arzela-Ascoli, every sequence of a given family of real-valued continuous functions defined on a closed and bounded interval has a uniformly convergent subsequence. We have to show that u in M we define $H\left(u\right)=\left\{\xi \in \mathrm{\Omega }:\gamma \left(x,u,\xi \right)\le 0\right\}$ from $u_n\in M,$ and $P\left\{\xi /H\left(u_n\right)\ge \alpha \right\}$. Boundness of $u_n$ by weak convergence implies that there is some closed ball B with sufficiently large radius such that u in $B$ for all n. From separability of $U^{\ast }$, the weak topology $B$ metrizable w.r.t. $\xi$ for fixed x. Regarding the finite-dimensional case, it has been proved in work [16].

Lemma 3.12. All the assumptions of Proposition (3.11) are true, there are constants $\epsilon >0,\sigma >0$, such that with $d$ referring to the Hausdorff distance from the point on $U_{\mathit{adm}}$ to the set in probabilistic set $P_x\left(u;\xi \right)$ such that $d(\in U_{\mathit{adm}},\left\{u\in \mathcal{P}:P\left(u,x\right)\ge \wp \right\}\le \mathit{\text{σmax}}\left\{\mathit{log}\left(\wp \right)-\mathit{log}\left(P_x\left(u,x\right)\right),0\right\},\forall \wp \in \left[\alpha -\epsilon ,\alpha +\epsilon \right]$.

For an infinite-dimensional problem of CCPDE, we have proved it in the previous work [14, 25]. This lemma has been verified for the finite-dimensional problem without the PDE system in [16]. It needs Arzela-Ascoli theorem, this CCPDE constrained function also holds bounded and continuous property [16, 21]. The Lax-Milgram theorem is sufficient for the bounded and continuous property from the continuous bilinear and coercivity form; see in the theorem (3.4).

4. Approximation for CCPDE by inner and outer functions

The approximation of the probability constrained function has been analyzed in the previous work [6, 14]. This proposed approach is a smooth parametric approximation for the nonsmooth and intractable probability function. This study will analyze some open issues related to the topological and structural properties of the CCPDE. Some of the issues are continuity, differentiability, closedness, compactness, and convexity of the inner and outer approximation functions in the infinite-dimensional Bochner space. They are significant for assuring the optimality criteria for the existence and uniqueness of the optimal control. Furthermore, the convergence approach and the numerical results are studied properly.

The smooth parametric inner and outer approximations are analyzed in the work [6]. This section briefly analyzes the parametric functions to define a smooth approximations to the optimal control of the boundary-value CCPDE. The optimal control of the CCPDEs is approximated by the family of the sequence of solutions of the inner-outer approximation problems, when the approximation parameter ${\tau }_k$ tends to zero as k goes to infinity. In case of convexity and norm convergence of the approximate solutions to a solution of CCPDE can be proved and to some extended the structural analysis of the proposed approach for guaranteeing the optimality criterion of CCPDE. For this purpose, we employ and extend our recent work [6, 13, 14, 25], where the inner-outer approximation methods, for finite-dimensional and smooth CCPDE, were proposed to solve the reduced CCPDE problem.

We consider the parametric Geletu-Hofmann function

$$\theta \left(\tau ,s\right)=\frac{1+m_1\tau }{1+m_2\tau exp\left(-\frac{s}{\tau }\right)},\mathit{\text{for}}\tau \in \left(0,1\right),s\in \mathbb{R}$$E47

to approximates a nonsmooth problem of the CCPDE in the work [13]. If we fix the parameters $m_1=0,m_2=1/\tau$, the parametric function $\theta \left(\tau ,s\right)$ is the same as the sigmoid function. Thus, we can approximate the probabilistic constraint by sigmoid function in terms of $\theta \left(\tau ,s\right).$ Unfortunately, the sigmoid function is not bound to the probabilistic constrained function from the below, and the computation is a too slowing in comparison with $\theta \left(\tau ,s\right).$ The advantage of theta is bound to the probabilistic function from above and below. Thus, the probabilistic constraint of CCPDE is bounded by the family of smooth functions of inner $E[\theta \left(\tau ,-\gamma \left(u,\xi ;t,x\right)]\right)$ and outer $1-E[\theta \left(\tau ,\gamma \left(u,\xi ;t,x\right)]\right)$ approximations from the lower and upper bound, respectively. We have a piecewise continuous function,

$$h\left(s\right)=\left\{\begin{array}{cc}1& \text{if}s\ge 0\\ 0& \text{if}s<0.\end{array}\right.$$E48

Multiplied by the discontinuous function, with jump nonlinear function and an infinitely differentiable function $E\left[h\left(s\right)\right]$, where $s=\gamma \left(x,u,\xi \right)$ is defined above in Eq. (48). Then, the product is infinitely differentiable, and the difference of approximation on the parametric functions in the region $0<x<\epsilon$ that is tiny for small positive number epsilon but the probabilistic constraint has measure zero property defined in Eq. (42). This makes the approximation functions a nice function property. The $m_1$ and $m_2$ are positive constants, the parametric family each property of $\theta$ function is written and proved in [6, 13] together with its continuity, differentiability, and convexity of its approximation functions. The following well-known identities are obtained

$$p\left(u;x\right)=E\left[h\left(-\gamma \left(u,\xi ;x\right)\right)\right]=1-E\left[h\left(\gamma \left(u,\xi ;x\right)\right)\right],$$E49

despite the appealing expected value $E\left[h\left(\gamma \left(u,\xi ;x\right)\right)\right]={\int }_{\mathrm{\Omega }}h\left(\gamma \left(u,\xi ;x\right)\right)\varphi \left(\xi \right)\mathit{d\xi }$ representation of probability functions in (49), the missing smoothness of the unit jump function does not provide computational advantages. Nevertheless, the function $h$ provides an insight for the construction of a smoothing approximation for the probability function $p$ and the internal function $\left\{\theta \left(\tau ,\gamma \right),\tau \in (0,1)\right\}$ of functions $\theta :\left(0,1\right)\times \mathbb{R}\to {\mathbb{R}}_{+}$ possessing the following strict monotonicity and uniform limit properties.

Assumption 4.1. Suppose there is a parametric family of a functions $\theta \left(\tau ,s\right)$_, which possess the following properties such as monotonic (strictly increasing) uniform limit properties

1. There is a constant C with $1<C<\infty$ such that $C>\theta \left(\tau ,s\right)>h\left(s\right),\forall s\in \mathbb{R}$ and $\forall \tau \in \left(0,1\right)$.

2. The parametric $\theta \left(\tau ,s\right)$ is strictly increasing in all $s\in \mathbb{R}$ and $\tau \in \left(0,1\right)$.

