On the Optimal Control Problem for Bilinear Systems with Bounded and Unbounded Controls

Abstract

In the present chapter, we address the problem of optimal control of bilinear systems with unbounded and bounded controls. The main purpose of this study is to establish the existence of an optimal control, which is characterized by a solution of an optimality system. Additionally, we provide a sufficient condition for the uniqueness of such a control. The results obtained are applied to various specific control sets. Furthermore, we present a numerical algorithm for computing the optimal control and successfully demonstrate its application through simulations involving heat equations.

1. Introduction

Bilinear systems form a significant subset of nonlinear systems, characterized by nonlinearity arising from state and control variables’ multiplication in dynamic equations. These systems have found extensive use in modeling various real-world physical processes where linear models fall short, providing a more effective alternative to linear systems. Notable examples include the basic law of mass action, heat exchanger dynamics with controlled flow, and quantum control.

Controllability analysis for bilinear systems has been explored by several researchers using diverse control techniques. Weak controllability of unidimensional beam bilinear equations was studied in [1], while multiple controllability of semi-linear parabolic and hyperbolic systems was established in [2]. Specific classes of bilinear parabolic systems adhering to Newton’s law were investigated for controllability in [3], and the exact controllability of parabolic equations with distributed controls was discussed in [4]. Other studies covered the controllability of unidimensional semilinear wave equations with Dirichlet boundary conditions [5].

Optimal control of bilinear systems has also received considerable attention in the literature. Works such as [6, 7] focused on the optimal control problem for convective-diffusive fluid bilinear systems and bilinear heat equations, respectively, with bounded controls. Unbounded optimal control for a class of bilinear systems was investigated in [8], while [9, 10] studied optimal control problems for wave equations and Kirchoff plate equations using distributed bounded controls. Bounded and unbounded controls were also considered in [11, 12] for bilinear wave and Kirchoff equations, respectively. Several other studies explored regional optimal control for bilinear systems with distributed controls [13, 14], infinite-dimensional bilinear systems with bounded and unbounded controls [15, 16], and constrained regional optimal control for bilinear plate equations [17, 18]. These works utilized Gateaux differentiability as the primary approach.

The present research focuses on optimal control for a wide range of infinite-dimensional bilinear systems, employing a cost function that includes the deviation between the desired and final states at time $T$, as well as effort and energy terms. The study establishes the existence of solutions to this problem and provides characterizations of optimal controls for both unbounded and bounded control sets. Frechet differentiability and the characterization of the normal vector to the set of admissible controls play a pivotal role in this approach, ensuring the uniqueness of such controls under an appropriate condition. In addition, we develop a computational method that leads to an algorithm and simulations.

Let $\mathrm{\Omega }$ be an open bounded domain of ${\mathrm{ℝ}}^n\left(n\ge 1\right)$, and we consider the following bilinear system

$$\left\{\begin{array}{ll}\dot{z}\left(t\right)=\mathcal{A}z\left(t\right)+u\left(t\right)\mathit{Bz}\left(t\right)& 0<t<T\\ z\left(0\right)=z_0\in L^2\left(\mathrm{\Omega }\right),& \end{array}\right.$$E1

with $\mathcal{A}:D\left(\mathcal{A}\right)\subset L^2\left(\mathrm{\Omega }\right)\to L^2\left(\mathrm{\Omega }\right)$ generates a strongly continuous semigroup ${\left(S\left(t\right)\right)}_{t\ge 0}$ on state space $L^2\left(\mathrm{\Omega }\right)$, whose norm and scalar product are denoted, respectively, by $\parallel .\parallel$ and $⟨.,.⟩$, the control space is $L^2\left(0,T\right)$, and $B:L^2\left(\mathrm{\Omega }\right)\to L^2\left(\mathrm{\Omega }\right)$ is a linear bounded operator.

For all $z_0\in L^2\left(\mathrm{\Omega }\right)$ and $u\in L^2\left(0,T\right)$, the system (1) has a unique weak solution in the space $C\left(0,T;L^2\left(\mathrm{\Omega }\right)\right)$ (see [19]) and $z$ is the solution of

$$z\left(t\right)=S\left(t\right)z_0+{\int }_0^{t}S\left(t-s\right)u\left(s\right)\mathit{Bz}\left(s\right)\mathit{ds}.$$E2

Our optimal control problem is given by

$$\left\{\begin{array}{c}min\mathcal{J}\left(u\right)\\ u\in U_{\mathit{ad}}.\end{array}\right.$$E3

where $U_{\mathit{ad}}$ is a closed and convex subet of $L^2\left(0,T\right)$. The functional $\mathcal{J}$ is given by

$$\mathcal{J}\left(u\right)=\frac{\alpha }{2}\parallel z\left(T\right)-z_d{\parallel }^2+\frac{\beta }{2}{\int }_0^T\parallel z\left(t\right)-z_d{\parallel }^2\mathit{dt}+\frac{\epsilon }{2}\parallel u{\parallel }_{L^2\left(0,T\right)}^2,$$E4

with $\alpha$, $\beta$, and $\epsilon$ are nonnegative constants, and $z_d\in L^2\left(\mathrm{\Omega }\right)$ is the desired state.

The paper is structured as follows: In Section 2, we demonstrate the existence of an optimal control solution for problem (3). Section 3 is dedicated to providing a characterization and discussing the uniqueness of this control. The results obtained are applied to various relevant scenarios. In Section 4, we present a numerical algorithm for computing the optimal control. The effectiveness of the approach is illustrated through successful simulations.

2. Existence of an optimal control

We will now discuss the existence of an optimal control solution to the problem (3).

Theorem 1.1.

There exists an optimal control $u\in U_{\mathit{ad}}$, solution of problem (3).

Proof:

The set $\left\{\mathcal{J}\left(u\right)|u\in U_{\mathit{ad}}\right\}$ is non-empty and nonnegative, and thus it has a nonnegative infimum.

Let ${\left(u_n\right)}_{n\in \mathrm{\mathbb{N}}}$ be a minimizing sequence in $U_{\mathit{ad}}$, as $\epsilon >0$ we have

$$\parallel u_n{\parallel }_{L^2\left(0,T\right)}^2\le \frac{2}{\epsilon }\mathcal{J}\left(u_n\right),\forall n\in \mathrm{\mathbb{N}}.$$

As a result, ${\left(u_n\right)}_{n\in \mathrm{\mathbb{N}}}$ is bounded. Then, there exists a subsequence, again denoted ${\left(u_n\right)}_{n\in \mathrm{\mathbb{N}}}$, which converges weakly to a limit $u^{\ast }$. As $U_{\mathit{ad}}$ is closed and convex, it is closed for the weak topology, which means that $u^{\ast }\in U_{\mathit{ad}}$.

