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.
- differential inclusion
- set-valued mapping
- polynomial differential operators
- linear problem
- Weierstrass-Pontryagin maximum principle
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  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  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 , the existence of viable solutions to the Cauchy problem is proved, where is a set-valued map defined on a locally compact set , contained in the Frechet subdifferential of a -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  and their references). The paper  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  deals with the optimization of the Bolza problem with third-order differential inclusions and arbitrary higher order discrete inclusions. The work  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  is concerned with the necessary and sufficient conditions of optimality for second-order polyhedral optimization described by polyhedral discrete and differential inclusions. The paper  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  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 ; 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  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 . We remind that a Hilbert space is 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 . By definition, every Hilbert space is also a Banach space. Furthermore, in every Hilbert space, the following
2. Preliminaries and problem statements
The basic concepts given in this section can be found in the book ; let be a -dimensional Euclidean space, be an inner product of elements and be a pair of . Let be a set-valued mapping from into the set of subsets of . Therefore is a convex set-valued mapping, if its graph is a convex subset of . A set-valued mapping is called closed if its is a closed subset in . The domain of a set-valued mapping is denoted by and is defined as . A set-valued mapping is convex-valued if is a convex set for each .
The Hamiltonian function and argmaximum set corresponding to a set-valued mapping are defined by the relations correspondingly:
We set if The interior and relative interior of a set are denoted by intand ri, respectively.
A convex cone , is a cone of tangent directions if from it follows that is a tangent vector to the set at a point , i.e., there exists such function that for sufficiently small and , as .
For a set-valued mapping , the set-valued mapping is defined by
It is called the LAM to at a point , where denotes the dual cone to the cone , as usual. Below by using the Hamiltonian function, associated to a set-valued mapping , we will define another LAM. Thus, the LAM to “nonconvex” mapping is defined as follows:
Clearly, for the convex mapping , the Hamiltonian function is concave, and the latter definition of LAM coincides with the previous definition of LAM (, 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  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:
where is a PLDO of degree with variable coefficients and is the operator of th-order derivatives. In what follows for each , a scalar function is th-order continuously differentiable function, on identically, is time-dependent set-valued mapping, is continuous function, are nonempty subsets of , and is an arbitrary fixed natural number. It is required to find an arc of the problem Eqs. (1)–(3) for the th-order differential inclusions satisfying Eq. (2) almost everywhere (a.e.) on a considered time interval and minimizing the functional . An arc is absolutely continuous and order differentiable function, where . 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 th-order discrete-approximate problem instead of the problem (PV):
Here th-order difference operator is defined as follows:
Thus by using the method of approximation [23, 26, 27], we can establish necessary and sufficient conditions for the rather complicated th-order discrete-approximate problem. Then by passing to the limit in necessary and sufficient conditions of this problem as , 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 th-order Euler-Lagrange type differential inclusion and the transversality conditions:
where is the adjoint PLDO of the primal operator .
Later on we assume that is absolutely a continuous function with the higher order derivatives until and . The following condition ensures that the LAM is nonempty:
, a.e. or, equivalently,
The following are sufficient optimality conditions for evolution differential inclusions with PLDOs.
Theorem 3.1. Let be a continuous and convex function and a convex set-valued mapping. Moreover, let
be convex sets. Then for optimality of the trajectory in the problem (PV) with evolution differential inclusions and PLDOs, it is sufficient that there exists an absolutely continuous function with the higher order derivatives until 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 and .
Proof. Using Theorem 2.1 (, p. 62), the definition of the Hamiltonian function and condition Eq. (i), we obtain
which can be rewritten as follows
Further using the definition of the Hamiltonian function, Eq. (4) can be converted to the inequality
Integrating Eq. (5) over the interval , we have
Let us denote
In what follows our approach lies in reducing in a relationship consisting of sums from to of suitable derivatives of scalar products; thus, after some transformations we can deduce an important representation for a first term of as follows:
Then in view of Eq. (7) in the definition of , we have an efficient formula:
Thus, integrating B, we can obtain
Here by suitable rearrangement and necessary simplification, we have
In order to make use of the transversality condition Eq. (ii), we rewrite it in a more relevant form:
Using the derivative operator , it is not hard to see that the relation described above can be expressed in a more compact form:
Furthermore, applying the definition of the transversality condition Eq. (iii) for all feasible arc , we have
Remark 3.1. It can be noted that in the particular case, if where , the second-order linear differential operator is a well-known self-adjoint Sturm-Liouville operator .
Corollary 3.1. Let be 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:
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 means 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, and in general are nonconvex function and set-valued mapping, respectively. Moreover, suppose that is the cones of tangent directions to .
Then for optimality of the trajectory , it is sufficient that there exists an absolutely continuous function , satisfying the following conditions:
, a.e. ,
, a.e. .
Proof. In the proof of Theorem 3.1, we have used the following inequality:
Hence, from the inequality Eq. (12), immediately we have the inequality Eq. (10). Moreover, setting for all feasible trajectories , it is not hard to see that for nonconvex the following inequality holds:
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 th-order PLDO with constant coefficients:
where is the th-order polynomial operator, are some real constants, is a convex set-valued mapping, is a continuous convex function, and are fixed -dimensional vectors. It is known that the multiplication operation is commutative for polynomial linear differential operators with constant coefficients. On the other hand, the th-order adjoint operator is defined as follows:
Corollary 4.1. Let and be a convex function and a set-valued mapping, respectively. Then, for the trajectory to be optimal in the problem (PC), it is sufficient that there exists an absolutely continuous function satisfying the Euler-Lagrange type differential inclusion.
and transversality condition at the endpoint .
