OPTIMAL CONTROL OF HYPERBOLIC TYPE DISCRETE AND DIFFERENTIAL INCLUSIONS DESCRIBED BY THE LAPLACE OPERATOR

. The paper is devoted to the optimization of a ﬁrst mixed initial-boundary value problem for hyperbolic diﬀerential inclusions (DFIs) with Laplace operator. For this, an auxiliary problem with a hyperbolic discrete inclusion is deﬁned and, using locally conjugate mappings, necessary and suﬃcient optimality conditions for hyperbolic discrete inclusions are proved. Then, using the method of discretization of hyperbolic DFIs and the already obtained optimality conditions for discrete inclusions, the optimality conditions for the discrete approximate problem are formulated in the form of the Euler-Lagrange type inclusion. Thus, using specially proved equivalence theorems, which are the only tool for constructing Euler-Lagrangian inclusions, we establish suﬃcient optimality conditions for hyperbolic DFIs. Further, the way of extending the obtained results to the multidimensional case is indicated. To demonstrate the above approach, some linear problems and polyhedral optimization with hyperbolic DFIs are investigated.


Introduction
Over the past decades, great progress has been made in various areas of optimal control problems described by ordinary [1,2,4,7,9,10,14,19,21,23,25,30,31,33,34,37,40] and partial differential equations/inclusions [3, 5, 11-13, 16, 18, 20, 22, 29, 32, 36, 39, 41]. The work [6] investigates boundary value problems for systems of Hamilton-Jacobi-Bellman first-order partial differential equations and variational inequalities, the solutions of which obey the viability constraints. In the paper [17], stabilization in a finite time of homogeneous quasilinear hyperbolic systems with one side controls and with a nonlinear boundary condition on the other hand is considered. Time-independent feedbacks are presented leading to stabilization in a finite time at any time, larger than the optimal time for null controllability of the linearized system if the initial condition is small enough. The paper [38] studies the behavior of solutions of a one-dimensional hyperbolic relaxation system at large times, which can be written in the form of a nonlinear equation of a damped wave. First, the global existence of a unique solution and its decay properties are proved for sufficiently small initial data. In [8], a general uniqueness theorem for nonlinear hyperbolic systems of partial differential equations in one-space dimension is established. The paper [35] considers optimal control problems for hyperbolic equations with controls in the Neumann boundary conditions with pointwise constraints on the control and state functions. New necessary optimality conditions are derived in the form of Pontryagin's pointwise maximum principle for the problem under consideration with state constraints. The approach is based on modern methods of variational analysis. In papers [11,12,19,37,40] various problems with ordinary differential equations/inclusions are considered. In [11], a strengthened version of Pontryagin's maximum principle is derived for general nonlinear optimal control problems on time scales in a finite-dimensional space. The finite time can be fixed or not, and in the case of general boundary conditions, the corresponding transversality conditions are derived, the proof of which is based on the Ekland's variational principle. The paper [40] develops the theory of tubes for solving differential inclusions within a given set of sets. The concept is defined as the minimum invariant tube with values in the collection. Under certain requirements on the set, the existence and Lipschitz property of "solution tubes" is proved. The paper [19] investigates infinite-dimensional differential inclusions. Sufficient conditions for nonemptiness of the viability kernels of such systems are obtained. In the paper [37], problems of optimal control of dynamical systems controlled by functional differential inclusions of neutral type with constraints are considered. Therein, to develop the method of discrete approximations and use advanced tools of generalized differentiation, a variational analysis of neutral functional-differential inclusions was carried out and new necessary optimality conditions were obtained, both of the Euler-Lagrange and Hamiltonian types. In [10], by using the Baire category method, an existence result is proven under Carath'eodory assumptions for boundary value problem of Dirichlet type with nonconvex ordinary differential inclusions. In [24] on the basis of the diffusion properties of weak solutions of stochastic partial DFIs, some existence theorems and some properties of solutions are given. In [15], an optimal control problem given by hyperbolic type DFIs with boundary conditions and endpoint constraints is studied.