Let $z_n$ and $z^{\ast }$ the unique solutions of systems (1), corresponding to $u_n$ and $u^{\ast }$ respectively. From (2), we have

$$\begin{array}{lll}{z}_n\left(t\right)-z^{\ast }\left(t\right)& =& {\int }_0^{t}S\left(t-s\right)\left[u_n\left(s\right){\mathit{Bz}}_n\left(s\right)-u^{\ast }\left(s\right){\mathit{Bz}}^{\ast }\left(s\right)\right]\mathit{ds}\\ & =& {\int }_0^{t}S\left(t-s\right)\left[\left(u_n\left(s\right)-u^{\ast }\left(s\right)\right){\mathit{Bz}}^{\ast }\left(s\right)+u_n\left(s\right)\left({\mathit{Bz}}_n\left(s\right)-{\mathit{Bz}}^{\ast }\left(s\right)\right)\right]\mathit{ds}.\end{array}$$

Then

$$\begin{array}{ll}\parallel z_n\left(t\right)-z^{\ast }\left(t\right)\parallel & \le ‖{\int }_0^{t}S\left(t-s\right)\left(u_n\left(s\right)-u^{\ast }\left(s\right)\right){\mathit{Bz}}^{\ast }\left(s\right)\mathit{ds}‖\\ & +{\int }_0^t\parallel S\left(t-s\right)\parallel \mid u_n\left(s\right)\mid \Vert B\Vert \parallel z_n\left(s\right)-z^{\ast }\left(s\right)\parallel \mathit{ds}.\end{array}$$

By employing Gronwall’s Lemma, the aforementioned inequality can be expressed as follows:

$$\parallel z_n\left(t\right)-z^{\ast }\left(t\right)\parallel \le ‖{\int }_0^{t}S\left(t-s\right)\left(u_n\left(s\right)-u^{\ast }\left(s\right)\right){\mathit{Bz}}^{\ast }\left(s\right)\mathit{ds}‖e^{\parallel B\parallel {\int }_0^t\parallel S\left(t-s\right)\parallel \mid u_n\left(s\right)\mid \mathit{ds}}.$$

There exist constant $M\ge 1$ and $\rho \in \mathrm{ℝ}$ such that $\parallel S\left(t\right)\parallel \le {\mathit{Me}}^{\mid \rho \mid t}$, then

$${\int }_0^t\parallel S\left(t-s\right)\parallel \mid u\left(s\right)\mid \mathit{ds}\le {\mathit{Me}}^{\mid \rho \mid T}{T}^{\frac{1}{2}}{\left({\int }_0^T{\left|u\left(s\right)\right|}^2\mathit{ds}\right)}^{\frac{1}{2}}.$$

The above inequality leads to the following result:

$$\parallel z_n\left(t\right)-z^{\ast }\left(t\right)\parallel \le \parallel {\mathrm{\Lambda }}_t\left(u_n-u^{\ast }\right)\parallel e^{{\mathit{Me}}^{\mid \rho \mid T}\left(\parallel B\parallel T^{\frac{1}{2}}\mu \right)},\forall n\in \mathrm{\mathbb{N}},$$E5

with $\mu =sup\parallel u_n{\parallel }_{L^2\left(0,T\right)}$, and the operator ${\mathrm{\Lambda }}_t:L^2\left(0,T\right)\to L^2\left(\mathrm{\Omega }\right)$ is given by

$${\mathrm{\Lambda }}_{t}u={\int }_0^{t}S\left(t-s\right)u\left(s\right){\mathit{Bz}}^{\ast }\left(s\right)\mathit{ds},\forall u\in L^2\left(0,T\right).$$

Allow us to show that for all $t\in \left[0,T\right]$, ${\mathrm{\Lambda }}_t$ is compact.

we will demonstrate that for every sequence ${\left({\phi }_n\right)}_n$ in $L^2\left(\mathrm{\Omega }\right)$ which weakly converges to $0$ in $L^2\left(\mathrm{\Omega }\right)$, ${\left({\mathrm{\Lambda }}_t^{\ast }{\phi }_n\right)}_n$ converge with norm to $0$ in $L^2\left(\mathrm{\Omega }\right)$. Compute the operator’s adjoint of${\mathrm{\Lambda }}_t$.

Let $u\in L^2\left(0,T\right)$ and $y\in L^2\left(\mathrm{\Omega }\right)$, we have

$$\begin{array}{ll}<{\mathrm{\Lambda }}_t\left(u\right),y>& ={\int }_0^t<S\left(t-s\right)u\left(s\right){\mathit{Bz}}^{\ast }\left(s\right),y\left(s\right)>\mathit{ds}\\ & & \\ & ={\int }_{\mathrm{\Omega }}<u\left(s\right),S^{\ast }\left(t-s\right)y\left(s\right)B^{\ast }{z}^{\ast }\left(s\right){>}_{L^2\left(0,T\right)}\mathit{dx}\\ & & \\ & =<u\left(.\right),{\int }_{\mathrm{\Omega }}{S}^{\ast }\left(t-s\right)y\left(x,.\right)B^{\ast }{z}^{\ast }\left(x,.\right)\mathit{dx}{>}_{L^2\left(0,T\right)}.\end{array}$$

So ${\mathrm{\Lambda }}_t^{\ast }:L^2\left(\mathrm{\Omega }\right)\to L^2\left(0,T\right)$ is given by

$$\left[{\mathrm{\Lambda }}_t^{\ast }z\right]\left(s\right)=\left\{\begin{array}{ll}<S\left(t-s\right){\mathit{Bz}}^{\ast }\left(.\right),z>& \mathit{\text{if}}0\le s<t\\ 0& \mathit{\text{if}}t<s\le T.\end{array}\right.$$

