Open access peer-reviewed chapter

Optimal Control of Evolution Differential Inclusions with Polynomial Linear Differential Operators

By Elimhan N. Mahmudov

Submitted: September 6th 2019Reviewed: December 18th 2019Published: February 13th 2020

DOI: 10.5772/intechopen.90888

Downloaded: 61

Abstract

In this chapter, we studied a new class of problems in the theory of optimal control defined by polynomial linear differential operators. As a result, an interesting Mayer problem arises with higher order differential inclusions. Thus, in terms of the Euler-Lagrange and Hamiltonian type inclusions, sufficient optimality conditions are formulated. In addition, the construction of transversality conditions at the endpoints of the considered time interval plays an important role in future studies. To this end, the apparatus of locally adjoint mappings is used, which plays a key role in the main results of this chapter. The presented method is demonstrated by the example of the linear optimal control problem, for which the Weierstrass-Pontryagin maximum principle is derived.

Keywords

  • Euler-Lagrange
  • differential inclusion
  • set-valued mapping
  • polynomial differential operators
  • linear problem
  • transversality
  • Weierstrass-Pontryagin maximum principle

1. Introduction

This chapter concerns with the special kind of optimal control problem with differential inclusions, where the left-hand side of the evolution inclusion is polynomial linear differential operators with variable coefficients; in fact, the main difficulty in the considered problems is to construct the Euler-Lagrange type higher order adjoint inclusions and the transversality conditions. That is why in the whole literature, only the qualitative properties of second-order differential inclusions are investigated (see [1, 2, 3] and references therein).

The paper [1] gives necessary and sufficient conditions ensuring the existence of a solution to the second-order differential inclusion with Cauchy initial value problem. Furthermore, second-order interior tangent sets are introduced and studied to obtain such conditions. The paper [2] studies, in the context of Banach spaces, the problem of three boundary conditions for both second-order differential inclusions and second-order ordinary differential equations. The results are obtained in several new settings of Sobolev type spaces involving Bochner and Pettis integrals. In the paper [3], the existence of viable solutions to the Cauchy problem xFxx,x0=x0,x0=y0is proved, where Fis a set-valued map defined on a locally compact set MR2n, contained in the Frechet subdifferential of a φ-convex function of order two.

Some qualitative properties and optimization of first-order discrete and continuous time processes with lumped and distributed parameters have been expanding in all directions at an astonishing rate during the last few decades (see [4, 5, 6, 7, 8, 9, 10, 11, 12, 13] and their references).

The optimization of higher order differential inclusions was first developed by Mahmudov in [14, 15, 16, 17, 18, 19, 20, 21]. Since then this problem has attracted many author’s attentions (see [22] and their references). The paper [14] studies a new class of problems of optimal control theory with Sturm-Liouville type differential inclusions involving second-order linear self-adjoint differential operators. By using the discretization method guaranteeing transition to continuous problem, the discrete and discrete-approximate inclusions are investigated. Necessary and sufficient conditions, containing both the Euler-Lagrange and Hamiltonian type inclusions, and “transversality” conditions are derived. The paper [15] deals with the optimization of the Bolza problem with third-order differential inclusions and arbitrary higher order discrete inclusions. The work [16] is devoted to the Bolza problem of optimal control theory given by second-order convex differential inclusions with second-order state variable inequality constraints. According to the proposed discretization method, problems with discrete-approximate inclusions and inequalities are investigated. Necessary and sufficient conditions of optimality including distinctive “transversality” condition are proved in the form of Euler-Lagrange inclusions. The paper [17] is concerned with the necessary and sufficient conditions of optimality for second-order polyhedral optimization described by polyhedral discrete and differential inclusions. The paper [18] is devoted to the study of optimal control theory with higher order differential inclusions and a varying time interval. Essentially, under a more general setting of problems and endpoint constraints, the main goal is to establish sufficient conditions of optimality for higher order differential inclusions. Thus with the use of Euler-Lagrange and Hamiltonian type of inclusions and transversal conditions on the “initial” sets, the sufficient conditions are formulated. The paper [21] studies a new class of problems of optimal control theory with state constraints and second-order delay discrete and delay differential inclusions. Under the “regularity” condition by using discrete approximations as a vehicle, in the forms of Euler-Lagrange and Hamiltonian type inclusions, the sufficient conditions of optimality for delay DFIs, including the peculiar transversality ones, are proved.

The present chapter is ordered in the following manner.