The present paper is dedicated to one of the difficult and interesting fields, where the main goal is to derive sufficient optimality conditions for problems with hyperbolic DFIs in twospatial dimension. Here we obtain the results in the same spirit as in [33], where the Mayer problem with higher order evolution differential inclusions and functional constraints of optimal control theory is studied. Recall that in [33] the obtained transversality and complementary slackness conditions for second-order ordinary differential inclusions play an essential role in further studies, where the results are generalized to a problem with an arbitrary higher-order differential inclusion.
Hyperbolic partial differential equations arise in many physics problems such as string vibration, acoustic modelling, and supersonic fluid flow used in high-speed applications such as rocket motion and supersonic jets. One of the most important hyperbolic equations is the wave equation, which is widely used in modelling electromagnetic waves, seismic waves and shock waves, earthquake engineering, ocean wave propagation and many others. In the mathematical model of hyperbolic partial differential equations, waves propagate at a finite speed, in contrast to parabolic partial differential equations, where the initial value immediately affects the solution at all other points in space. The paper is organized in the following order.
In Section 2, the needed concepts and results, such as, Hamiltonian function H F and argmaximum sets of a set-valued mapping F , the locally adjoint mappings (LAMs), etc., from the book of Mahmudov [28], are given and the problem with discrete and continuous hyperbolic inclusions are stated. Moreover, some "nondegeneracy condition", that is, the standard condition for convex analysis about the existence of an interior point is formulated.
In Section 3, for a problem with discrete inclusions of hyperbolic type, we prove a theorem on necessary and sufficient conditions for optimality. It is shown that under non-degeneracy condition the necessary conditions are also sufficient for the optimality. In the proof of this theorem, the main role is played by the reduction of the problem posed to the problem of mathematical programming and, as a consequence, the application of its methods.
In Section 4 we introduce difference operators defined on three-point models, and using difference approximations of the Laplace operator and grid functions on a uniform grid, we approximate the problem with hyperbolic DFIs. In what follows, with a skillful definition of an auxiliary set-valued mapping G(·, x, y, t) : R 6n ⇒ R n involving the original set-valued mapping F (·, x, y, t) : R n ⇒ R n we reduce this problem to a problem with a discrete inclusion and apply the results obtained in Section 3 and obtain a necessary and sufficient optimality condition for a discrete approximate problem. But, in turn, the latter is possible due to the LAM equivalence theorems that provide the transition from LAM G * to LAM F * , which, in our opinion, are the only tool for constructing Euler-Lagrangian inclusions for problem (PC) with hyperbolic DFIs. For this, auxiliary results are obtained on the equivalence of the cone of tangent directions and the argmaximum sets.
In Section 5, we use the results of Section 4 to obtain sufficient optimality conditions for hyperbolic DFIs with Laplace operator. The formulation of sufficient conditions is realized by going over to the formal limit in the discrete-approximate problem, as discrete steps tend to zero. Further, for the generalized initial-boundary value problem, by analogy, we indicate the ways of extending Theorem 5.1 to the multidimensional case. Moreover, in the proof of the optimality condition, the Green's formula is used in the multidimensional case. At the end of section some linear problem with hyperbolic DFIs is considered. Our methodology with a transition to a discrete, and then a discrete-approximate problem guarantees to construct step by step the hyperbolic type Euler-Lagrange adjoint DFIs and the "endpoint" and boundary conditions for the problem with hyperbolic DFIs.
Section 6 is devoted to the problem with hyperbolic polyhedral DFIs. Here, the integrand of the Lagrange functional is also a polyhedral function. The Euler-Lagrange type inclusion is constructed, and necessary and sufficient optimality condition for problem with hyperbolic polyhedral DFIs involving Laplace operator are proved.