Consider a sequence ${\left({\varphi }_l\right)}_{l\in \mathrm{\mathbb{N}}}$ be a sequence in $L^2\left(\mathrm{\Omega }\right)$ such that ${\varphi }_l\underset{l\to \infty }{\rightharpoonup }0$. Without loss of generality, we can assume that $\parallel {\varphi }_l{\parallel }_{L^2\left(\mathrm{\Omega }\right)}\le l,\forall l\in \mathrm{\mathbb{N}}$, Then, the following holds:

$${\varphi }_l\underset{l\to \infty }{\rightharpoonup }0\Rightarrow \parallel \left[{\mathrm{\Lambda }}_t^{\ast }{\varphi }_l\right]\left(s\right)\parallel \underset{l\to \infty }{\to }0a.e\mathrm{on}\left[0,T\right].$$

Thus, for any $s\in \left[0,T\right]$, we have

$$\mid \left[{\mathrm{\Lambda }}_t^{\ast }{\varphi }_l\right]\left(s\right)\mid \le \parallel S\left(t-s\right)\parallel \Vert B\Vert \parallel z^{\ast }\left(s\right)\parallel \parallel {\varphi }_l\left(s\right)\parallel \le {\mathit{Me}}^{\mid \rho \mid T}\Vert B\Vert \parallel z^{\ast }\left(s\right)\parallel ,\forall s\in \left[0,T\right].$$

By employing the Dominated Convergence Theorem, we can deduce that:

$$\parallel {\mathrm{\Lambda }}_t^{\ast }{\varphi }_l\left(s\right)\parallel \underset{l\to \infty }{\to }0,\text{in}{L}^2\left(0,T\right).$$

Then, ${\mathrm{\Lambda }}_t$ is compact.

It follows from the weak convergence $\left(u_n-u^{\ast }\right)\underset{n\to \infty }{\rightharpoonup }0$ that

$$\underset{n\to \infty }{lim}\parallel {\mathrm{\Lambda }}_t\left(u_n-u^{\ast }\right){\parallel }_{L^2\left(\mathrm{\Omega }\right)}=0.$$

Therefore, by the inequality (5), we obtain

$$\underset{n\to \infty }{lim}\parallel z_n\left(t\right)-z^{\ast }\left(t\right){\parallel }_{L^2\left(\mathrm{\Omega }\right)}=0,a.e\mathrm{on}t\in \left[0,T\right].$$

We have $$\parallel z^{\ast }\left(T\right)-z_d{\parallel }_{L^2\left(\mathrm{\Omega }\right)}^2=\underset{n\to \infty }{lim}inf\parallel z_n\left(T\right)-z_d{\parallel }_{L^2\left(\mathrm{\Omega }\right)}^2.$$

Applaying Fatou Lemma, gives

$${\int }_0^T\parallel z^{\ast }\left(T\right)-z_d{\parallel }^2\mathit{dt}\le \underset{n\to \infty }{lim}inf{\int }_0^T\parallel z_n\left(T\right)-z_d{\parallel }^2\mathit{dt}.$$

Since the norms are lower semi-continuous for the weak topology, it follows that the weak convergence of $\left(u_n\right)\to u^{\ast }$ yields that $\parallel u^{\ast }{\parallel }_{L^2\left(0,T\right)}\le \underset{n\to \infty }{lim}inf\parallel u_n{\parallel }_{L^2\left(0,T\right)}$.

Thus

$$\mathcal{J}\left(u^{\ast }\right)\le \underset{n\to \infty }{lim}inf\mathcal{J}\left(u_n\right)={\mathcal{J}}^{\ast }.$$

Then $u^{\ast }$ an optimal control.

3. Characterization of an optimal control

In which we give a characterization of an optimal control solution to the problem (3).

3.1. Optimality conditions

First, we characterize the differential of the functional cost (4).

Proposition 1.

The functional (4) is differentiable in the Frechet sense, and its differential for all $u\in L^2\left(0,T\right)$ is given by

$$\left\{\begin{array}{l}{D}_u\mathcal{J}\cdot h={\int }_0^T{\mathcal{J}}^{\prime }\left(u\right).\mathit{hdt}\\ {\mathcal{J}}^{\prime }\left(u\right)\left(t\right)=<\phi \left(t\right),\mathit{Bz}\left(t\right)>+\mathit{\epsilon u}\left(t\right),\end{array}\right.$$E6

with $z$ is solution of system (1), and $\phi$ is solution of the following adjoint equation

$$\left\{\begin{array}{l}\dot{\phi }\left(t\right)=-{\mathcal{A}}^{\ast }\phi \left(t\right)-u\left(t\right)B^{\ast }z\left(t\right)\phi \left(t\right)-\beta \left(z\left(T\right)-z_d\right)\\ \phi \left(T\right)=\alpha \left(z\left(T\right)-z_d\right).\end{array}\right.$$E7

Proof:

The operator ${\mathcal{A}}^{\ast }$ is the adjoint operator of $\mathcal{A}$, then generates a $C_0$-semigroup ${\left(S^{\ast }\left(t\right)\right)}_{t>0}$ on $L^2\left(\mathrm{\Omega }\right)$.

For $u\in U_{\mathit{ad}}$, the eq. (7) has a unique weak solution $\phi \in \mathcal{C}\left(\left[0,T\right],L^2\left(\mathrm{\Omega }\right)\right)$, which is a solution of the following equation

$$\phi \left(t\right)=S^{\ast }\left(T-t\right)\alpha \left(z\left(T\right)-z_d\right)+{\int }_t^T{S}^{\ast }\left(s-t\right)\left[u\left(t\right)B^{\ast }z\left(t\right)\left(\phi \left(t\right)\right)-\beta \left(z\left(T\right)-z_d\right)\right]\mathit{ds}.$$E8

Let us show that $\mathcal{J}$ is Frechet differentiable.