In Section 2 the necessary facts and supplementary results from the book of Mahmudov are given [23]; Hamiltonian function and locally adjoint mapping are introduced, and the problems with initial point constraints for polynomial linear differential operators governed by time-dependent set-valued mapping are formulated. In Section 3, we present the main results; on the basis of “transversality” conditions at the endpoints of the considered time interval, the sufficient conditions of optimality for differential inclusions with polynomial linear differential operators and with initial point constraints are proved. In particular, it is shown that our problems involve optimization of the so-called Sturm-Liouville type differential inclusions. To the best of our knowledge, there is no paper which considers optimality conditions for these problems in the literature, and we aim to fill this gap. Therefore, the novelty of our formulation of the problem is justified. To establish the Euler-Lagrange and Hamiltonian inclusions and the transversality conditions, we use the construction of a suitable rewriting of the primal polynomial linear differential operator and the rearrangement of its integration. The case of variable coefficients of polynomial linear differential operators turns out to be more complicated, unless transversality assumptions at the endpoints of the considered time interval are applied. It should be noted that the main proof can be easily generalized to the nonconvex case. Then, using the new approach given in Section 4 of this chapter, we construct the Weierstrass-Pontryagin maximum condition [24] for the linear optimal control problem. Consequently, in the particular case, the maximum principle follows from the Euler-Lagrange inclusion.

In Section 5 the optimality conditions are given for convex problem with second-order differential inclusions and endpoint constraints. By using second-order suitable Euler-Lagrange type adjoint inclusions and transversality conditions, Theorem 5.1 is proved.

The main results in this section can be extended to the case of Hilbert spaces 2,L2n. We remind that a Hilbert space His a real or complex inner product space that is also a complete metric space with respect to the distance function induced by the inner product [2]. By definition, every Hilbert space is also a Banach space. Furthermore, in every Hilbert space, the followingparallelogram identityx+y2+xy2=2x2+y2holds. Conversely, every Banach space in which the parallelogram identity holds is a Hilbert space. Remember that 2is a space of numerical sequences, such that if x=xi, then i=1xi2<. In fact 2is an infinity dimensional coordinate-wise Hilbert space with the corresponding inner product xy=i=1xiyi. Endowing a relevant norm, we have a Banach space. Obviously, optimization of problem with PLDOs can be reduced to problem with geometric constraints in such finite-dimensional Hilbert space. As is known with all the pairs of elements of this space, a certain finite number is associated, i.e., inner product, existence of which is guaranteed by applying the familiar Cauchy Schwarz-Bunyakovskii [25] inequality. We remark that in our case for x=x0x1x22and x=x0x1x22, the inner product xx=i=1xixiis finite numbers since this series is convergent. Besides it is known [25] that 2is a self-adjoint space, i.e., p=q,and 1/p+1/q=1, and so 2=2for p=2. Thus a dual cone constructed can be defined. The set of square integrable functions L2n01is a Hilbert space with inner product xtyt=01xtytdt.

2. Preliminaries and problem statements

The basic concepts given in this section can be found in the book [23]; let Rnbe a n-dimensional Euclidean space, xvbe an inner product of elements x,vRn,and xvbe a pair of x,v. Let F:RnRnbe a set-valued mapping from Rninto the set of subsets of Rn. Therefore Fis a convex set-valued mapping, if its graph gphF=xv:vFxis a convex subset of R2n. A set-valued mapping Fis called closed if its gphFis a closed subset in R2n. The domain of a set-valued mapping Fis denoted by domFand is defined as domF=x:Fx. A set-valued mapping Fis convex-valued if Fxis a convex set for each xdomF.

The Hamiltonian function and argmaximum set corresponding to a set-valued mapping Fare defined by the relations correspondingly:

HFxv=supvvv:vFx,vRn,FAxv=vFx:vv=HF(xv)

We set HFxv=if Fx=.The interior and relative interior of a set MR2nare denoted by intMand riM, respectively.

A convex cone KMz0, z0Mis a cone of tangent directions if from z¯=x¯v¯KMz0it follows that z¯is a tangent vector to the set Mat a point z0M, i.e., there exists such function q:R1R2nthat z0+αz¯+qαMfor sufficiently small α>0and α1qα0, as α0.

For a set-valued mapping F, the set-valued mapping F:RnRnis defined by

Fvxvx:xvKgphF(xv)
KgphFxv=conegphFxv,x1v1gphF.

It is called the LAM to Fat a point xvgphF, where K=z:z¯z0z¯Kdenotes the dual cone to the cone K, as usual. Below by using the Hamiltonian function, associated to a set-valued mapping F, we will define another LAM. Thus, the LAM to “nonconvex” mapping Fis defined as follows:

Fvxvx:HFx1vHF(xv)xx1x,x1Rn,xvgphF,vFAxv.