Necessary facts and and problem statements
Although all the necessary definitions and concepts can be found in the book [28], for the convenience of readers, they are given in this section. Let (u, v) and u, v be a pair and inner product of elements u, v in n-dimensional Euclidean space R n , respectively. A set-valued mapping F : For a set-valued mapping, we introduce the Hamiltonian function and the notation of the argmaximum set: The cone of tangent directions at the point (u, v) ∈ gphF will be denoted by K F (u, v) ≡ K gphF (u, v) and for a convex set-valued mapping F it is defined at a point (u, v) ∈ gph F as follows The cone K A (u, v) of tangent directions of nonconvex set A ⊂ R 2n at (u, v) ∈ A is defined as the set of (ū,v) for which there exists a function ϕ : (0, +∞) → R 2n with ϕ(γ) → 0 as γ ↓ 0 such that (u, v) + (ū,v) + ϕ(γ) ∈ A for sufficiently small γ.
Recall that a set-valued mapping F * (·; (u, v)) : R n ⇒ R n defined by In what follows, we mainly use the equivalent definition of LAM in terms of the Hamiltonian function A set-valued mapping is called the LAM to "nonconvex" mapping F at a point (u, v) ∈ gph F : Since for a convex set-valued mapping H F (·, v * ) is concave function, by Theorem 2.1 [28, p. 62] the following assertion holds In fact, the given in the paper notion LAM is closely related to the coderivative concept of Mordukhovich [37], which is essentially different for nonconvex mappings. The notion of coderivative has been introduced for set-valued mappings in terms of the basic normal cone to their graphs and is defined as In the most interesting settings for the theory and applications, coderivatives are nonconvex-valued and hence are not tangentially/derivatively generated. This is the case of the first coderivative for general finite dimensional set-valued mappings for the purpose of applications to optimal control. The main advantage of the definition of LAM is its simplicity.
The cone K A (z) of tangent directions of the set A at a point z = (u, v) ∈ A ⊂ R 2n is called a local tent [28 , p. 120] if for eachz ∈ ri K A (z) there exists a convex cone K ⊆ K A (z) and a continuous mapping ψ(z) defined in a neighborhood of the origin such that (1)z ∈ ri K, Lin K = Lin K A (z), where Lin K is the linear span of is the ball of radius ε and with center the origin.
A function g is said to be proper if it does not take the value −∞ and is not identically equal to +∞. First, consider the following optimization problem with a discrete analogue of the so-called first mixed hyperbolic DFIs, labelled by (PD): 2) that minimizes (2.1). Section 4 of the present paper is devoted to the study of a first mixed boundary value problem for hyperbolic DFIs in a two-spatial dimension, containing Laplace operator in a bounded rectangular parallelepiped that is sometimes referred to simply as a cuboid: where F (·, x, y, t) : R n ⇒ R n is a convex set-valued mapping, g(·, x, y, t) is proper convex function, ∆ is Laplace's operator: ∆ = ∂ 2 ∂x 2 + ∂ 2 ∂y 2 and α 1 , α 2 and β 0 , β S , γ 0 , γ L are given continuous functions, T ] → R n ; L, S, T are real positive numbers. Since hyperbolic inclusion (2.5) is second order in time, values of u(x, y, t) and its partial derivative on t must be specified along the initial time. On the other hand, since equation (2.5) is the second order in space, the values of u(x, y, t) must be specified along boundaries of the region D × [0, T ]. In fact, D × [0, T ] is a rectangular parallelepiped that is sometimes referred to simply as a cuboid. We label this problem (PC). The problem consists in finding a solutionũ(x, y, t) of a first mixed initial-boundary value problem (PC) that minimizes (2.4). For convenience, we assume throughout the context that feasible solutions are classical solutions; let , and the boundary conditions in Γ T is called the classical solution of the initial-boundary value problem (2.4)-(2.6). Here C 2 (P T ) is the space of continuous functions having in P T all continuous partial derivatives of the first and second order. It should be noted that the definition of a solution in one sense or another (classical, generalized, almost everywhere, etc.) in no way hinders the implementation of this method for the class of problems under consideration.