3. The parametric $\theta \left(\tau ,s\right)$ is continuous and infinitely differentiable smooth function.

4. The infimum ${\mathit{inf}}_{\tau \in \left(0,1\right)}\left(\theta \left(\tau ,s\right)\right)=h\left(s\right),\forall s$.

5. The ${\mathit{limsup}}_{\tau \to 0+}\left(\theta \left(\tau ,s\right)\right)-h\left(s\right)=0,\tau \in \left(0,1\right)$ and $s\in \left(-\infty ,0\right)\cup \left(0,\infty \right)$.

6. The parametric $\theta \left({\tau }_k,s\right)$ uniform convergence and ${\mathit{lim}}_{k\to \infty }\mathit{sup}\frac{\partial \theta \left({\tau }_k,s\right)}{\partial s}=0,$ for ${\tau }_k\to 0$ as $k\to \infty$.

Thus, from the above boundedness property of jump function in above assumption we have,

$$1-\theta \left(\tau ,s\right)\le h\left(-s\right)\le \theta \left(\tau ,-s\right),$$E50

It follows that, $E\left(1-\theta \left(\tau ,s\right)\right)\le E\left(h\left(-s\right)\right)\le E\left(\theta \left(\tau ,-s\right)\right),$ from property of expectation. Hence, ${\left.p\left(u;x\right)≔\mathit{Pr}\left\{y\right(u,\xi ,x)\le y_{\mathit{max}}\}\ge \alpha ,\forall x\in D\Rightarrow \mathit{Pr}\{\xi \in \mathrm{\Omega }:\gamma (x,u,\xi )\le 0\}=E\left(h\left(s\right)\right)\right|}_{s=\gamma \left(x,u,\xi \right)\le 0}\ge \alpha ,\forall x\in D.$ Now, based on the parametric function $\theta$_, the following functions are defined

$$\phi \left(\tau ,u;t,x\right)≔E[\theta \left(\tau ,-\gamma \left(u,\xi ;t,x\right)]\right),$$E51

$$\psi \left(\tau ,u;t,x\right)≔1-E\left[\theta \left(\tau ,\gamma \left(u,\xi ;t,x\right)\right)\right],\tau \in \left(0,1\right).$$E52

Under the measure zero property and smoothness properties of $\gamma \left(u,\xi ;t,x\right)$, the functions $\psi \left(\tau ,u;t,x\right)$ and $\phi \left(\tau ,u;t,x\right)$ can be shown to be smoothing approximations of $1-p\left(u;t,x\right)$ and $p\left(u;t,x\right)$, respectively (see Geletu et al. [6]). Moreover, the following convergence properties

$$\underset{\tau \in \left(0,1\right)}{inf}\phi \left(\tau ,u;t,x\right)=p\left(u;t,x\right);$$E53

$$\underset{\tau \in \left(0,1\right)}{sup}\left(1-\psi \left(\tau ,u;t,x\right)\right)=p\left(u;t,x\right);$$E54

and the Lebesgue dominance convergence properties analyze the almost indeed convergence properties of the inner and outer function sequence. The detail convergence approach of the outer approach to CCPDE has been studied in the work [6, 13]. However, the convergence analysis for the inner approximation to the optimal solution of CCPDE of the smooth function ${\mathit{IA}}_{\tau }$ has not been properly analyzed in the previous work [13] because the study needs the convex approximations and subdifferentiability for probabilistic constraints as the convergence of stationary points of ${\mathit{IA}}_{\tau }$ is very relevant for the existence of the optimal solution.

For several chance constraints $p_i\left(u\right)\ge \alpha ,i=1,2,\dots ,m$, the regularity is given by $\left\{u\in U|p_i\left(u\right)\ge \alpha ,i=1,2,\dots ,m\right\}=\mathit{cl}\left\{u\in U|p_i\left(u\right)>\alpha ,i=i=1,2,\dots ,m\right\}$ [29]. For continuously differentiable probability functions $p_i$, a sufficient condition for the validity of the regularity assumption is the satisfaction of the Mangasarian-Fromowitz constraint qualification (MFCQ) at the active points, [16, 29] Proposition 3.7.

The respective feasible sets of inner and outer approximations are defined as follows:

$$\mathcal{M}\left(\tau \right)≔\left\{u\in U|\psi \left(\tau ,u;x\right)\le 1-\alpha ,\left(x\right)\in D\right\};$$E55

$$\mathcal{S}\left(\tau \right)≔\left\{u\in U|\phi \left(\tau ,u;x\right)\ge \alpha ,\left(x\right)\in D\right\}.$$E56

As a consequence of the properties of the functions $\psi \left(\tau ,u;x\right)$ and $\phi \left(\tau ,u;x\right)$, we have the following relations among the feasible sets of CCPDE, IA_τ, and OA_τ. That is,

$$M\left(\tau \right)\subset \mathcal{P}\subset S\left(\tau \right),\text{for}\tau \in \left(0,1\right).$$E57

The tightness property of the relaxation of IA and compression of OA sequentially analyzed in the previous work [29], if $0<{\tau }_2\le {\tau }_1<1$, then.

$$\mathcal{M}\left({\tau }_2,x\right)\subset \mathcal{M}\left({\tau }_1\right)\subset \mathcal{P}\left(x\right)\subset \mathcal{S}\left({\tau }_1,x\right)\subset \mathcal{S}\left({\tau }_2\right).$$E58

It leads ${\mathit{lim}}_{\tau \to 0+}S\left(\tau \right)={\cap }_{\tau \in \left(0,1\right)}S\left(\tau \right)=P$ and ${\mathit{lim}}_{\tau \to 0+}M\left(\tau \right)=\mathit{Cl}\left\{{\cup }_{\tau \in \left(0,1\right)}M\left(\tau \right)\right\}=P.$ Both $M\left(\tau \right)$ and $S\left(\tau \right)$ are closed sets as $\tau \to 0+$; see the property in proposition 5.1 and 5.2 in the next section.