Clearly, for the convex mapping F, the Hamiltonian function HFvis concave, and the latter definition of LAM coincides with the previous definition of LAM ([23], p. 62). Note that prior to the LAM, the notion of coderivative has been introduced for set-valued mappings in terms of the basic normal cone to their graphs by Mordukhovich [26] and for the smooth convex maps, the two notions are equivalent.

The aim of Section 3 is to obtain the Euler-Lagrange type adjoint inclusion and sufficient optimality conditions for a problem with polynomial linear differential operators:

Minimizeφx1x1xs11,E1
(PV)LxtFxtt,a.e.t01,E2
x0Q0,x0Q1x0Q2,,xs10Qs1E3

where Lx=k=1spktDkxis a PLDO of degree swith variable coefficients pk:01R1and Dk,k=1,,sis the operator of kth-order derivatives. In what follows for each k, a scalar function pkis kth-order continuously differentiable function, pst0on 01identically, Ft:RnRnis time-dependent set-valued mapping, φ:RnsR1is continuous function, QjRn,j=0,1,,s1are nonempty subsets of Rn, and ss2is an arbitrary fixed natural number. It is required to find an arc x˜tof the problem Eqs. (1)(3) for the sth-order differential inclusions satisfying Eq. (2) almost everywhere (a.e.) on a considered time interval and minimizing the functional φx1xs11. An arc xis absolutely continuous and s1order differentiable function, where xsdsxdtsL1n01. Obviously, such class of functions is a Banach space, endowed with the different equivalent norms.

Remark 2.1. Notice that to get sufficient condition of optimality for the Mayer problem (PV) described by ordinary evolution differential inclusions with PLDOs and with initial point constraints, using the discretized method, we consider the following sth-order discrete-approximate problem instead of the problem (PV):

minimizeφx1s1hΔx1s1hΔs1x1s1h,
k=1spktΔkxtFxtt,t=0,h,2h,,1sh;
Δkx0Qk,k=0,,s1.

Here kth-order difference operator is defined as follows:

Δkxt=1hks=0k1sCksxt+ksh,Cks=k!s!ks!, t=0,h,,1h.

Thus by using the method of approximation [23, 26, 27], we can establish necessary and sufficient conditions for the rather complicated sth-order discrete-approximate problem. Then by passing to the limit in necessary and sufficient conditions of this problem as h0, we can construct the optimality condition for the Mayer problem (PV) described by higher order differential inclusions with PLDOs and with initial point constraints. But in this chapter to avoid long calculations, derivations of these conditions are omitted.

3. Optimization of evolution differential inclusions with PLDOs

In the present section, we study sufficient optimality conditions for the problem (PV). Before all, we formulate the so-called sth-order Euler-Lagrange type differential inclusion and the transversality conditions:

  1. LxtFxtx˜tLx˜tt,a.e. t01,

    where Lxt=k=1s1kDkpktxtis the adjoint PLDO of the primal operator L.

  2. k=0s11skDsk1psk0x0KQ0x˜0;

    k=0s21sk1Dsk2psk0x0KQ1x˜0;

    Dps0x0ps10x0KQs2x˜s20;

    ps0x0KQs1x˜s10.

  3. k=0s11skDsk1psk1x1, k=0s21sk1Dsk2psk1x1,, Dps1x1ps11x1ps1x1φx˜1x˜1x˜s11.

    Later on we assume that xt,t01is absolutely a continuous function with the higher order derivatives until s1and xsL1n01. The following condition ensures that the LAM Fis nonempty:

  4. Lx˜tFAx˜txtt, a.e. t01or, equivalently,

    Lx˜txt=HFx˜txt, Lx˜tFx˜tt.

The following are sufficient optimality conditions for evolution differential inclusions with PLDOs.

Theorem 3.1. Let φbe a continuous and convex function and Fta convex set-valued mapping. Moreover, let

Qj,j=0,,s1

be convex sets. Then for optimality of the trajectory x˜in the problem (PV) with evolution differential inclusions and PLDOs, it is sufficient that there exists an absolutely continuous function xwith the higher order derivatives until s1,satisfying a.e. the Euler-Lagrange type differential inclusion with PLDOs Eqs. (i) and (iv) and transversality conditions Eqs. (ii) and (iii) at the endpoints t=0and t=1.