With respect to ( [28], p.122) h(ū, u) is called a CUA of the function g : Here the exterior supremum is taken on all r(α) such that α −1 r(α) → 0 as α ↓ 0. In addition, let the function g(u) admit CUA h(ū, u) at point u that are continuous with respect toū. The latter means that the subdifferentials ∂g(u) := ∂h(0, u) is defined.
For the functions g(·, x, y, t), (x, y, t) ∈ D × [0, T ] and the set-valued mapping Q(·, x, y, t) we impose the following condition. It should only be noted that, since a non-smooth and nonconvex function cannot be approximated in a neighbourhood of a point of non-smoothness with positively homogeneous functions, we use the concept of CUAs.
Condition N Let in problem (P D) the mapping Q(·, x, y, t) be such that the cone of tangent directions K Q(,x,y,t) (u x−1,y,t , u x,y−1,t , u x,y,t , u x,y+1,t , u x,y,t+1 , u x,y,t−1 , u x+1,y,t ) is a local tent. In addition, let the functions g(·, x, y, t) admit CUA h x,y,t (ū, u x,y,t ) at points u x,y,t that are continuous with respect toū. The latter means that the subdifferentials ∂g (u x,y,t , x, y, t) := ∂h x,y,t (0, u x,y,t ) are defined. The problem (2.1)-(2.3) is called convex, if the mapping Q(·, x, y, t) is convex and the functions g(·, x, y, t), , (x, y, t) ∈ L 0 × S 0 × T 0 , (x, y) = (0, 0), (L, S) are proper convex functions. The following condition guarantees for the problem (PD) the existence of a standard interior point of convex analysis. As usual, ri A be the relative interior of a set A.
is the fixed triple and g(·, x, y, t) are continuous at the point u x,y,t .

Optimization of first mixed initial-boundary value problem for hyperbolic type discrete inclusions
We first consider the convex problem (PD). In order to use convex programming results, we form the m = n(L + 1)(S + 1)(T + 1) dimensional vector w = (u 0 , u 1 , . . . , u L ) ∈ R m , where u x = (u x0 , u 1 , . . . , u xS ) ∈ R n(S+1)(T +1) and u xy = (u x,y,0 , u x,y,1 , . . . , u x,y,T ) ∈ R n(T +1) . Let us consider the following convex sets defined in the space R m : M x,y,t = {w = (u 0 , u 1 , . . . , u L ) : (u x−1,y,t , u x,y−1,t , u x,y,t , u x,y+1,t , u x,y,t+1 , u x,y,t−1 , u x+1,y,t ) ∈ gph Q(·, x, y, t)} , we reduce the convex problem (PD) to the following convex minimization problem in the space R m : We apply Theorem 3.4 ( [28], p. 99) to the convex minimization problem (3.1). For this, it is necessary to calculate the dual cones K * Mx,y,t (w), Mx,y,t (w) to the cone of tangent directions K Mx,y,t (w) has a form: This means that w + µw ∈ M x,y,t for sufficiently small µ > 0, which is the same as On the other hand, w * ∈ K * Mx,y, (w) is equivalent to the condition where the componentsū x,y,t of thee vectorw(see (3.2)) are arbitrary. Therefore, the last relation is valid only for This ends the proof of the lemma.