The new monotonic properties of the objective function $E\left(J\left(T\left(u,.\right)\right)\right)$ with respect to $\tau$ are a continuous function on $u\in U_{\mathit{adm}}$. If $0<{\tau }_1<{\tau }_2<1$ with $\mathit{inf}\left[\varnothing \right]=\infty ,$

$$\begin{array}{c}{\mathit{inf}}_{u\in M\left({\tau }_2\right)}\left[E\left(J\left(T\left[u\right]\right)\right)\right]\ge {\mathit{inf}}_{u\in M\left({\tau }_1\right)}\left[E\left(J\left(T\left[u\right]\right)\right)\right]\ge {\mathit{inf}}_{u\in P}\left[E\left(J\left(T\left[u\right]\right)\right)\right]\\ \ge {\mathit{inf}}_{u\in S\left({\tau }_1\right)}\left[E\left(J\left(T\left[u\right]\right)\right)\right]\ge {\mathit{inf}}_{u\in S\left({\tau }_2\right)}\left[E\left(J\left(T\left[u\right]\right)\right)\right],\forall \tau \in \left(0,1\right).\end{array}$$E59

It implies the following compact form conditions of nondegenerate fuzzy optimality,

$$\begin{array}{r}{\mathit{inf}}_{u\in M\left({\tau }_2\right)}[E\left(J\left(T\left[u\right]\right)\right)+{\mu }_i(\alpha -\left(1-{\psi }_i\left(u_i\right)\right)]\ge {\mathit{inf}}_{u\in M\left({\tau }_1\right)}[E\left(J\left(T\left[u\right]\right)\right)+{\mu }_i(\alpha -\left(1-{\psi }_i\left(u_i\right)\right)]\\ \ge {\mathit{inf}}_{u\in P}[E\left(J\left(T\left[u\right]\right)\right)+{\mu }_i(\alpha -{\mathcal{P}}_i\left(u_i\right)]\le {\mathit{inf}}_{u\in S\left({\tau }_1\right)}[E\left(J\left(T\left[u\right]\right)\right)+{\mu }_i(\alpha -{\phi }_i\left(u_i\right)]\\ \ge {\mathit{inf}}_{u\in S\left({\tau }_2\right)}[E\left(J\left(T\left[u\right]\right)\right)+{\mu }_i(\alpha -{\phi }_i\left(u_i\right)],\forall \tau \in \left(0,1\right).\end{array}$$E60

The complementary property of the probability function $1-p\left(u;x\right)\le \psi \left(\tau ,u;x\right)\le 1-\alpha$ and $\phi \left(\tau ,u;x\right)\ge p\left(x\right)\ge \alpha ,\text{for}x\in D$ hold true. Now, using the parametric functions $\phi \left(\tau ,\cdot \right)$ and $\phi \left(\tau ,\cdot \right)$, we define the following problems with the same objective function $q$ in the equation given (38) as in CCPDE.

$$\begin{array}{lll}& & \underset{u}{min}q\left(u\right)\left({\mathit{IA}}_{\tau }\right)\\ & \mathrm{s}.\mathrm{t}.& \\ & & \psi \left(\tau ,u;x\right)\le 1-\alpha ,x\in D_c\\ & & u\in U,\end{array}\begin{array}{lll}& & \underset{u}{min}q\left(u\right)\left({\mathit{OA}}_{\tau }\right)\\ & \mathrm{s}.\mathrm{t}.& \\ & & \phi \left(\tau ,u;x\right)\ge \alpha ,\\ & & u\in U.\end{array}$$E61

The feasible set of CCPDE defined in the above set $P$ in Eq. (41) has a property that satisfies Mangasarian-Fromowitz constraint qualification (MFCQ). It is an important prerequisite for applying the necessary optimality criteria in nonlinear optimization. The MFCQ is a condition for the regularity of a permissible point. The MFCQ is in one of point x, and if this point is a local minimum, then the Karush-Kuhn-Tucker conditions are met at this point. If the MFCQ is valid, it is easy to check whether a given point is optimal or not; see the work [13]. From Section 6 proposition (3.10), it follows that the convexity of inner approximation is convex conservative, $\psi$ is concave function or all fixed x in D and $\xi \in \mathrm{\Omega }$. The convexity of $\phi$ analyzed in the work [13]. The Slater condition is a sufficient condition for strong duality to hold for a convex optimization problem named after Slater; for further analysis, please see the work in [13].

The necessary optimality condition is not valid for expressing local optimal solution in this technique, but can be shown to hold true for generalized stationary points of Fritz–John types [27, 29]. This essentially requires the uniform convergence of partial gradients of inner and outer functions ${\phi }_i\left(\tau ,u,\xi \right)$ and $1-{\psi }_i\left(\tau ,u,\xi \right)$ of $\nabla {\phi }_i\left(\tau ,u,\xi \right)=0$ and $\nabla \left(1-{\psi }_i\left(\tau ,u,\xi \right)\right)=0$, for ${\tau }_k\to 0+$, as $k\to \infty$ on bounded subsets W of $U_{\mathit{adm}}$ [29]. In addition, each strict local minimum of CCPDE is a cluster point of local minima of the inner approximation problems ${\mathit{IA}}_{\tau }$ under the satisfaction of tightness or the outer approximation problems ${\mathit{OA}}_{\tau }$; see ([29], Proposition 3.3–3.7). The following sections are more about the structural properties of approximation functions such as closedness, convergence, and differentiability of ${\mathit{IA}}_{\tau }$ and ${\mathit{OA}}_{\tau }$; those properties are not properly analyzed.

5. Closedness, convergence, and differentiability of ${{IA}}_{\tau }$ and ${{OA}}_{\tau }$

5.1 Closedness property of M set of ${{IA}}_{\tau }$ and S set of ${{OA}}_{\tau }$

The nice properties of the parametric function are given in assumption (4.1); closedness property is defined by the distance from a particular point on the $P$ to set $\mathit{\text{MorS}}$ (this distance is close to zero). It is called Hausdorff distance (point to set distance). The specific point of $P$ is closed under the arbitrary sequence of the set $M_n$ and $S_n.$ This property is an essential property for the compactness of the specified $M$ and $S$. In this case, compactness is boundedness, closedness, and any convergent sequence has a convergent subsequence with the same limit point. The boundness and subsequential convergence property have been clear from (4.1) and tightness property.

Let $u\in U_{\mathit{adm}}$ be separable Banach space and ${U_{\mathit{adm}}}^{\ast }$ is dual space of $U_{\mathit{adm}}$, assume that the $\gamma \left(x,.,.\right)$ are the continuous differentiable functions as shown in proportions (3.8) and (3.9) for the continuity property of $P_x\left(u\right)$ for any fixed $x\in D$. Then, it satisfies the compactness of set $P$, which is a feasible set of CCPDE and the conditions as shown (proposition 3.7 in the work [11]). The $\gamma \left(x,u,.\right)$ is a Borel measurable w.r.t. $\xi \in \mathrm{\Omega }$, thus, $P=\left\{u\in U_{\mathit{adm}}:P\left(u\right)\ge \alpha \right\}$ is weakly closed in the proposition (3.11), it is from the property of the wssc of $P$. Now, the closedness property of $M$ and $S$ proved in the following proposition.