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
It follows that . On the other hand, since , and are constants, by sequentially substitution in the transversality condition Eq. (iii), we derive that
and therefore for .
Suppose now that we have the so-called linear Mayer problem with PLDOs:
where differentiable convex function; and are and continuous matrices, respectively; U is a convex compact of ; , are constant vectors. It is required to find a control function such that the corresponding trajectory minimizes the Mayer functional .
In fact, this is optimization of Cauchy problem for “linear” differential inclusions with PLDO. The controlling parameter is called admissible if it only takes values in the given control set which is a nonempty, convex compact.
Theorem 4.1. The arc corresponding to the controlling parameter is a solution to Eqs. (14) and (15) if there exists an absolutely continuous function , satisfying the Euler-Lagrange type differential equation, the transversality condition, and Weierstrass-Pontryagin maximum principle:
Proof. Obviously, the Hamiltonian is
where the argmaximum inclusion implies that and . Thus, applying Theorem 3.1, we obtain
Consequently, the transversality condition Eq. (ii) of Theorem 3.1 is unnecessary and by Corollary 4.1 , . □
Remark 4.1. Suppose that in the definition of th-order PLDO (see Eq. (13)) , and is a convex closed polyhedron. Then we have linear equations with variable coefficients in the finite time interval . Obviously, for such problems an adjoint Euler-Lagrange type differential equation and transversality condition at a point consist of the following: . We remind that along with Pontryagin’s maximum principle (see, e.g., ) 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 :
Here and .
It should be noted that substituting into Newton’s second law , we have .
Obviously, in this problem , .
Then Eq. (16) has the form:
It can be easily seen that in the adjoint inclusion Eq. (i).
of Corollary 3.1 and , and so we have
Now, it is not hard to see that
Then taking into account , as a result of Theorem 3.1 (see also Corollary 3.1) from Eq. (19), we deduce that
for which the solution is a linear function of the form , where are arbitrary constants. Then Eq. (18) implies that or
Further, from the linearity of and 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.
. On the other hand, it is not hard to see that is a convex function; in fact, the Hessian matrix
is a positive semidefinite, that is, all eigenvalues of are nonnegative. Indeed, denoting this matrix by , we see that the characteristic equation is a unique square matrix) has two real nonnegative eigenvalues . Consequently, is convex and . It follows that . Comparing this relation with , we immediately have Then from a general solution of the adjoint Euler-Lagrange type inclusion (equation) , we have (are arbitrary constants), and so , whence , if . Therefore, Eq. (20) implies that for optimal control , there are four possibilities:
(observe that is a point of discontinuity of and the values of the control functions at a point of discontinuity are unessential). As a consequence, it follows that either the sign of the linear function does not change for the whole interval or for a some in the interval (the case Eq. (24) is excluded). Therefore, since is 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
By analogy, for we have.
Now, let and be parabolas of Eqs. (26) and (27), respectively. Here, in the case Eq. (21), , and so from Eq. (26), we have ; . Consequently, the value of problem Eq. (16) is , if . By a similar way, for a control function from Eq. (27), we obtain that and .
On the other hand, in the case Eq. (23), the control function first is equal to and then equal to , and the trajectory consists of two pieces of parabolas and (is continuous and piecewise smooth on the interval ). Then the solution of the equation Eq. (25) on the interval is given by Eq. (26); at a point are satisfied . Consider now the initial value problem:
It is clear that and from which we obtain that the solution of the initial value problem Eq. (28) is Substituting the value into equation , we have (it follows that ). Moreover, and Thus, the value of our Mayer problem is where is defined as in Eq. (23). Comparing the values , we believe that the value of Mayer problem is
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:
where is a convex continuous function, is convex set-valued mapping, and are convex sets.
The following adjoint inclusion is the second-order Euler-Lagrange type inclusion for the problem (PM):
b1. , a.e. .
In what follows we assume that , is absolutely continuous function together with the first-order derivatives for which . Besides the auxiliary function is absolutely continuous and .
The transversality conditions at the endpoint consist of the following:
Now we are ready to formulate the following theorem of optimality.
Theorem 5.1. Suppose that is a continuous and convex function, is a convex set-valued mapping, and are convex sets. Then for optimality of the feasible trajectory in the problem (PM), it is sufficient that there exists a pair of absolutely continuous functions:
satisfying a.e. the second-order Euler-Lagrange type inclusions Eqs. (a1) and (b1) and the transversality condition Eq. (c1) at the endpoint .
Proof. By the proof idea of Theorem 3.1 from Eqs. (a1) and (b1), we obtain the adjoint differential inclusion of second order:
On the definition of subdifferential set of the Hamiltonian function for all feasible trajectory , we rewrite the last relation in the equivalent form:
Now by using definition of the Hamiltonian function, the inequality Eq. (29) can be reduced to the inequality
Integrating of the inequality Eq. (30) over the interval , we derive that
For convenience we transform the expression in the square parentheses on the right-hand side of Eq. (31) as follows
Thus by elementary property of the definite integrals, we can compute the integral on the right-hand side of Eq. (31):
Now, remember that are feasible trajectories and and .
. Then it follows from Eq. (33) that
Now, thanking to the transversality conditions Eq. (c1) at the endpoint , we can rewrite
or, in other words
or . □
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.