Based on Lemma 3.1 and the idea of the proof of Theorem 3.1 [3] and Proposition 3.2 [32], it is easy to establish the validity of the following theorem, the proof of which is omitted. Theorem 3.2. Suppose that Q(·, x, y, t), (x, y, t) ∈ L 1 × S 1 × T 1 are convex set-valued mappings, and g(·, x, y, t) are convex proper functions at the points of some feasible solution {u x,y,t }. Then for the {ũ x,y,t } to be an optimal solution of the problem (P D) it is necessary that there exist a number λ ∈ {0, 1} and six vectors ψ * x,y,t , ξ * x,y,t , u * x,t , η * x,y,t , ϕ * x,y,t , ζ * x,y,t , not all equal to zero, such that: Moreover, under the nondegeneracy condition λ = 1 and therefore, these conditions are also sufficient for the optimality of {ũ x,y,t }. Theorem 3.4. Suppose that conditions N are satisfied for the non-convex problem (PD). Then, for the optimality of the solution {ũ x,y,t } in the nonconvex problem with a hyperbolic discrete inclusions, it is necessary that there exist a number λ ∈ {0, 1} and six of vectors ψ * x,y,t , ξ * x,y,t , u * x,t , η * x,y,t , ϕ * x,y,t , ζ * x,y,t , not all zero, satisfying the conditions (i) and (ii) of Theorem 3.2.
Proof. Here condition N guarantees the fulfillment of the conditions of Theorem 3.24 ( [28], p. 133). Therefore, in view of this theorem, we have a necessary condition (may be insufficient), as in Theorem 3.2, written out for a non-convex problem.

Optimization of first mixed problem for hyperbolic type discrete-approximate inclusions
Here, to obtain optimality conditions for the problem (PC), as an intermediate stage, we use the difference derivatives to approximate the problem (PC) and, using Theorem 3.2, we formulate a necessary and sufficient condition for it. We choose steps δ, σ and h on the x, y-and t-axis, respectively, using the grid function u x,y,t = u δσh (x, y, t) on a uniform grid on D × [0, T ]. We introduce the following difference operators, defined on the three-point models ( [29], p. 319), [28], A 1δ = A 1 and A 2σ = A 2 Therefore, we establish the following difference first mixed hyperbolic type initial-boundary value problem (PDA): minimize (x,y,t)∈L0×S0×T0 (x,y) =(0,0),(L,S) δσhg(u(x, y, t), x, y, t), Recall that (4.1) is the Riemann sum [29] of the triple integral of a continuous function of three variables g over a bounded rectangular parallelepiped in three-dimensional space. Introducing a new mapping, we reduce the problem (4.1) and (4.2) to a problem of the form (PD). To do this, rewrite discrete-approximate inclusion (4.2) in the equivalent form, where
Transforming this inclusion, we get Recall that, by the definition of a set-valued mapping G, the inclusion v ∈ G (u 1 , u 2 , u, u 3 , u 4 , u 5 , x, y, t) implies Therefore, from this relation it is clear that (4.7) By going in the reverse direction, it is also not hard to see from (4.7) that (ū 1 ,ū 2 ,ū,ū 3 ,ū 4 ,ū 5 ,v) ∈ K G(,x,y,t) (u 1 , u 2 , u, u 3 , u 4 , u 5 , v). This completes the proof of the theorem.
Proposition 4.2. Let a convex-valued mapping G(·, x, y, t) be defined as in Theorem 3.1. Then G A (u 1 , u 2 , u, u 3 , u 4 , u 5 , v * , x, y, t) = F A (u, −v * , x, y, t) and the following statements for argmaximum sets are equivalent (1) v ∈ G A (u 1 , u 2 , u, u 3 , u 4 , u 5 , v * , x, y, t) ; (2) 2 1 Proof. First of all, we will find the connection between the Hamilton functions H G and H F .
Therefore, keeping in mind this formula for H G and H F we get the proof of proposition immediately as follows x, y, t) : . The needed proof is ended.