Proposition 5.1. Assume that the $\gamma \left(u,.,x\right)$ are Borel measurable for all $u\in U$ and $x\in D$ and $\gamma \left(.,x,\xi \right)$ are weakly sequentially semicontinuous (wss) for $x\in D$ and $\xi \in \mathrm{\Omega }$. Then $\phi \left(\gamma \left(x,u,\xi \right),\tau \right)$ and $\psi \left(\gamma \left(x,u,\xi \right),\tau \right)$ are wssc and $\gamma \left(x,u,\xi \right)=y-y_{\mathit{max}}\le 0,\forall x\in D$.

Proof. This is extension of the proposition 3.1 and 3.2. The prove is directly related with proofs of those propositions. Observe first, that $\rho$ is well defined by Borel measurability of $\gamma$ in the second argument. This $\gamma$ is wssc, then the parametric function $\theta \left(\gamma ,\tau \right)$ is wssc. It satisfies the six properties of monotonicity, smoothness, boundedness, and uniform convergence property are expressed in Assumption 4.1. The inner function and outer functions are continuous.

Proposition 5.2. The function $1-\psi \left(u,x\right)$ of M of ${\mathit{IA}}_{\tau }$ and $\phi \left(u,x\right)$ of ${\mathit{OA}}_{\tau }$ are continuous from the continuity property $\theta \left(\tau ,\gamma \right)$. The set M of ${\mathit{IA}}_{\tau }$ and the set S of ${\mathit{OA}}_{\tau }$ are closed if the following Hausdorff distance holds true, and there are constants $\forall \epsilon >0,\sigma >0$, such that with $d$ referring to the Hausdorff distance $d\left(u,\left\{u\in U_{\mathit{adm}}|\phi \left(u,x\right)\ge \alpha \right\}\right)\le \mathit{\text{σmax}}\left\{\mathit{log}\left(\alpha \right)-\mathit{log}\left(\phi \left(u,x\right)\right),0\right\},\forall \alpha \in \left[1-\epsilon ,1\right]$. The same case for IA, there are constants $\forall \epsilon >0,\sigma >0$, such that with d referring to the Hausdorff distance $d\left(u,\left\{u\in U_{\mathit{adm}}|1-\psi \left(u,x\right)\ge \alpha \right\}\right)\le \mathit{\text{σmax}}\left\{\mathit{log}\left(\alpha \right)-\mathit{log}\left(1-\psi \right),0\right\},\forall \alpha \in \left[1-\epsilon ,1\right]$.

Proof. The proof directly follows from the nice property of expectation in the given assumptions (4.1, i–vi). Moreover, the proposition (3.11), the inequality $E\left(h\left(\gamma \right)\right)\ge \alpha$ is bounded by the two parametric functions $\phi \left(u,x\right)$, and $1-\psi \left(u,x\right)$. This is a direct extension of the lemma (3.12), we have shown that, for arbitrary $u_1,u_2\in U_{\mathit{adm}}$ and $\lambda \in \left[0,1\right]$ the inequality $\phi \left(\lambda u_1+\left(1-\lambda \right)u_2\right)\le \mathit{\lambda \phi }\left(u_1\right)+\left(1-\lambda \right)\phi \left(u_2\right)$ and $\left(1-\psi \left(\lambda u_1+\left(1-\lambda \right)u_2\right)\right)\ge \lambda \left(1-\psi \right)\left(u_1\right)+\left(1-\lambda \right)\left(1-\psi \left(u_2\right)\right)$ hold true from convexity property in (3.10), so $\psi \left(u\right)$ is a concave function. This means that $\mathit{\text{logφ}},\mathit{log}\left(1-\psi \right)$ are the log-convex functions and, hence, the inequality $\mu \left(\phi \left(u\right)\right)\ge \alpha$ and $\mu \left(\psi \left(u\right)\right)\le 1-\alpha$ is equivalent with $\widehat{P}\left(u\right)\le -\mathit{log}\left(\alpha \right)$, where $\widehat{P}\left(u\right)≔-\mathit{\text{logP}}$ is a log-convex function. The Proposition (3.11), P(u) is wsus, the expectation of P(u) is $\widehat{P}$, is wsls from continuity property in proposition (3.8) and (3.9). The function given in proportion (3.10) is a convex function. From Robison-Ursescu theorem ([16], Lemma 4), the continuous convex function is closed. Therefore, the probabilistic constrained function P(u) is closed.

We have proved that the sets M and S are defined in the following section, are the convex feasible sets. It has a closed and convex graph. Consider an arbitrary sequence $\left(u_n,t_n\right)\to \left(\widehat{u},\widehat{t}\right)$ with $u_n\in M$ of IA and $\left(u_n,t_n\right)\to (\widehat{u},\widehat{t}$) with $u_n\in S$ of OA. Then, $u_n\in U_{\mathit{adm}}$ and, hence,$u\in U_{\mathit{adm}}$ by closedness of $U_{\mathit{adm}}$ see in Proposition (3.11).

Moreover, $\widehat{\phi }\left(u_n\right)\le t_n$ and $\widehat{\psi }\left(u_n\right)\le t_n$. Since $\widehat{\phi }$ and $\widehat{\psi }$ are wsls, we derive that $\widehat{\phi }\left(\widehat{u}\right)\le {\mathit{liminf}}_{n\to \infty }\widehat{\phi }\left(u_n\right)\le {\mathit{liminf}}_{n\to \infty }{t}_n=\widehat{t}$ and $\widehat{\psi }\left(\widehat{u}\right)\le {\mathit{liminf}}_{n\to \infty }\widehat{\psi }\left(u_n\right)\le {\mathit{liminf}}_{n\to \infty }{t}_n=\widehat{t}$. So, $\widehat{t}\in M\left(\widehat{u}\right)$ and $\widehat{t}\in S\left(\widehat{u}\right)$ implying that the graph of M is closed.

The same case for OA, we have that $\widehat{\psi }\left(u_1\right)\le t_1$ and $\widehat{\psi }\left(u_2\right)\le t_2$. Then, convexity of $\mathit{\text{hatφ}}$ yields that $\widehat{\psi }\left(\lambda u_1+\left(1-\lambda \right)u_2\right)\le \lambda t_1+\left(1-\lambda \right)t_2$. In other words, $\lambda t_1+\left(1-\lambda \right)t_2\in S\left(\lambda u_1+\left(1-\lambda \right)u_2\right)$, proving that the graph of S is also convex. Further analysis on the convexity of IA is in the following section.

For the properties of closedness of IA and OA in the infinite-dimensional space, we need further analysis of Robinson- Ursescu theorem; see [16]. It is depend on the Hausdorff distance between the point of the set and the set of probabilistic constrained function expressed in Eq. (41). Thus, closedness and boundedness are sufficient for compactness in the finite-dimensional case.