The mapping $u\mapsto z_u$ is Frechet differentiable in $L^2\left(0,T\right)$ and $D_{u}z\left(t\right)$ its differential at $u$. We have $z_h\left(t\right)=D_{u}z\left(t\right)h$ is the solution of the integral equation

$$z_h\left(t\right)={\int }_0^{t}S\left(t-s\right)h\left(s\right){\mathit{Bz}}_u\left(s\right)+u\left(s\right){\mathit{Bz}}_u\left(s\right)z_h\left(s\right)\mathit{ds}.$$E9

For all $u,u+h\in U_{\mathit{ad}}$, we have

$$\begin{array}{lll}\parallel z_{u+h}\left(T\right)-z_d{\parallel }_{L^2\left(\mathrm{\Omega }\right)}^2& =& <z_{u+h}\left(T\right)-z_d,z_{u+h}\left(T\right)-z_d{>}_{L^2\left(\mathrm{\Omega }\right)}\\ & =& <z_u\left(T\right)+z_h\left(T\right)-z_d,z_u\left(T\right)+z_h\left(T\right)-z_d{>}_{L^2\left(\mathrm{\Omega }\right)}\\ & =& <z_u\left(T\right)-z_d,z_u\left(T\right)+z_h\left(T\right)-z_d{>}_{L^2\left(\mathrm{\Omega }\right)}+<z_h\left(T\right),z_u\left(T\right)\\ & +& z_h\left(T\right)-z_d{>}_{L^2\left(\mathrm{\Omega }\right)}\\ & =& <z_u\left(T\right)-z_d,z_u\left(T\right)-z_d{>}_{L^2\left(\mathrm{\Omega }\right)}+<z_h\left(T\right),z_u\left(T\right)-z_d{>}_{L^2\left(\mathrm{\Omega }\right)}\\ & +& <z_h\left(T\right),z_u\left(T\right)-z_d{>}_{L^2\left(\mathrm{\Omega }\right)}+<z_h\left(T\right),z_h\left(T\right){>}_{L^2\left(\mathrm{\Omega }\right)}\\ & =& \parallel z_u\left(T\right)-z_d{\parallel }_{L^2\left(\mathrm{\Omega }\right)}^2+2<z_h\left(T\right),z_u\left(T\right)-z_d{>}_{L^2\left(\mathrm{\Omega }\right)}+o\parallel h{\parallel }_{L^2\left(0,T\right)}.\end{array}$$

Then

$$\parallel z_{u+h}\left(T\right)-z_d{\parallel }_{L^2\left(\mathrm{\Omega }\right)}^2-\parallel z_u\left(T\right)-z_d{\parallel }_{L^2\left(\mathrm{\Omega }\right)}^2=2<z_h\left(T\right),z_u\left(T\right)-z_d{>}_{L^2\left(\mathrm{\Omega }\right)}+o\parallel h{\parallel }_{L^2\left(0,T\right)}.$$

By employing similar calculations as those provided above, we obtain:

$${\int }_0^T\parallel z_{u+h}\left(t\right)-z_d{\parallel }_{L^2\left(\mathrm{\Omega }\right)}^2-\parallel z_u\left(t\right)-z_d{\parallel }_{L^2\left(\mathrm{\Omega }\right)}^2\mathit{dt}=2{\int }_0^T<z_h\left(t\right),z_u\left(t\right)-z_d{>}_{L^2\left(\mathrm{\Omega }\right)}\mathit{dt}+o\parallel h\parallel ,$$

and it is easy to see that $\parallel u+h{\parallel }_{L^2\left(0,T\right)}^2-\parallel u{\parallel }_{L^2\left(0,T\right)}^2=2<u,h{>}_{L^2\left(0,T\right)}+o\parallel h{\parallel }_{L^2\left(0,T\right)}.$ So, $\mathcal{J}$ is Frechet differentiable over $U_{\mathit{ad}}$, and its derivative at $u$ is given by

$${\mathcal{J}}^{\text{'}}\left(u\right)\cdot h=<z_h\left(T\right),\mathit{\alpha z}\left(T\right)-z_d{>}_{L^2\left(\mathrm{\Omega }\right)}+{\int }_0^T<z_h\left(t\right),\beta \left(z\left(T\right)-z_d\right){>}_{L^2\left(\mathrm{\Omega }\right)}\mathit{dt}+\epsilon {\int }_0^{T}u\left(t\right)h\left(t\right)\mathit{dt},$$

with $z_h$ solution of (9) is the solution of the equation

$$\left\{\begin{array}{l}{\dot{z}}_h\left(t\right)=\mathcal{A}{z}_h\left(t\right)+h\left(t\right)\mathit{Bz}\left(t\right)+u\left(t\right)\mathit{Bz}\left(t\right)z_h\left(t\right)\\ z_h\left(0\right)=0.\end{array}\right.$$E10

Let $n\in \rho \left(\mathcal{A}\right)$, where $\rho \left(\mathcal{A}\right)$ represents the resolving set of $\mathcal{A}$. We define ${\mathcal{A}}_n=n\mathcal{A}{\left(\mathit{nI}-\mathcal{A}\right)}^{-1}$, and ${\mathcal{A}}_n^{\ast }=n{\mathcal{A}}^{\ast }{\left(\mathit{nI}-{\mathcal{A}}^{\ast }\right)}^{-1}$ as the Yosida’s approximations of $\mathcal{A}$ and ${\mathcal{A}}^{\ast }$, respectively.

Next, we consider the solutions ${\phi }_n$ and $z_n$ the solutions of eqs. (7) and (10), respectively, where we use ${\mathcal{A}}_n$ and ${\mathcal{A}}_n^{\ast }$ in place of $\mathcal{A}$ and ${\mathcal{A}}^{\ast }$. Since both ${\mathcal{A}}_n$ and ${\mathcal{A}}_n^{\ast }$ are bounded, we have ${\dot{z}}_n\in L^2\left(0,T;L^2\left(\mathrm{\Omega }\right)\right)$ and ${\dot{\phi }}_n\in L^2\left(0,T;L^2\left(\mathrm{\Omega }\right)\right)$. Thus

$$\begin{array}{ll}{\int }_0^T<\beta \left(z\left(T\right)-z_d\right),z_n\left(t\right)>\mathit{dt}=-{\int }_0^T<{\dot{\phi }}_n\left(t\right)+\left({\mathcal{A}}_n^{\ast }+u\left(t\right){\left[B\left(z\left(t\right)\right)\right]}^{\ast }\right){\phi }_n\left(t\right),z_n\left(t\right)>\mathit{dt}& \\ =-{\int }_0^T\left[<{\dot{\phi }}_n\left(t\right),z_n\left(t\right)>+<{\phi }_n\left(t\right),{\mathcal{A}}_n{z}_n\left(t\right)u\left(t\right)B\left(z\left(t\right)\right)z_n\left(t\right)>\right]\mathit{dt}& \\ =-{\int }_0^T<{\dot{\phi }}_n\left(t\right),z_n\left(t\right)>\mathit{dt}-{\int }_0^T<{\phi }_n\left(t\right),{\dot{z}}_n\left(t\right)-h\left(t\right)\mathit{Bz}\left(t\right)>\mathit{dt}.& \end{array}$$