Suppose that a mapping G (u 1 , u 2 , u, u 3 , u 4 , u 5 , x, y, t) is such that the cones of tangent directions K G(,,x,y,t) (u 1 , u 2 , u, u 3 , u 4 , u 5 , v) determine a local tent. Then the following inclusions are equivalent under the conditions that u * Proof. Suppose that the condition (1) of theorem is fulfilled. By the definition of LAM, this means that in the case of (1) we have (4.8) Using the equivalence of the inclusions (1) and (2) for cones K G(·,x,y,t) and K F (,,x,y,t) , we rewrite this inequality in a convenient form as follows where ω and ψ are to be determined. Carrying out the necessary transformations in (4.9), we get Then comparing (4.8), (4.10) it is not hard to see that we obtain the following formulas for ω and ψ : Thus (4.9) means that Then from Theorem 3.2 we obtain condition (2), i.e., (1) ⇒ (2). By analogy, it can be shown that (2) ⇒ (1). As for the conditions concerning the argmaximum sets G A and F A , then by Theorem 2.1 ( [28], p.62), they simply guarantee the non-emptiness of the LAMs G * and F * . The proof of the theorem is completed.
Remark 4.6. If we consider the nonconvex problem (2.1)-(2.3) for a first mixed hyperbolic discrete inclusions, then it is easy to see that, under the condition N, the Euler-Lagrange type inclusion (4.15) and the initialboundary values (4.16) may not be sufficient optimality conditions. This is explained by the fact that when forming the optimality condition in the nonconvex case, instead of Theorem 3.2 [28, p. 98], we use Theorems 3.24 [28, p. 133], where the equality λ = 1 is optional.

Sufficient conditions of optimality for the hyperbolic DFIs
In this section, based on the results obtained in Section 3, we formulate a sufficient optimality condition for the continuous problem (PC). To do this, recall that the limits of A 1 v(x, y, t), A 2 v(x, y, t) and Bv(x, y, t), are second order partial derivatives with respect to x, y and t, respectively, i.e., lim δ→0 . Thus, setting λ = 1 in (4.6), (4.7) and passing to the formal limit, as the discrete steps δ, σ and h tend to 0 , in term of Laplace operator we find that: In fact, the following condition ensures that the LAM F * is nonempty [Thm. 2.1 ( [28], p. 62)] that: Here we assume that feasible solutions u * (x, y, t) are classical solutions satisfying an adjoint hyperbolic inclusion of the Euler-Lagrange type (i) and homogeneous end-point and boundary conditions (ii).
As expected, in the next theorem we prove that conditions (i)-(iii) are sufficient conditions of optimality in the first mixed problem (PC) with hyperbolic DFIs in a two spatial dimension.
Theorem 5.1. Suppose that g(·, x, y, t) is a continuous convex proper function and F (·, x, y, t) : R n ⇒ R n is a convex set-valued mapping. Then for the optimality of the solutionũ(x, y, t) in the first mixed problem (P C) with hyperbolic DFIs and Laplace operator it is sufficient that there exists a classical solutions u * (x, y, t), satisfying the conditions (i)-(iii).
Remark 5.2. It is well known [34] that in the problem with ordinary polynomial linear differential operators Ax = s k=1 p k (t)D k x of the s-th order with variable coefficients p k : [0, T ] → R 1 and with the operator of derivatives D k (k = 1, . . . , s) of the k-th order, the EulerLagrange inclusion includes the adjoint operator , where the sign of the corresponding term does not change for even k. The Euler-Lagrange inclusion (i) shows that the same property holds for the second-order hyperbolic operator.
Remark 5.3. The stated problem (2.4)-(2.6) can be generalized for hyperbolical differential inclusions second order; let D be some bounded convex closed region of R n and Q T ⊂ R n+1 is a bounded cylinder Q T = {x ∈ D, 0 < t < T } with height T > 0 and base D. Let Γ T be the lateral surface of the cylinder Q T , i.e. Γ T = {x ∈ ∂D, 0 < t < T }, where ∂D is a piecewisesmooth boundary of D and D 0 = {x ∈ D, t = 0} be lower base of Q T .
Note that the same result can be obtained from (5.9) and (5.11), assuming that A is a zero matrix, B is n × n unit matrix and U ⊂ R n .