Lemma 5.3. Under the assumptions of Proposition 3.7, there are constants $\epsilon >0,\sigma >0$, such that with d referring to the Hausdorff distance $d\left(u\in M,u\in U_{\mathit{adm}}|\psi \left(u,x\right)\le 1-\tau \right)\le \mathit{\text{σmax}}\left\{\mathit{log}\left(\tau \right)-\mathit{log}\left(\psi \left(u,x\right)\right),0\right\}\forall \tau \in \left[\alpha -\epsilon ,\alpha +\epsilon \right]$ for IA and $d\left(u\in S,u\in U_{\mathit{adm}}|\phi \left(u,x\right)\ge \tau \right)\le \mathit{\text{σmax}}\left\{\mathit{log}\left(\tau \right)-\mathit{log}\left(\phi \left(u,x\right)\right),0\right\}\forall \tau \in \left[\alpha -\epsilon ,\alpha +\epsilon \right]$.

The prove of this lemma is analyzed in [16].

5.2 Convergence of the stationary points and differentiability of ${\mathit{IA}}_{\tau }$ and ${{OA}}_{\tau }$

The continuity and continuous differentiability of the functions $\psi \left(s,\tau \right),\phi \left(s,\tau \right)$ are directly from the continuity and continuous differentiability of the parametric function $\theta \left(\tau ,s\right)$. The expectation of a continuous function is also continuous. Additionally, the expectation of a continuously differentiable function is continuously differentiable.

The associated probability functions $P_x\left(u\right)\ge \alpha ,\forall x\in D$, in the chance constraints, are allowed to be lower semicontinuous or continuous in Proposition (3.8). Also, it cannot be differentiable. Hence, the characterization of optimality properties of CCPDE calls for generalized subdifferentiation such as Fréschet, Clarke, and Mordukhovich subderivative and implicit formula of gradient computations on the reflexive and separable Bochner space. It is analyzed in the previous submitted work in [25]. This concept is applicable to lower semicontinuous (lsc) functions. Furthermore, the epigraph of lsc function is closed everywhere. These subdifferentials assure the optimality criteria for the CCPDE. Also, the $P_x\left(u\right)\ge \alpha ,x\in D$ are Lipschitzian functions, assuring the explicit formula for the Clarke subgradient under special conditions.

For each $u\in U_{\mathit{adm}}$, there is some neighborhood $W\subset U_{\mathit{adm}}$ and some measurable function $q:{\mathbb{R}}^n\to \mathbb{R}+$ such that, for continuously partially differentiable up to order $b$ on $u_o\in U_{\mathit{adm}}$ for $\xi \in \mathrm{\Omega }$ almost surely (a.s.),

$${\mathit{sup}}_{u_j\in W}{\mathit{max}}_{{\sum }_{j=1}^n{m}_j}\mid \frac{{\partial }^{\sum l_i}\gamma }{\partial u_1^{m_1}\partial u_2^{m_2}\partial u_3^{m_3}.\dots \partial u_j^{m_j}}\mid \le q\left(\xi \right);$$E62

$$E\left(q\right)={\int }_{\mathrm{\Omega }}q\left(\xi \right)\varphi \left(\xi \right)\mathit{d\xi }<\infty ,$$E63

for $b={\sum }_{j=1}^n{l}_j\ge {\sum }_{i=1}^j{m}_i$. Suppose each property above holds true. Then, for the parametric function $\theta =\mathcal{H}$_, the functions $\phi \left(\gamma ,\tau \right)$ and $\psi \left(\gamma ,\tau \right)$ are continuously partially differentiable w.r.t. u up to order b on $U_{\mathit{adm}}$, for all $\tau \in \left(0,1\right)$. The higher derivatives

$$\frac{{\partial }^{{\sum }_{j=1}^n{l}_j}(1-\psi \left(\mathcal{H}\left(\tau ,u\right)\right)}{\partial u_1^{m_1}\partial u_2^{m_2}\partial u_3^{m_3}.\dots \partial u_j^{m_j}}={\int }_{\mathrm{\Omega }}\frac{{\partial }^{{\sum }_{j=1}^n{l}_j}\psi \left(\gamma ,\tau ,u\right)\varphi \left(\xi \right)\mathit{d\xi }}{\partial u_1^{m_1}\partial u_2^{m_2}\partial u_3^{m_3}.\dots \partial u_j^{m_j}}\varphi \left(\xi \right)\mathit{d\xi };$$E64

for the case for outer approximation,

$$\frac{{\partial }^{{\sum }_{j=1}^n{l}_j}\phi (\mathcal{H}\left(\tau ,u\right)}{\partial u_1^{m_1}\partial u_2^{m_2}\partial u_3^{m_3}.\dots \partial u_j^{m_j}}={\int }_{\mathrm{\Omega }}\frac{{\partial }^{{\sum }_{j=1}^n{l}_j}\phi \left(\gamma ,\tau ,u\right)\varphi \left(\xi \right)\mathit{d\xi }}{\partial u_1^{m_1}\partial u_2^{m_2}\partial u_3^{m_3}.\dots \partial u_j^{m_j}}\varphi \left(\xi \right)\mathit{d\xi };$$E65

for ${\sum }_{j=1}^n{l}_j\ge {\sum }_{i=1}^j{m}_i$. Further analysis of convergence approach and uniform convergence gradient of the approximation functions are seen in [29, 30] of proposition 4.2. All the above properties ensure the application of Lebesgue’s dominating convergence theorem [29] for interchanging the integration and differentiation operations. Observe the chain rule yields powers $E(J\left(T\left(u\right)\right)$ till order $l=k$ in the upper estimation of the integrand. The differentiability up to order $k$ of the lower and upper approximating functions $E\left(\mathcal{H}\left(\tau ,\gamma \right)\right)$ and $E\left(\mathcal{H}\left(\tau ,-\gamma \right)\right)$ converges to the corresponding order derivative of $p\left(x,u\right)\forall t\in \mathit{Ia}.s.$ It depends mainly on the existence of the corresponding derivatives of the internal function $\gamma$ w.r.t.$u$ and the finite expectation of these derivatives, uniformly on some neighborhood of $W=B\left(u,r\right),$ whenever the family $\mathcal{H}$ is suitably chosen. Thus, it needs the newly generalized derivative in the neighborhood set of $W,$ such as Fréchet and the weak derivatives (Clarke and Mordukhovich derivatives) for computing gradient, stationary points, and their conveniences; see in the previous work [25].