Integrating by part, as ${\dot{z}}_n\in L^2\left(0,T;L^2\left(\mathrm{\Omega }\right)\right),\dot{\phi }\in L^2\left(0,T;L^2\left(\mathrm{\Omega }\right)\right)$, gives

$${\int }_0^T<{\dot{\phi }}_n\left(t\right),z_n\left(t\right)>\mathit{dt}+{\int }_0^T<{\phi }_n\left(t\right),{\dot{z}}_n\left(t\right)>\mathit{dt}=<{\phi }_n\left(T\right),z_n\left(T\right)>-<\phi \left(0\right),z_n\left(0\right)>.$$

As $z_n\left(0\right)=0$ and ${\phi }_n\left(T\right)=\alpha \left(z\left(T\right)-z_d\right)$, the above equality yields

$${\int }_0^T<{\dot{\phi }}_n\left(t\right),z_n\left(t\right)>\mathit{dt}+{\int }_0^T<{\phi }_n\left(t\right),{\dot{z}}_n\left(t\right)>\mathit{dt}=<\alpha \left(z\left(T\right)-z_d\right),z_n\left(T\right)>.$$

Thus

$${\int }_0^T<\beta \left(z\left(t\right)-z_d\right),z_n\left(t\right)>\mathit{dt}=-<\alpha \left(z\left(t\right)-z_d\right),z_n\left(T\right)>+{\int }_0^T<{\phi }_n\left(t\right),h\left(t\right)\mathit{Bz}\left(t\right)>\mathit{dt}.$$

Or again

$$<\alpha \left(z\left(T\right)-z_d\right),z_n\left(T\right)>+{\int }_0^T<\beta \left(z\left(t\right)-z_d\right),z_n\left(t\right)>\mathit{dt}={\int }_0^T<{\phi }_n\left(t\right),h\left(t\right)\mathit{Bz}\left(t\right){>}_{L^2\left(\mathrm{\Omega }\right)}\mathit{dt}.$$

Let $\phi$ be the solution of the adjoint eq. (7), we have $\underset{n\to +\infty }{lim}\parallel z_n-z_h{\parallel }_{\mathcal{C}\left(\left[0,T\right],L^2\left(\mathrm{\Omega }\right)\right)}=0$, and $\underset{n\to +\infty }{lim}\parallel {\phi }_n-\phi {\parallel }_{\mathcal{C}\left(\left[0,T\right],L^2\left(\mathrm{\Omega }\right)\right)}=0.$.

The above calculation allows

$$<\alpha \left(z\left(T\right)-z_d\right),z_h\left(T\right)>+{\int }_0^T<\beta \left(z\left(t\right)-z_d\right),z_h\left(t\right)>\mathit{dt}=<B^{\ast }z\left(t\right)\phi \left(t\right),h\left(t\right){>}_{L^2\left(0,T\right)},$$

and

$$<\alpha \left(z\left(T\right)-z_d\right),z_h\left(T\right)>+{\int }_0^T<\beta \left(z\left(t\right)-z_d\right),z_h\left(t\right)>\mathit{dt}={\int }_0^T<{\phi }_n\left(t\right),\mathit{Bz}\left(t\right){>}_{L^2\left(\mathrm{\Omega }\right)}\mathit{dt}.$$

Hence, the derivative of $\mathcal{J}$ can be expressed as follows:

$${\mathcal{J}}^{\text{'}}\left(u\right)\cdot h={\int }_0^T\left[<\phi \left(t\right),\mathit{Bz}\left(t\right){>}_{L^2\left(\mathrm{\Omega }\right)}+\mathit{\epsilon u}\left(t\right)\right]h\left(t\right)\mathit{dt}.$$

Then $D_u\mathcal{J}\cdot h=<{\mathcal{J}}^{\text{'}}\left(u\right),h{>}_{L^2\left(0,T\right)}.$

Next, we present the necessary optimality conditions.

Proposition 2.

Let $u^{\ast }$ be an optimal control solution of problem (3). Then $u^{\ast }$, must satisfy the following optimality conditions:

$$<{\mathcal{J}}^{\prime }\left(u^{\ast }\right),v-u^{\ast }{>}_{L^2\left(0,T\right)}\ge 0,\forall v\in U_{\mathit{ad}},$$E11

where ${\mathcal{J}}^{\prime }\left(u^{\ast }\right)$ is the Frechet derivative of $\mathcal{J}$ at $u^{\ast }$, given by (6).

Proof:

Let $u^{\ast }$ be an optimal control, and $v\in U_{\mathit{ad}}$, the convexity of $U_{\mathit{ad}}$ leads to the following conclusion:

$$u^{\ast }+\lambda \left(v-u^{\ast }\right)\in U_{\mathit{ad}},\forall \lambda \in ]0,1[.$$

Thus

$$\mathcal{J}\left(u^{\ast }\right)\le \mathcal{J}\left(u^{\ast }+\lambda \left(v-u^{\ast }\right)\right),$$E12

So

$$\mathcal{J}\left(u^{\ast }\right)\le \mathcal{J}\left(u^{\ast }\right)+\lambda <{\mathcal{J}}^{\prime }\left(u^{\ast }\right),v-u^{\ast }>+\parallel \lambda \left(v-u^{\ast }\right)\parallel \theta \left(\lambda \left(v-u^{\ast }\right)\right),$$

with $\underset{\parallel z\parallel \to 0}{lim}\theta \left(z\right)=0$.

$$<{\mathcal{J}}^{\prime }\left(u^{\ast }\right),v-u^{\ast }>\ge -\parallel v-u^{\ast }\parallel \theta \left(\lambda \left(v-u^{\ast }\right)\right),\forall \lambda \in ]0,1[.$$

Then

$$<{\mathcal{J}}^{\text{'}}\left(u^{\ast }\right),v-u^{\ast }>\ge -\parallel v-u^{\ast }\parallel \underset{\lambda \to 0}{lim}\theta \left(\lambda \left(v-u^{\ast }\right)\right).$$

Since ${lim}_{\lambda \to 0}\theta \left(\lambda \left(v-u^{\ast }\right)\right)=0$ we obtain $<{\mathcal{J}}^{\prime }\left(u^{\ast }\right),v-u^{\ast }>\ge 0.$

Proposition 3.