Proof. Using Theorem 2.1 ([23], p. 62), the definition of the Hamiltonian function and condition Eq. (i), we obtain

HFxtxttHFx˜txttLxtxtx˜t

which can be rewritten as follows

HFxtxttHFx˜txtt
k=1s1kpktxtkxtx˜t.E4

Further using the definition of the Hamiltonian function, Eq. (4) can be converted to the inequality

0LxtLx˜txtLxtxtx˜t

or

0k=1spktxtx˜tkxtk=1s1kpktxtkxtx˜t.E5

Integrating Eq. (5) over the interval 01, we have

001k=1spktxtx˜tkxt
k=1s1kpktxtkxtx˜tdtE6

Let us denote

B=k=1sxtx˜tkpktxtk=1s1kpktxtkxtx˜t

In what follows our approach lies in reducing Bin a relationship consisting of ssums from kk=1sto sof suitable derivatives of scalar products; thus, after some transformations we can deduce an important representation for a first term of Bas follows:

k=1sxtx˜tkpktxt=k=1sddtxk1tx˜k1tpktxt
k=2sddtxk2tx˜k2tpktxt
+k=3sddtxk3tx˜k3tpktxt
+k=s2sddtxks+2tx˜ks+2t1m3pktxts3E7
+k=s1sddtxks+1tx˜ks+1t1s2pktxts2
+ddtxtx˜t1m1pstxts1
+k=1sxtx˜t1kpktxtk.

Then in view of Eq. (7) in the definition of B, we have an efficient formula:

B=k=1sddtxk1tx˜k1tpktxt
k=2sddtxk2tx˜k2tpktxtE8
+k=3sddtxk3tx˜k3tpktxt
+k=s2sddtxks+2tx˜ks+2t1s3pktxts3
+k=s1sddtxks+1tx˜ks+1t1s2pktxts2
+ddtxtx˜t1s1pstxts1.

Then taking into account the structure of Bin Eq. (8), we can compute the integral on the right-hand side of Eq. (6) as follows:

01Bdt=k=1s01dxk1tx˜k1tpktxt
k=2s01dxk2tx˜k2tpktxt
+k=3s01dxk3tx˜k3tpktxt
+k=s2s01dxks+2tx˜ks+2t1s3pktxts3
+k=s1s01dxks+1tx˜ks+1t1s2pktxts2
+01dxtx˜t1s1pstxts1.

Thus, integrating B, we can obtain

01Bdt=k=1sxk11x˜k11pk1x1
xk10x˜k10pk0x0
k=2sxk21x˜k21pk1x1xk20x˜k20pk0x0
+k=3sxk31x˜k31pk1x1
xk30x˜k30pk0x0
+k=s2sxks+21x˜ks+211s3pk1x1s3
xks+20x˜ks+201s3pk0x0s3
+k=s1sxks+11x˜ks+111s2pk1x1s2
xks+10x˜ks+101s2pk0x0s2
+x1x˜11s1ps1x1s1
x0x˜01s1ps0x0s1.

Here by suitable rearrangement and necessary simplification, we have

01Bdt=k=1sxk11x˜k11pk1x1
k=2sxk21x˜k21pk1x1
+k=3sxk31x˜k31pk1x1
+k=s2sxks+21x˜ks+211s3pk1x1s3
+k=s1sxks+11x˜ks+111s2pk1x1s2E9
+x1x˜11s1ps1x1s1+x0x˜0k=0s11skDsk1psk0x0
+x0x˜0k=0s21sk1Dsk2psk0x0
+x0x˜0k=0s31sk2Dsk3psk0x0
+xs20x˜s20Dps0x0+ps10x0xs10x˜s10ps0x0.

In order to make use of the transversality condition Eq. (ii), we rewrite it in a more relevant form:

x0x˜0k=0s11skDsk1psk0x0

+x0x˜0k=0s21sk1Dsk2psk0x0+x0x˜0k=0s31sk2Dsk3psk0x0

+xs20x˜s20Dps0x0+ps10x0xs10x˜s10ps0x00;xk0KQkx˜k0,

k=0,,s1.Thus, from Eqs. (6) and (9), we have

0k=1sxk11x˜k11pk1x1
k=2sxk21x˜k21dpk1x1dt
+k=3sxk31x˜k31d2pk1x1dt2+k=s2sxks+21x˜ks+211s3ds3pk1x1dts3
+k=s1sxks+11x˜ks+111s2ds2pk1x1dts2
+x1x˜11s1ds1ps1x1dts1.

Using the derivative operator D, it is not hard to see that the relation described above can be expressed in a more compact form:

0k=0s11sk1Dsk1psk1x1x1x˜1
+k=0s21sk2Dsk2psk1x1x1x˜1+E10
Dps1x1ps11x1xs21x˜s21
+ps1x1xs11x˜s11.