Assumptions on the properties $P_1\mathit{\text{to}}{P}_5$, together with additional properties in above $\gamma$, it is smooth function w.r.t. $u$. Because the random state solution $T$ is smooth w.r.t $u$. The differential operator $A$ computed from coercivity properties of $a$ and $l$, $A$ is invertible and $\frac{\partial T}{\partial u}=A$.

Assumption 5.4. Assume that $u\in \mathcal{P},$ is a neighborhood of the set $M$ and super-set $S$ in Eq. (55). This set $\mathcal{P}$ is a connected neighborhood super set $S$ of the ${\mathit{OA}}_{\tau }$ and inside the interior set $M$ of ${\mathit{IA}}_{\tau }$, the following conditions hold true for the approximation smooth functions.

1. The support of the probability density function (pdf) $\mathit{supp}\left\{\varphi \right\}$ is compact.

2. The pdf is continuous on some set $\mathcal{P}$.

3. The internal function $\gamma$ and ${\nabla }_u\left(\gamma \right)=A$ are continuous on the set $\mathcal{P}\times \mathit{supp}\left\{\varphi \right\}$. The functions $\nabla \gamma$ and $\varphi \left(\xi \right)$ vanish over $\mathcal{P}\times \mathrm{\Omega }/\mathit{supp}\left\{\varphi \right\}$.

4. The $\nabla \gamma$ is continuous at $\mathrm{\Gamma }\left(u\right)$ the boundary of S and M, and $\nabla \gamma \ne 0$ for any $\xi \in \mathrm{\Gamma }\left(u\right)$ in boundary of M and S.

Theorem 5.5. If the assumption (5.4) holds true for any family of the parametric function ${\left.\left\{\theta \right(s_i\left(u\right),\tau \left)\right\}\right|}_{s_i={\gamma }_i}$ for any $i\in I$, the probabilistic function $p\left(u\right)$ has a continuous differentiable or a continuous gradient in some open ball W around point $u\in U_{\mathit{adm}}$ and the gradient of inner and outer approximation functions respectively converges uniformly to the gradient of $p\left(u\right)$ on W for any decreasing sequence ${\left\{{\tau }_k\right\}}_{k\in N}\in \left(0,1\right)$ for any $i\in I$ such that

$${\mathit{lim}}_{k\to \infty }{\mathit{inf}}_{u\in W}{\nabla }_u\left(1-{\psi }_i\left({\tau }_k,u\right)\right)={\nabla }_u{p}_i\left(u\right);$$E66

$${\mathit{lim}}_{k\to \infty }{\mathit{sup}}_{u\in W}{\nabla }_u{\phi }_i\left({\tau }_k,u\right)={\nabla }_u{p}_i\left(u\right);$$E67

for a particular $\tau \to 0+,$ all of the gradient of the functions is zero. Thus, as $k\to \infty ,$

$${\mathit{lim}}_{{\tau }_k\to 0+}{\mathit{inf}}_{u\in W}{\nabla }_u\left(1-{\psi }_i\left({\tau }_k,u\right)\right)={\nabla }_u{p}_i\left(u\right)=0;$$E68

$${\mathit{lim}}_{{\tau }_k\to 0+}{\mathit{sup}}_{u\in W}{\nabla }_u{\phi }_i\left({\tau }_k,u\right)={\nabla }_u{p}_i\left(u\right)=0.$$E69

This theorem has been proved in the previous submitted work [25]. Suppose each property above assumption (5.4) holds true. Then, for the appropriate parametric function $\theta \equiv \mathcal{H}$ the functions $\phi \left(\gamma ,\tau \right)$ and $\psi \left(\gamma ,\tau \right)$ are continuously partial differentiable w.r.t. u up to order b on $U_{\mathit{adm}}$, for all $\tau \in \left(0,1\right)$. The higher derivatives,

$${\partial }^{{\sum }_{j=1}^n{l}_j}E(\mathcal{H}\left(\tau ,\gamma \right)/\partial u_1\partial u_2\partial u_3.\dots \partial u_j={\int }_{\mathrm{\Omega }}{\partial }^{{\sum }_{j=1}^n{l}_j}\psi \left(\gamma ,\tau \right)/\partial u_1\partial u_2\partial u_3.\dots \partial u_j\varphi \left(\xi \right)\mathit{d\xi }.$$E70

And the case for outer approximation

$${\partial }^{{\sum }_{j=1}^n{l}_j}E(\mathcal{H}\left(\tau ,-\gamma \right)/\partial u_1\partial u_2\partial u_3.\dots \partial u_j={\int }_{\mathrm{\Omega }}{\partial }^{{\sum }_{j=1}^n{l}_j}\phi \left(\gamma ,\tau ,u\right)/\partial u_1\partial u_2\partial u_3.\dots \partial u_j\varphi \left(\xi \right)\mathit{d\xi }.$$E71

Further analysis of convergence approach and uniform convergence gradient of the approximation functions are seen in [29] of proposition 4.2.; thus, the equation given in (70) and (70) are equivalent.

6. Convexity property of inner and outer approximations

We notice that the coefficient parameter is either constant or depends on the $x\in D,$ and the convexity issues depend on the following three cases.

Case 1 $\kappa$ is constant dependent of x.

Then $\left(u,\xi \right)\mapsto y\left(u,x,\xi \right):U\times \mathrm{\Omega }\to \mathbb{R}$ is linear by see Eq. (16), hence, jointly convex w.r.t. $\left(u,\xi \right)$, for each $x\in D$ (see Theorem 5.4 of the continuous dependent of the solution on the random parameter.

Case 2 $\kappa$ is a positive independent $x$ but depends on random variable, and concave function.

Case 3 $\kappa$ depends on x, positive and nonlinear; we have to use linear approximation of order 1 such as Taylor approximation of order one for k.