Let $u^{\ast }$ be an optimal control, solution of problem (3), then

$$\left\{\begin{array}{l}{\mathcal{J}}^{\prime }\left(u^{\ast }\right)=0\\ \text{or}{u}^{\ast }\in \partial U_{\mathit{ad}}\text{and}-\frac{{\mathcal{J}}^{\prime }\left(u^{\ast }\right)}{\parallel {\mathcal{J}}^{\prime }\left(u^{\ast }\right)\parallel }\text{is}\mathrm{a}\text{normal}\text{vector}\text{to}{U}_{\mathit{ad}}\mathit{at}{u}^{\ast }.\end{array}\right.$$E13

Proof:

Let $u^{\ast }$ be an optimal control. We suppose that ${\mathcal{J}}^{\prime }\left(u^{\ast }\right)\ne 0$, then $u^{\ast }\in \partial U_{\mathit{ad}}$. Indeed, if $u^{\ast }\in U_{\mathit{ad}}$, then there exists $r>0$ such that $B\left(u^{\ast },r\right)\subset U_{\mathit{ad}}$, where $B\left(u^{\ast },r\right)$ the open ball with centre $u^{\ast }$ and radius $r$.

Applying the inequality (11) to the elements of $B\left(u^{\ast },r\right)$, we obtain ${\mathcal{J}}^{\prime }\left(u^{\ast }\right)=0$ which is absurd. Then $u^{\ast }\in \partial U_{\mathit{ad}}$.

Using (11), we deduce that $-\frac{{\mathcal{J}}^{\prime }\left(u^{\ast }\right)}{\parallel {\mathcal{J}}^{\prime }\left(u^{\ast }\right)\parallel }$ is the normal vector to $U_{\mathit{ad}}$ at $u^{\ast }$.

The following result gives a sufficient condition ensuring the uniqueness of solution of problem (3).

Proposition 4.

Assume that $U_{\mathit{ad}}$ is bounded. There exists a constant $\gamma \ge 0$, depending on the parameters of system (1), such that

$$\epsilon >\gamma T^{\frac{1}{2}}\left(\alpha +\beta T^{\frac{1}{2}}\right),$$E14

holds, then the optimal control solution of problem (3) is unique.

Proof:

Let $u_1$ and $u_2$ be two optimal controls.

Using (11), we have

$$⟨{\mathcal{J}}^{\prime }\left(u_1\right),u_1-u_2⟩\le 0\mathit{\text{and}}⟨{\mathcal{J}}^{\prime }\left(u_2\right),u_2-u_1⟩\le 0.$$

Then

$$⟨{\mathcal{J}}^{\prime }\left(u_2\right)-{\mathcal{J}}^{\prime }\left(u_1\right),u_1-u_2⟩\ge 0.$$

We know that

$$\begin{array}{lll}{\mathcal{J}}^{\prime }\left(u_2\right)-{\mathcal{J}}^{\prime }\left(u_1\right)& =& ⟨p_{u_2},{\mathit{Bz}}_{u_2}⟩-⟨p_{u_1},{\mathit{Bz}}_{u_1}⟩+\epsilon \left(u_2-u_1\right)\\ & =& ⟨p_{u_2},{\mathit{Bz}}_{u_2}-{\mathit{Bz}}_{u_1}⟩+⟨p_{u_2}-p_{u_1},{\mathit{Bz}}_{u_1}⟩+\epsilon \left(u_2-u_1\right).\end{array}$$

So, the above equality yields

$$\epsilon \parallel u_1-u_2{\parallel }^2\le \parallel u_1-u_2\parallel \left(\parallel B\parallel \parallel p_{u_1}\parallel \parallel z_{u_1}-z_{u_2}\parallel +\parallel p_{u_1}-p_{u_2}\parallel \parallel {\mathit{Bz}}_{u_2}\parallel \right).$$E15

Denote $\mu ={sup}_{u\in U_{\mathit{ad}}}\parallel u\parallel$, and let $M\ge 0$ and $\rho \in \mathrm{ℝ}$ such that $\parallel S\left(t\right)\parallel \le {\mathit{Me}}^{\mathit{\rho t}}$.

By applying Gronwall’s Lemma to $z_{u_2},z_{u_1}-z_{u_2}$, and $p_{u_1}-p_{u_2}$, we deduce

$$\parallel {\mathit{Bz}}_{u_2}\parallel \le K_1\parallel B\parallel T^{\frac{1}{2}}$$E16

$$\parallel z_{u_1}-z_{u_2}\parallel \le K_{2}T\parallel u_1-u_2\parallel$$E17

$$\parallel p_{u_1}\parallel \le K_3\left(\alpha +\beta T^{\frac{1}{2}}\right)$$E18

$$\parallel p_{u_1}-p_{u_2}\parallel \le K_0\left(K_2+\parallel B\parallel K_3+\mu {\mathit{kK}}_2{K}_3\right)\left(\alpha +\beta T^{\frac{1}{2}}\right)\parallel u_1-u_2\parallel ,$$E19

where

$$K_0\le {\mathit{Me}}^{\mid \rho \mid T}{e}^{\parallel B\parallel \mu T^{\frac{1}{2}}{\mathit{Me}}^{\mid \rho \mid T}}$$E20

$$K_1\le K_0\parallel z_0\parallel$$E21

$$K_2\le K_0{K}_1\parallel B\parallel$$E22

$$K_3\le K_0\left(K_1+\parallel z_d\parallel \right).$$E23

Then, by (15), there exists a constant $\gamma >0$ that does not depend on the parameters $\alpha ,\beta$ and $\epsilon$ such that

$$\epsilon \parallel u_1-u_2{\parallel }^2\le \gamma T^{\frac{1}{2}}\left(\alpha +\beta T^{\frac{1}{2}}\right)\parallel u_1-u_2{\parallel }^2.$$

Hence, by choosing $\epsilon >\gamma T^{\frac{1}{2}}\left(\alpha +\beta T^{\frac{1}{2}}\right)$, we obtain $\parallel u_1-u_2{\parallel }^2=0$, then the optimal control is unique.

3.2. Particular controls sets

In this subsection we apply the optimality condition (13), to characterize the optimal control for special cases of constraints.

Proposition 5.