Furthermore, applying the definition of the transversality condition Eq. (iii) for all feasible arc x, we have

φx1x1xs11φx˜1x˜1x˜s11
k=0s11skDsk1psk1x1x1x˜1
+k=0s21sk1Dsk2psk1x1x1x˜1+
+Dps1x1ps11x1xs21x˜s21
ps1x1xs11x˜s11E11

Then from the last two inequalities Eqs. (10) and (11) for all feasible arc x, we have immediately φx1x1xs11φx˜1x˜1x˜s11, that is, x˜is an optimal trajectory. □

Remark 3.1. It can be noted that in the particular case, if p2t=pt,p1t=pt,where p:010, the second-order linear differential operator Lx=p2tx+p1txis a well-known self-adjoint Sturm-Liouville operator Lxpx.

Corollary 3.1. Let Ftbe a closed set-valued mapping. Then under the assumptions of Theorem 3.1, the conditions Eqs. (i) and (iii) can be rewritten in terms of Hamiltonian function as follows:

LxtxHFx˜txtt;.Lx˜tvHx˜txtt,a.e.t01

Proof. Indeed, by Theorem 2.1 ([23], p. 62) and Lemma 5.1 [14], we can write.

Fvxvt=xHFxvt,andFAxvt=vHFxvt

respectively. Then it is easy to see that the result of corollary are equivalent with the conditions Eqs. (i) and (iv) of Theorem 3.1. □

Below nonconvexity of a set-valued mapping Ftmeans that its Hamilton function in general is a nonconcave function satisfying the condition Eq. (a).

Theorem 3.2. Suppose that we have the “nonconvex” problem (PV), that is, φ:RnsR1and Ft:RnRnin general are nonconvex function and set-valued mapping, respectively. Moreover, suppose that KQjx˜j0,x˜j0Qjis the cones of tangent directions to Qj,j=0,,s1.

Then for optimality of the trajectory x˜, it is sufficient that there exists an absolutely continuous function x, satisfying the following conditions:

  1. LxtFxtx˜tLx˜tt, a.e. t01,

  2. k=0s1j1skjDsk1jpsk0x0KQjx˜j0,j=0,,s1,

  3. φν0v1vs1φx˜1x˜1x˜s11

    j=0s1k=0s1j1skjDsk1jpsk1x1νjx˜j1,

    vjRn,j=0,,s1,

  4. Lx˜txt=HFx˜txtt, a.e. t01.

Proof. In the proof of Theorem 3.1, we have used the following inequality:

HFxtxttHFx˜txtt
k=1s1kpktxtkxtx˜t.E12

Hence, from the inequality Eq. (12), immediately we have the inequality Eq. (10). Moreover, setting νj=x˜j1j=0s1for all feasible trajectories x, it is not hard to see that for nonconvex φthe following inequality holds:

φx1x1xs11φx˜1x˜1x˜s11j=0s1k=0s1j1skjDsk1jpsk1x1xj1x˜j1,j=0,1,,s1,

Then for the furthest proof, we proceed by analogy with the preceding derivation of Theorem 3.1. □

4. Some applications to optimal control problems with PLDOs

In this section we give two applications of our results. The first one is the particular Mayer problem for differential inclusions involving PLDOs with constant coefficients, and the second one concerns optimization of “linear” differential inclusions with PLDOs and constant coefficients. Thus, suppose now we have the following optimization problem (for simplicity we consider a convex problem) with sth-order PLDO with constant coefficients:

Minimizeφ0x1,
PCLxtFxtt,a.e.t01,Lx=Dsx+p1Ds1x++ps1Dx
x0=α0,x0=α1,x0=α2,,xs10=αs1,.E13

where Lis the sth-order polynomial operator, pk,k=1,,s1are some real constants, Ft:RnRnis a convex set-valued mapping, φ0:RnR1is a continuous convex function, and αjRn,j=0,,s1are fixed n-dimensional vectors. It is known that the multiplication operation is commutative for polynomial linear differential operators with constant coefficients. On the other hand, the sth-order adjoint operator is defined as follows:

Lx=1sDsx+1s1p1Ds1x+ps1Dx.

Corollary 4.1. Let φ0and Ftbe a convex function and a set-valued mapping, respectively. Then, for the trajectory x˜to be optimal in the problem (PC), it is sufficient that there exists an absolutely continuous function xsatisfying the Euler-Lagrange type differential inclusion.

LxtFxt(x˜tLx˜t)t;Lx˜txt=HFx˜txt,
a.e.t01,Lx=1sxs+1s1p1xs1+ps1x

and transversality condition at the endpoint t=1.