For the above two cases, $\kappa \in \mathcal{K},$ the elliptic PDE system defined in Eqs. (3)–(4) becomes

$$-\mathit{\Delta y}\left(x,\xi \right)=\frac{1}{\kappa }f\left(u,x\right),\text{for}x\in D,$$E72

$$y\left(x,u,\xi \right)=g\left(x,u,\xi \right),\text{for}x\in \partial D,\xi \in \mathrm{\Omega }\mathrm{a}.\mathrm{s}.$$E73

For all three cases, the $\kappa$ is independent or depends on $x$ we can solve the PDE system by stochastic finite difference method (SFDM). It follows ${\left(\mathcal{A}\left(\kappa \right)\right)}^{-1}=\frac{1}{\kappa }{\mathcal{A}}^{-1}$ and thus

$$y\left(u,x,\xi \right)=\frac{1}{\kappa }{\mathcal{A}}^{-1}\left[j_{\mathcal{H}}\left(f\left(\cdot \right)\right)+\mathit{kg}(\cdot ,\xi \left)\right)\right]\left(x\right)\le y_{\mathit{max}}.$$E74

It is equivalent to

$$\left[{\mathcal{A}}^{-1}{j}_{\mathcal{H}}\left(f_{\mathit{ij}}\left(\cdot \right)\right)+{\mathit{kg}}_{\mathit{ij}}(\cdot ,\xi \left)\right)\right]\left(x\right)-\kappa y_{\mathit{max}}\left(x\right)\le 0.$$E75

The expression on the left-hand side of the last inequality is jointly convex w.r.t. $\left(u,\xi \right)$ because of the linearity of $f$ w.r.t. $\left(u,\xi \right)$, the concavity of $\kappa$ and the nonnegativity of $y_{\mathit{max}}$. Since

$$\mathit{Pr}\left\{y\left(u,x,\xi \right)\le y_{\mathit{max}},\forall x\in D\right\}=\mathit{Pr}\left\{{\mathcal{A}}^{-1}\left[j_{\mathcal{H}}\left(f\left(x\right)\right)+\mathit{kg}(\cdot ,\xi \left)\right)\right]-\kappa y_{\mathit{max}}\left(x\right)\le 0,\forall x\in D\right\}.$$E76

is valid, in Proposition 4.2 [13] yields the following result with measure zero property. Hence, for any arbitrary random function $g\in \mathcal{B},$ the internal part of the probabilistic function is a linear, it implies the quasi-concavity/quasi-convexity of $\gamma$. Hence, the probabilistic uniform-constrained function is a convex; see proposition (3.10). For any general case, the convexity of the proposed approach is proved in the following proposition.

Proposition 6.1. Let $\gamma \left(x,u,\xi \right)=y-y_{\mathit{\text{maxd}}}\le 0,\forall \xi \in \mathrm{\Omega },u\in U_{\mathit{adm}}$ with a measure zero property see in the proposition 4.2. and $x\in D.$ If $\gamma \left(x,u,\xi \right)$ is quasi convex or quasi concave, then the approximation functions ${\psi }_i\left(\tau ,u,\xi \right)=1-E\left(\theta \right(\gamma \left(x,u,\xi \right)\le 1-\alpha$ and ${\phi }_i\left(\tau ,u,\xi \right)=E(\theta \left(\tau ,-\gamma \left(x,u,\xi \right)\right)\ge \alpha$ are convex.

Proof. The outer approximation convexity is proved in [13, 29]. Let $m\left(s\right)=e^{-s/\tau }$ is a quasi-convex function for all $s\in \mathbb{R}$ and $\tau \in \left[0,1\right]$. Then, the sum and the constant multiple of quasi-convex functions are a quasi-convex. Also, based on this statement $l\left(s\right)=1+m_2\tau e^{-s/\tau }$ is a quasi-convex function. The reciprocal of $l\left(s\right)$ which is

$${\left(l\left(s\right)\right)}^{-1}={\left(1+m_2\tau e^{-s/\tau }\right)}^{-1},$$E77

is said to be reciprocally quasi-concave [31]. It is from

$$\frac{\partial \left(l{\left(s\right)}^{-1}\right)}{\partial s}=m_2\tau e^{-s/\tau }{\left[1+m_2\tau e^{-s/\tau }\right]}^{-2},$$E78

$\frac{\partial {\left(l\left(s\right)\right)}^{-1}}{\partial s}=0,\mathrm{as}\tau \to 0+$. The second derivative with respect to $s$ is negative at the stationary point $\tau \to 0+$. To check convexity property through the second-order derivative test

$$\frac{{\partial }^2\left(l{\left(s\right)}^{-1}\right)}{\partial s^2}=-\left(m_2\tau e^{-s/\tau }\right){\left[1+m_2\tau e^{-s/\tau }\right]}^{-2}\left(1-2m_2\tau e^{-s/\tau }\right)<0,$$E79

not valid for $0<m_2\le m_1/\left(1+m_1\right)\text{and}2m_2\tau e^{-s/\tau }>1$. The function $l$ is a strictly concave function for every $\tau \in \left(0,1\right].$ Every concave function is a quasi-concave. Therefore, ${\left(l\left(s\right)\right)}^{-1}={\left(1+m_2\tau e^{-s/\tau }\right)}^{-1}$ is a concave function. Also, it holds the quasi-concavity property. It follows that $\theta \left(\tau ,s\right)=\left(1+m_1\tau \right).{}^{\ast }\left(l{\left(s\right)}^{-1}\right)$ is a constant multiple of concave function and a concave function. The negative of concave function is a convex, it implies that $-\theta \left(\tau ,s\right)=-\left(1+m_1\tau \right).{}^{\ast }l\left(s\right))$ is a convex function. Since, the integral of the convex function is convex, $-E(\theta \left(\tau ,s\right)=-{\int }_{\mathrm{\Omega }}\left(1+m_1\tau .{}^{\ast }l\left(s\right)\right)\varphi \left(\xi \right)\mathit{d\xi }$ is a convex function. Furthermore, $\psi \left(\tau ,s\right)=1-E\left(\theta \left(\tau ,s\right)\right)\ge \alpha$ is a convex function, based on the convexity property, which stipulates that the sum of convex functions is convex. Therefore, the inner approximation is generally a convex function. □

Theorem 6.2. The properties of the functions $E\left(\theta \left(\tau ,s\right)\right)=1-\psi \left(\tau ,s\right)\ge \alpha$ and $\phi \left(\tau ,u,\xi \right)=E\left(\theta \left(\tau ,-s\right)\right)\ge \alpha$ are convex in the proposition (6.1). The regularization parameter $\rho \ge 0,$ the objective function $E\left(J\left(y\left(.\right)\right)\right)$ is convex in Eq. (38), If P(u,x)is a continuous and convex function w.r.t. $u$, then the ${\mathit{IA}}_{\tau }$ and the ${\mathit{OA}}_{\tau }$ have the unique optimal solutions as $\tau \to 0+.$ The optimal solutions of the approximations converge to the optimal solution of CCPDE.

This theorem has been proved in the work [13].

7. Numerical implementation and case study

This work considers the stationary boundary value elliptic PDEs, where the randomness comes from the boundary conditions of the PDE system. It analyzes the observed temperature variation and distribution processes widely used in biological applications, such as hyperthermia treatment of cancer tissue in human body organs. This work is an extension of the previous works Kibru et al. [13, 14]. The optimal heating of an enclosed, thermally well-insulated, spatial domain $D_c\subset D$ to the desired temperature $y_d$ is given. The heat injection is elected through a distributed stationary heat source ($y\left(u^{\ast }\right)=T\left(u^{\ast }\right)$) [20, 32].