For $U_{\mathit{ad}}=\left\{u\in L^2\left(0,T\right):{\rho }_1\left(t\right)\le u\left(t\right)\le {\rho }_2\left(t\right)\right\}$ with ${\rho }_1,{\rho }_2\in L^2\left(0,T\right)$ such that ${\rho }_1\left(t\right)\le {\rho }_2\left(t\right)$ a.e on $]0,T[$. Then, an optimal control is given by

$$u^{\ast }\left(t\right)=max\left({\rho }_1\left(t\right),\mathit{min}\left({\rho }_2\left(t\right),\frac{-1}{\epsilon }<\phi \left(t\right),\mathit{Bz}\left(t\right)>\right)\right).$$E24

Proof:

• If ${\rho }_1\left(t\right)<u^{\ast }\left(t\right)<{\rho }_2\left(t\right)$ on a set $I\subset ]0,T[$ of positive Lebesgue measure.

For $h\in D\left(I\right)$ the ensemble of functions ${\mathcal{C}}^{\infty }$ over the interval $I$ with compact support, $h\in L^{\infty }\left(I\right)$ small enough and null outside $I$, we have: $u^{\ast }+h,u^{\ast }-h\in U_{\mathit{ad}}$, then $<{\mathcal{J}}^{\text{'}}\left(u^{\ast }\right),h>=0.$.

Since $D\left(I\right)$ dense in in $L^2\left(I\right)$, we have ${\mathcal{J}}^{\text{'}}\left(u^{\ast }\right)=0$ a.e on $I$.

we derive that $u^{\ast }\left(t\right)=-\frac{1}{\epsilon }<\phi \left(t\right),\mathit{Bz}\left(t\right)>.$ This is equivalent to (16) on $I$.

• If $u^{\ast }\left(t\right)={\rho }_2\left(t\right)$ on an open $I\subset ]0,T[$. Let consider $h\in D\left(I\right)$ such that $h\le 0$ in $I$, if $\parallel h{\parallel }_{L^{\infty }\left(I\right)}$ sufficiently small, we have: $u^{\ast }+h,u^{\ast }-h\in U_{\mathit{ad}}$. Since $<{\mathcal{J}}^{\prime }\left(u^{\ast }\right),h>\ge 0$, it follows that ${\mathcal{J}}^{\prime }\left(u^{\ast }\right)\le 0$ on $I$. Consequently $u^{\ast }\left(t\right)={\rho }_2\left(t\right)\le -\frac{1}{\epsilon }<\phi \left(t\right),\mathit{Bz}\left(t\right)>.$ Which is equivalent to (24) on $I$.

• The case $u^{\ast }\left(t\right)={\rho }_1\left(t\right)$ is similar to the previous case.

If $U_{\mathit{ad}}$ is unbounded, we have the following result.

Proposition 6.

If $U_{\mathit{ad}}=L^2\left(0,T\right)$, then the optimal control is given by

$$u^{\ast }\left(t\right)=-\frac{1}{\epsilon }<\phi \left(t\right),\mathit{Bz}\left(t\right)>.$$E25

Proof:

Let $u^{\ast }$ be an optimal control, then $u^{\ast }\in \mathit{Int}\left(U_{\mathit{ad}}\right)=U_{\mathit{ad}}$, so, the condition (13) becomes ${\mathcal{J}}^{\prime }\left(u^{\ast }\right)=0$. We deduce that $u^{\ast }\left(t\right)=-\frac{1}{\epsilon }<\phi \left(t\right),\mathit{Bz}\left(t\right)>$.

4. Algorithm and simulations

Through our observations, it has been established that the optimal control solution for problem (3) corresponds to a solution of eq. (13), which provides formula (24) for bounded control sets and formula (25) for unbounded control sets. To compute this control, you can follow the algorithm outlined below.

4.1. Algorithm

4.2. Simulations

4.2.1. One-dimensional

Let $\mathrm{\Omega }=]0,1[$, we consider the following bilinear heat equation

$$\left\{\begin{array}{ll}\frac{\partial z\left(x,t\right)}{\partial t}=c\frac{{\partial }^{2}z\left(x,t\right)}{\partial x^2}+u\left(t\right)z\left(x,t\right)& x\in ]0,1\left[,t\in \right]0,T[\\ z\left(0,t\right)=z\left(1,t\right)=0& t\in ]0,T[\\ z\left(x,0\right)=z_0& x\in ]0,1[,\end{array}\right.$$E26

where $\mathcal{A}=c\frac{{\partial }^2}{\partial x^2},c\in \mathrm{ℝ}$ with domain $D\left(\mathcal{A}\right)=H^2\left(0,1\right)\cap H_0^1\left(0,1\right)$ and control operator $B=I$. The operator $\mathcal{A}$ generates a $C_0$-semigroup ${\left(S\left(t\right)\right)}_{t\ge 0}$ on $L^2\left(\mathrm{\Omega }\right)$ and is given by

$$S\left(t\right)z=\sum _{n=0}^{\infty }{e}^{-{\left(\mathit{n\pi }\right)}^2\mathit{ct}}⟨{\varphi }_n,z⟩{\varphi }_n,$$

where ${\varphi }_0=1$, ${\varphi }_n\left(x\right)=\sqrt{2}sin\left(\mathit{n\pi x}\right)$, $\forall n\in {\mathrm{\mathbb{N}}}^{\ast }$.

Consider the problem (3) with

$$\mathcal{J}\left(u\right)=\frac{\alpha }{2}\parallel z\left(.,T\right)-z_d{\parallel }_{L^2\left(0,1\right)}^2+\frac{\beta }{2}{\int }_0^T\parallel z\left(.,t\right)-z_d{\parallel }_{L^2\left(0,1\right)}^2\mathit{dt}+\frac{\epsilon }{2}{\int }_0^T{\left|u\left(t\right)\right|}^2\mathit{dt},$$E27

and with $U_{\mathit{ad}}=L^2\left(0,T\right)$.

The differential of the functional cost is given by

$${\mathcal{J}}^{\prime }\left(u\right)\left(t\right)=<\phi \left(t\right),z\left(t\right){>}_{L^2\left(0,1\right)}+\mathit{\epsilon u}\left(t\right),$$E28

where $\phi$ is the solution of

$$\left\{\begin{array}{l}\dot{\phi }\left(t\right)=\left[-c\frac{{\partial }^2}{\partial x^2}-u\left(t\right)z\left(t\right)\right]\phi \left(t\right)-\beta \left(z\left(T\right)-z_d\right)\\ \phi \left(T\right)=\alpha \left(z\left(T\right)-z_d\right).\end{array}\right.$$E29

For simulations we apply the following data.