1sxs11φ0x˜1,xj1=0,j=0,,m2.

Proof . We conclude this proof by returning to the conditions Eqs. (i)–(iii) of Theorem 3.1. Clearly, a problem (PC) can be reduced to the problem of form (PV), where

φx1x1xs11φ0x1.

It follows that φx1x1xs11=xφ0x1×(0,,0s1). On the other hand, since pst1,pjt=psj, and j=1,,s1are constants, by sequentially substitution in the transversality condition Eq. (iii), we derive that

k=0s1j1skjDsk1jpsk1x1=xs1j1=0,j=1,,s1,

and therefore for j=0.

k=0s11skDsk1psk1x1=1sxs11φ0x˜1.□

Suppose now that we have the so-called linear Mayer problem with PLDOs:

Minimizeφ0x1,E14
LxtFxtt,a.e.t01,
xj0=αj,j=0,,s1,Fxt=Atx+BtUE15

where φ0differentiable convex function; Atand Btare n×nand n×rcontinuous matrices, respectively; U is a convex compact of Rr; αj, j=0,,s1are constant vectors. It is required to find a control function u˜such that the corresponding trajectory x˜minimizes the Mayer functional φ0x1.

In fact, this is optimization of Cauchy problem for “linear” differential inclusions with PLDO. The controlling parameter uis called admissible if it only takes values in the given control set Uwhich is a nonempty, convex compact.

Theorem 4.1. The arc x˜tcorresponding to the controlling parameter u˜tis a solution to Eqs. (14) and (15) if there exists an absolutely continuous function x, satisfying the Euler-Lagrange type differential equation, the transversality condition, and Weierstrass-Pontryagin maximum principle:

Lxt=Atxt,a.e.t01,
1sxs11=φ0x˜1,.xj1=0,j=0,,s2
Btu˜txt=maxuUBtuxt.

Proof. Obviously, the Hamiltonian is

HFxvt=Atxv+maxuUBtuv.

Hence,

Fvxv˜t=xHFxvt=Atv,v˜FAxvtv˜=Atx+Btu˜

where the argmaximum inclusion v˜FAxvtimplies that Btu˜v=maxuUBtuvand Fvxv˜t. Thus, applying Theorem 3.1, we obtain

Lxt=Atxt,Lx˜tFAx˜txtt,
Btu˜txt=maxuUBtuxt.

Consequently, the transversality condition Eq. (ii) of Theorem 3.1 is unnecessary and by Corollary 4.1 1sDs1x1=φ0x˜1, Djx1=0,j=0,,s2. □

Remark 4.1. Suppose that in the definition of sth-order PLDO (see Eq. (13)) s=1, pi=0,i=1,..,s1and Uis a convex closed polyhedron. Then we have linear equations with variable coefficients x=Atx+Btu,uUin the finite time interval t01. Obviously, for such problems an adjoint Euler-Lagrange type differential equation and transversality condition at a point t=1consist of the following: xt=Atxt,x1=φ0x˜1. We remind that along with Pontryagin’s maximum principle (see, e.g., [24]) under the condition for generality of position for time-optimal problem, the existence results of optimal control are proved.

Example 4.1. Let us consider the following Mayer problem with second-order PLDO Lx=D2x=x:

Infimumφx1x1is subject tox=u,u11,x0Q0,x0Q1.E16

Here φx1x1=x21x1and Q0=0,Q1=1.

It should be noted that substituting Ft=ut,xt=at,m=1into Newton’s second law Ft=mat, we have x=u.

Obviously, in this problem FxtFx=u:u1, s=2.

Then Eq. (16) has the form:

Infimumφx1x1is subject toLxFx,t01,x0=0,x0=1.E17

It can be easily seen that in the adjoint inclusion Eq. (i).

Dp1txt+D2p2txtFxtx˜tLx˜t

of Corollary 3.1 p2t0and p2t1, and so we have

xtFxtx˜tx˜t.

Now, it is not hard to see that

HFxv=maxuuv:u1=vE18

and

Fvxv=xHFxv0,vFAxv=1+1.E19

Then taking into account Lx=d2x/dt2, as a result of Theorem 3.1 (see also Corollary 3.1) from Eq. (19), we deduce that

x=0,t01,

for which the solution is a linear function of the form xt=C1t+C2, where C1,C2are arbitrary constants. Then Eq. (18) implies that u˜txt=xtor

u˜t=sgnxt,ifxt0,u011,ifxt=0.E20

Further, from the linearity of xand from Eq. (20), we insure that each optimal control function is a piecewise constant function.