The heat source is assumed to be highly affected by uncertainties, e.g., due to inaccuracies arising from heating devices and inlet heating processes. The boundary condition is supposed to be nonhomogeneous and nonlinear depending on random parameters so that the thermal conductivity is spatially constant, which is not precisely known or position-dependent $x\in D$. Furthermore, despite the specified overall desired temperature $y_d$ is deterministic. The required state solution is $y\left(x,u^{\ast },\xi \right)$ is random, and it should be kept below a maximum allowed value $y_{\mathit{max}}$ with a high-reliability level $\alpha \le 0.95$ in a given subset $\mathit{Dc}$ of $D$. More practical applications are studied in the work [14, 33].

The numerical random state solution $y$ is obtained from the stochastic finite difference method (SFDM [34]). We have discretized 10,000 points from the $x_1$ and $x_2$ axes with the mesh generation. After the solution $y\left(x=\left(x_1,x_2\right),u,\xi \right)$ obtained from the infinite-dimensional space $D$ of the PDEs, the optimization problem is reduced to finite-dimensional reduced ${\mathit{\text{CCPDE}}}_{\mathit{red}}$. The nonsmooth analysis is relevant for solving this nondifferentiable CCPDE where the random fluctuation comes from the system’s boundary. We have developed a smooth parametric approximation called IA and OA in equations.

The variables $\xi \in \mathrm{\Omega }$ are identical independently distributed (iid) as standard normal distributed random variables. The samples for the random variable are generated by using the multilevel Monte Carlo method (ML-MCM) sampling approach. Subsequently, the PDE system is solved through the SFDM using a MATLAB implementation at each step of the optimization algorithm. After discretization, the inner and outer approximation problems are solved using the MATLAB Optimization function fmincon.m, each decreasing values of $\tau ={10}^{-k},k=1,\dots ,4$.

In our previous work, we have considered practical applications on hyperthermia treatment (HT) and planning as a case study Kibru et al. [14]. The HT and planning are used as an accompanying strategy in modern clinical cancer therapy [20]. Hyperthermia treatment consists of the heating of tumor tissue in order to subdue or eradicate the growth of tumor cells from a given organ. The hyperthermia treatment procedure aims to heat the tumor tissue in the human body to a given temperature without causing damage to healthy surrounding sensitive tissue due to overheating.

The heating is usually done through multiple electromagnetic (EM) sources, where each EM source generates an electric field $G\left(x\right)$ of the heat capacity $c$ and density $\rho$ the phases and amplitude p. As a result, the electric fields facilitate a net power deposition on the tumor region given by [20] (e.g., $Q\left(x\right)=\frac{\sigma \left(\xi \right)}{2}{\left|G\left(x\right)\right|}^2,$ where $G\left(x\right)={\sum }_{j=1}^N{p}_j{G}_{j}x$ is a linear superposition of the individual fields and $\sigma \left(x\right)$ the electric conductivity. In general, the phases and the power $Q$, corresponding to each individual antenna, are not known in advance.

Figure 1 displays the surface of the state solution $y\left(x,u^{\ast },\xi \right)$ and the adjoint operator $P$ for the optimality criteria of the infinite-dimensional CCPDE problem where

Figure 1.

The solution of PDEs and adjoint P. (a) State y. (b) Adjoint P.

$$S^{\ast }\left(y-y_d\right)+\rho \left(u-u^{\ast }\right))\ge 0,\forall u\in U_{\mathit{adm}},$$E80

where $S:U_{\mathit{adm}}\mapsto \mathcal{H}$ is a control to state map,

$$y^{\ast }=S\left(u^{\ast }\right),$$E81

is the optimal state of CCPDE, since CCPDE is a convex optimization problem w.r.t. $u$ in the proposition expressed (3.10), and

$$P=S^{\ast }\left(y-y_d\right)=S^{\ast }\left(S\left(u\right)-y_d\right)$$E82

are displayed Figure 1 in (a) and (b), respectively. Hence, the state and adjoint are depending on the random variable. So, they are the random state and adjoint functions. The solution of boundary value PDEs is displayed in (a) and (b) from SFDM in this Figure 1.

The controls obtained from IA and OA are displayed in Figures 2 and 3 with different values of $\tau ={10}^k$ for $k=1,\dots ,4.$ the optimal controls from the optimization approach of the IA and OA are displayed in (a) to (d) respectively.

Figure 2.

The control ${\mathit{IA}}_{\tau }$. (a) IA, tau=0.1 (b) IA, tau=0.01 (c) IA, tau=0.001 (d) IA, tau=0.0001.

Figure 3.

The control ${\mathit{OA}}_{\tau }$. (a) OA, tau=0.1 (b) OA, tau=0.01 (c) OA, tau=0.001 (d) OA, tau=0.0001.

The error between the IA and OA is displayed in (a–d) Figure 4, when $\tau$ is reduced, that mean $\tau \to 0+$ the error is near to zero. This shows that the controls of inner and outer approximation are equal at $\tau =0.001$.

Figure 4.

The errors of ${\mathit{IA}}_{\tau },$ ${\mathit{OA}}_{\tau }$, Obj. (a) Error of IA and OA, tau=0.1; (b) Error of IA and OA, tau=0.01; (c) Error of IA and OA, tau=0.001; (d) Error of IA and OA, tau=0.0001; (e) Error of IA and OA, objective of OA and IA.

Finally, the Figure 4e shows the objective functions of $J_{\mathit{IA}}\left(\tau \right)$ and $J_{\mathit{OA}}\left(\tau \right)$ when $\tau \to 0+,$ the $\mathit{IA}\equiv \mathit{OA},\tau ={10}^{-4}$. The error is zero when tau is less than 0.001 in Figure 4a.

Example: To solve the following CCPDE, where $\xi$ are iid $\mu =0$ and $\sigma =1$:

$$-\nabla \left(\kappa \left(x\right)\nabla y\right)=\Delta \left(y\right),\kappa \left(x\right)=1,f\left(x\right)=0,g\left(u,\xi ,x\right)=\left(u\left(x\right)-y\right)\xi ,y_{\mathit{max}}=2.999;$$E83

$$\mathit{yd}=2-2.{}^{\ast }\left(x1.{}^{\ast }\left(x1-1\right)+\left(x2.{}^{\ast }\left(x2-1\right)\right)\right),u_{\mathit{min}}=0.5,u_{\mathit{max}}=4.5;$$E84

$$\rho ={10}^{-3},\alpha =0.95;$$E85

$$D=\left[0,1\right]\times \left[0,1\right].$$E86

Acknowledgments

This work is financially supported by KAAD and DFG.