$z_0\left(x\right)=x\left(1-x^3\right),z_d\left(x\right)=0.2x\left(1-x\right)\left(x^2+x+1\right),c=0.1,T=1,\alpha =1,\beta =1,\epsilon =1$ and $\kappa ={10}^{-4}$.

The optimal control is given by $u^{\ast }\left(t\right)=-\frac{1}{\epsilon }<\phi \left(t\right),z\left(t\right){>}_{L^2\left(0,1\right)}$ and the simulations give the following figures.

Figure 1 shows that the final state is very close to the desired one with error $\parallel z\left(T\right)-z_d{\parallel }_{L^2\left(\mathrm{\Omega }\right)}=1.0412\times {10}^{-4}$ and the evolution of control is given by Figure 2.

Figure 1.

Final state on $]0,1[$.

Figure 2.

Evolution of the control function.

4.2.2. Two-dimensional

On $\mathrm{\Omega }=]0,1\left[\times \right]0,1[$, we consider the following bilinear heat equation

$$\left\{\begin{array}{ll}\frac{\partial z\left(x,t\right)}{\partial t}=c\left(\frac{{\partial }^2}{\partial x_1^2}+\frac{{\partial }^2}{\partial x_2^2}\right)z\left(x,t\right)+u\left(t\right)z\left(x,t\right)& x\in \mathrm{\Omega },t\in ]0,T[\\ z\left(0,t\right)=z\left(1,t\right)=0& t\in ]0,T[\\ z\left(x,0\right)=z_0& x\in \mathrm{\Omega },\end{array}\right.$$E30

where $x=\left(x_1,x_2\right)$, $\mathcal{A}=c\left(\frac{{\partial }^2}{\partial x_1^2}+\frac{{\partial }^2}{\partial x_2^2}\right),c\in \mathrm{ℝ}$ with domain $D\left(\mathcal{A}\right)=H^2\left(\mathrm{\Omega }\right)\cap H_0^1\left(\mathrm{\Omega }\right)$ and control operator $B=I$. The operator $\mathcal{A}$ generates a $C_0$-semigroup ${\left(S\left(t\right)\right)}_{t\ge 0}$ on $L^2\left(\mathrm{\Omega }\right)$ and is given by

$$S\left(t\right)z=\sum _{n=0}^{\infty }{e}^{-{\left(\left(n+m\right)\pi \right)}^2\mathit{ct}}<{\varphi }_{n,m},z>{\varphi }_{n,m},$$

where ${\varphi }_{n,m}\left(x\right)=e_n\left(x_1\right)e_m\left(x_2\right)$, such that $e_0=1$, $e_n\left(x_i\right)=\sqrt{2}cos\left(\mathit{n\pi }{x}_i\right)$, $\forall n\in {\mathrm{\mathbb{N}}}^{\ast }$.

Consider the problem (3) with

$$\mathcal{J}\left(u\right)=\frac{\alpha }{2}\parallel z\left(.,T\right)-z_d{\parallel }_{L^2\left(\mathrm{\Omega }\right)}^2+\frac{\beta }{2}{\int }_0^T\parallel z\left(.,t\right)-z_d{\parallel }_{L^2\left(\mathrm{\Omega }\right)}^2\mathit{dt}+\frac{\epsilon }{2}{\int }_0^T{\left|u\left(t\right)\right|}^2\mathit{dt},$$E31

and $U_{\mathit{ad}}=\left\{u\in L^2\left(0,T\right):-1\le u\left(t\right)\le 1\right\}$.

The differential of the functional cost is given by

$$\begin{array}{l}{\mathcal{J}}^{\prime }\left(u\right)\left(t\right)=<\phi \left(t\right),z\left(t\right){>}_{L^2\left(\mathrm{\Omega }\right)}+\mathit{\epsilon u}\left(t\right),\end{array}$$E32

where $\phi$ is the solution of

$$\left\{\begin{array}{l}\dot{\phi }\left(t\right)=\left[-c\left(\frac{{\partial }^2}{\partial x_1^2}+\frac{{\partial }^2}{\partial x_2^2}\right)z\left(t\right)-u\left(t\right)z\left(t\right)\right]\phi \left(t\right)-\beta \left(z\left(T\right)-z_d\right)\\ \phi \left(T\right)=\alpha \left(z\left(T\right)-z_d\right).\end{array}\right.$$E33

The optimality conditions are

$$\left\{\begin{array}{llll}& {\mathcal{J}}^{\prime }\left(u^{\ast }\right)\left(t\right)=0,& \mathit{\text{if}}& -1\le u^{\ast }\left(t\right)\le 1\\ \mathit{\text{or}}& {\mathcal{J}}^{\prime }\left(u^{\ast }\right)\left(t\right)<0,& \mathit{\text{if}}& u^{\ast }\left(t\right)=1\\ \mathit{\text{or}}& {\mathcal{J}}^{\prime }\left(u^{\ast }\right)\left(t\right)>0,& \mathit{\text{if}}& u^{\ast }\left(t\right)=-1.\end{array}\right.$$

For simulations we take.

$z_0\left(x\right)=x_1{x}_2\left(1-x_1\right)\left(1-x_2\right)$, $z_d\left(x\right)=0$, $c=0,1$, $T=1$, $\alpha =1$, $\beta =1$, $\epsilon =1$, and $\kappa ={10}^{-4}$.

The optimal control is given by $u^{\ast }\left(t\right)=max\left(-1,min\left(1,-\frac{1}{\epsilon }<\phi \left(t\right),z\left(t\right){>}_{L^2\left(\mathrm{\Omega }\right)}\right)\right).$.

Applaying the above algorithm we obtain the following figures (Figure 3).

Figure 3.

Initial state on $\mathrm{\Omega }$.

Figure 3 gives initial state on $\mathrm{\Omega }$ and Figure 4 shows that the final state is very close to the desired one with error equals $\parallel z\left(T\right){\parallel }_{L^2\left(\mathrm{\Omega }\right)}=2.8351\times {10}^{-4}.$ The control function is given by the Figure 5.

Figure 4.

Final state on $\mathrm{\Omega }$.

Figure 5.

Evolution of the control function.

5. Conclusion

In this context, optimal control for a class of bilinear systems is investigated, encompassing both cases of bounded and unbounded controls. The existence of an optimal control is established and further characterized by optimality conditions. The results obtained are demonstrated through examples and validated successfully via simulations. Questions are still open, for instance the case of optimal control of linear systems acted by bilinear boundary controls.