In addition, by the transversality condition Eq. (iii) of Corollary 3.1, we can write.

x1x1φx˜1x˜1. On the other hand, it is not hard to see that φxy=y2xis a convex function; in fact, the 2×2Hessian matrix

φxy=φxxxyφxyxyφyxxyφyyxy=0002

is a positive semidefinite, that is, all eigenvalues of φxyare nonnegative. Indeed, denoting this matrix by A, we see that the characteristic equation AλE=λ22λ=0(Eis a 2×2unique square matrix) has two real nonnegative eigenvalues λ1=0,λ2=2. Consequently, φxyis convex and φxy=12y. It follows that φx˜1x˜1=12x˜1. Comparing this relation with x1x1φx˜1x˜1, we immediately have x1=1,x1=2x˜1.Then from a general solution of the adjoint Euler-Lagrange type inclusion (equation) xt=C1t+C2, we have 2x˜1=x1=C1+C2,1=x1=C1(C1,C2are arbitrary constants), and so xt=1t2x˜1, whence xt0, if tτ=12x˜1. Therefore, Eq. (20) implies that for optimal control u˜, there are four possibilities:

u˜t=1,xt>0,t01.E21
u˜t=1,xt<0,t01.E22
u˜t=1,if0t<τ,1,ifτ<t1.E23
u˜t=1,if0t<τ,1,ifτ<t1.E24

(observe that τis a point of discontinuity of u˜and the values of the control functions u˜at a point of discontinuity τare unessential). As a consequence, it follows that either the sign of the linear function xtdoes not change for the whole interval 01or xt>0,0t<τ;xt<0,τ<t1for a some τin the interval 0<τ<1(the case Eq. (24) is excluded). Therefore, since u˜tis a piecewise constant function, having not more than two intervals of constancy, we have either the cases Eqs. (21) and (22) or the case Eq. (23). In general, using Eqs. (21)(23), by solving the Cauchy problem

xt=ut,x0=0,x0=1E25

we have a unique solution of the initial value problem Eq. (25). Thus for the time interval on which u=1, we have xt=t+c1;xt=t2/2+c1t+c2(c1,c2are constants). From Eq. (25) we obtain.

xt=t+1;xt=t2/2+t.E26

By analogy, for u=1we have.

xt=1t;xt=t2/2+t.E27

Now, let x1tand x2tbe parabolas of Eqs. (26) and (27), respectively. Here, in the case Eq. (21), u˜t=1,t01, and so from Eq. (26), we have x˜11=0.5+1=1.5; x˜11=2. Consequently, the value of problem Eq. (16) is φx˜11x˜11=x˜121x˜11=221.5=2.5, if u˜t=1,t01. By a similar way, for a control function u˜t=1,t01from Eq. (27), we obtain that x˜21=12/2+1=0.5,x˜21=0and φx˜21x˜21=020.5=0.5.

On the other hand, in the case Eq. (23), the control function u˜tfirst is equal to +1and then equal to 1, and the trajectory x˜tconsists of two pieces of parabolas x˜1tand x˜2t(x˜tis continuous and piecewise smooth on the interval 0t1). Then the solution of the equation Eq. (25) on the interval 0tτis given by Eq. (26); at a point τare satisfied x1τ=τ2/2+τ,x1τ=1+τ. Consider now the initial value problem:

x2t=1,x2τ=τ2/2+τ,x2τ=1+τ,tτ1.E28

It is clear that τ2/2+τ=x2τ=τ2/2+c1τ+c2and 1+τ=x2τ=τ+c1from which we obtain that the solution of the initial value problem Eq. (28) is x˜2t=t2/2+1+2τtτ2.Substituting the value τ=12x˜1into equation x˜2t=1t+2τ, we have 5x˜21=2,x˜21=x˜1=0.4(it follows that τ=0.2). Moreover, x˜21=2ττ2+0.5and x˜21=x˜1=3/24x˜21=0.86.Thus, the value of our Mayer problem is φx˜1x˜1=x˜21x˜1=2/5243/50=0.7,where u˜tis defined as in Eq. (23). Comparing the values 2.5,0.5,0.7, we believe that the value of Mayer problem is 0.7.

5. Sufficient conditions of optimality for second-order evolution differential inclusions with endpoint constraints

Note that in this section the optimality conditions are given for second-order convex differential inclusions (PM) with convex endpoint constraints. These conditions are more precise than any previously published ones since they involve useful forms of the Weierstrass-Pontryagin condition and second-order Euler-Lagrange type adjoint inclusions. In the reviewed results, this effort culminates in Theorem 5.1:

Minimizegx1x1,
PMxtFxtxtt,a.e.t01,
x0=x0,x0=x1;x1M0,x1M1,

where gis a convex continuous function, Ft:R2nRnis convex set-valued mapping, and M0,M1Rnare convex sets.

The following adjoint inclusion is the second-order Euler-Lagrange type inclusion for the problem (PM):

a1. xt+vtvtFxtx˜tx˜tx˜tt,a.e.t01,

where

b1. x˜tFAx˜tx˜txtt, a.e. t01.

In what follows we assume that xt, t01is absolutely continuous function together with the first-order derivatives for which xL1n01. Besides the auxiliary function vt,t01is absolutely continuous and vL1n01.

The transversality conditions at the endpoint t=1consist of the following:

c1. v1+x1x1xugx˜1x˜1KM0x˜1×KM1x˜1.

Now we are ready to formulate the following theorem of optimality.

Theorem 5.1. Suppose that gis a continuous and convex function, Ftis a convex set-valued mapping, and M0,M1are convex sets. Then for optimality of the feasible trajectory x˜tin the problem (PM), it is sufficient that there exists a pair of absolutely continuous functions:

xtvt,t01

satisfying a.e. the second-order Euler-Lagrange type inclusions Eqs. (a1) and (b1) and the transversality condition Eq. (c1) at the endpoint t=1.

Proof. By the proof idea of Theorem 3.1 from Eqs. (a1) and (b1), we obtain the adjoint differential inclusion of second order:

xt+vtvtxuHFx˜tx˜txtt,t01.

On the definition of subdifferential set of the Hamiltonian function HFtfor all feasible trajectory xt,t01, we rewrite the last relation in the equivalent form:

HFxtxtxttHFx˜tx˜txtt
xt+vtxtx˜t+vtxtx˜t.E29

Now by using definition of the Hamiltonian function, the inequality Eq. (29) can be reduced to the inequality

0xtx˜txtxtxtx˜tddtvtxtx˜t.E30

Integrating of the inequality Eq. (30) over the interval 01, we derive that

001xtx˜txtxtxtx˜tdt
+v0x0x˜0v1x1x˜1.E31

For convenience we transform the expression in the square parentheses on the right-hand side of Eq. (31) as follows

xtx˜txtxtxtx˜t
=ddtxtx˜txtddtxtxtx˜t.

Thus by elementary property of the definite integrals, we can compute the integral on the right-hand side of Eq. (31):

01xtx˜txtxtxtx˜tdt
=x1x˜1x1x0x˜0x0x1x1x˜1+x0x0x˜0.E32

Then substituting Eq. (32) into Eq. (31), we have

0x1x˜1x1x0x˜0x0v1+x1x1x˜1+v0+x0x0x˜0.E33

Now, remember that x,x˜are feasible trajectories and x0=x˜0=x0and x0=x˜0.

=x1. Then it follows from Eq. (33) that

0x1x˜1x1v1+x1x1x˜1.E34

Now, thanking to the transversality conditions Eq. (c1) at the endpoint t=1, we can rewrite

gx1x1gx˜1x˜1v1+x1+x1x1x˜1+x1x1x1x˜1,x1KM0x˜1,x1KM1x˜1

or, in other words

gx1x1gx˜1x˜1
v1+x1x1x˜1x1x1x˜1.E35

Thus, summing the inequalities Eqs. (34) and (35) for all feasible trajectories x,satisfying the initial conditions x0=x0and x0=x1and endpoint constraints x1M0,x1M1, we have the needed inequality:

gx1x1gx˜1x˜10or gx1x1gx˜1x˜1. □

6. Conclusion

According to proposed method, the problem with the differential inclusions described by polynomial linear differential operators is investigated. Obviously, this problem is an important generalization of problems with first-order differential inclusions. Thus, sufficient conditions of optimality for such problems are deduced. Here the existence of nonfunctional initial point or endpoint constraints generates different kinds of transversality conditions. Besides, there can be no doubt that investigations of optimality conditions of problems with second- and fourth-order Sturm-Liouville type differential inclusions can play an important role in the development of modern optimization and there is every reason to believe that this role will be even more significant in the future. Thus, the suggested problem with linear differential operators and variable coefficients can be used in various forms in applied problems.

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Elimhan N. Mahmudov (February 13th 2020). Optimal Control of Evolution Differential Inclusions with Polynomial Linear Differential Operators, Functional Calculus, Kamal Shah and Baver Okutmuştur, IntechOpen, DOI: 10.5772/intechopen.90888. Available from:

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