The optimality principle for second-order discrete and discrete-approximate inclusions

Article History: Received 14 December 2020 Accepted 21 February 2021 Available ** May 2021 This paper deals with the necessary and sufficient conditions of optimality for the Mayer problem of second-order discrete and discrete-approximate inclusions. The main problem is to establish the approximation of second-order viability problems for differential inclusions with endpoint constraints. Thus, as a supplementary problem, we study the discrete approximation problem and give the optimality conditions incorporating the Euler-Lagrange inclusions and distinctive transversality conditions. Locally adjoint mappings (LAM) and equivalence theorems are the fundamental principles of achieving these optimal conditions, one of the most characteristic properties of such approaches with second-order differential inclusions that are specific to the existence of LAMs equivalence relations. Also, a discrete linear model and an example of second-order discrete inclusions in which a set-valued mapping is described by a nonlinear inequality show the applications of these results.


Introduction
Optimal control problems with discrete and differential inclusions are increasingly studied in mathematical theory [1,2]. Also, discrete and continuous-time processes have wide applications in the field of mathematical economics and the problems of control dynamic system optimization [3][4][5][6]. Thus, the optimal control problems described by discrete inclusions with endpoint constraints and approximation play a very significant role in both theory and applications of control theory [7,8]. Second-order discrete and differential inclusions have been studied by many authors when the setvalued mapping is both convex and nonconvex valued (see [9][10][11] and references therein). Auslender and Mechler [12] establish necessary and sufficient conditions to ensure the existence of solutions to the second-order differential inclusions with state constraints via interior tangent sets.
An approximation to a linear differential inclusion using N-stage single step discrete inclusions is described in the paper [13]. The result is applied to the discretization of control constrained optimal control problems in the second-order and the use of dynamic programming for approximate feedback design.
The paper [14] deals with the discrete approximations of nonconvex differential inclusions in Hilbert spaces and dynamic optimization-optimal control problems concerning such differential inclusions.
Agarwal and O'Regan [15] present new fixedpoint theorems for weakly sequentially upper semicontinuous maps. These results are used to establish existing principles for second-order differential equations and inclusions.
The optimal control of discrete-time systems are given in the book of Boltyanskii [16]. That book explores some results from the classical control *Corresponding author point of view for a linear, time-invariant, discretetime, optimal control system with infinite-time case.
In general, several studies on the differential inclusions of the second-order are devoted to problems of existence and viability. In the case where the multifunction is upper semicontinuous and has compact convex values, Haddad and Yarou [17] provided the first viability result for secondorder differential inclusions. The Cauchy problem for the infinite-dimensional case and second-order differential inclusions are considered in that paper. As is pointed out in the paper [18], Marco and Murillo analyzed the existence of Lyapunov functions for second-order differential inclusions by using the methods of the viability theory. A necessary assumption on the initial states and sufficient conditions are obtained for the existence of local and global Lyapunov functions. Lupulescu [19] demonstrated the existence of viable solutions for autonomous second-order functional-differential inclusions in the case where the multifunction defining the inclusion is upper semicontinuous, compact valued, and contained in the subdifferential of a proper lower semicontinuous convex function. Due to the higher-order derivatives and their discrete analogs, the problems accompanied by the higher-order discrete and differential inclusions are more complicated. A convenient procedure for eliminating this complication in optimal control theory involving higher-order derivatives is a formal reduction of the problems by substitution to the system of first-order differential inclusions or equations. But in practice, returning to the higher-order problem and expressing the arising optimality conditions in terms of the original problem data, in general, is very difficult. Although the construction of adjoint inclusions and transversality conditions is more complicated, Mahmudov formulates the conditions of optimality for the optimal control problem of higher-order differential inclusions with functional constraints in the paper [20]. The principle approach we use is that of locally adjoint mapping(LAM ), which facilitates obtaining necessary and sufficient conditions for all types of discrete and differential inclusions. Optimization of various forms of discrete inclusions can be reduced to finite-dimensional problems of mathematical programming, namely, to the minimization of functions on the intersection of a finite number of sets. We use difference approximations of ordinary derivatives and grid functions on a uniform grid to approximate differential inclusions and to derive necessary and sufficient conditions of optimality for discrete-approximation problems. It turns out that this requires some particular equivalence theorems of a LAM , which arise in discrete and discrete-approximation problems. One of the central objects of this paper is the relationship between continuous and discrete systems. By using particular equivalence theorems of the LAM , which play a significant role in the following investigations and without which few necessary or sufficient conditions would be obtained, the transition to the optimal conditions for discrete-approximation problems from their discrete counterparts is realized. The point argument is that discrete and discrete-approximation problems naturally are described by different setvalued mappings (say F and G, respectively). Therefore, we have to express the LAM G * by F * to formulate the optimality conditions for each discrete and discrete-approximation problem associated with the continuous problem. The rest of this paper is organized as follows. For the convenience of the reader, the necessary facts and supplementary results from the book of Mahmudov [1] are summarized in Section 2. In particular, the Hamiltonian function, argmaximum set of a set-valued mapping, and the locallyadjoint mapping are introduced and the viability problems for second-order discrete and differential inclusions are described with endpoint constraints. In Section 3, the discrete problem with the second-order discrete inclusions posed in Section 2 is reduced to a convex problem of a finite number of geometric constraints. We prove the necessary and sufficient conditions of optimality in terms of LAM s by using constructions of convex and nonsmooth analysis. In Section 4, we use difference approximations of derivatives and grid functions on a uniform grid to approximate problems with second-order differential inclusions. Then we derive the necessary and sufficient condition for optimality in both forms of Euler-Lagrange inclusions and transversality conditions for the discrete-approximation problem. In Section 5, we present some applications of the results obtained for problems with second-order discrete inclusions and set-valued mappings.

Needed facts and problem statement
For the convenience of the reader, the books Mahmudov [1] and Mordukhovich [2] contain all the necessary notions and results from set-valued analysis theory. Let R n be an n-dimensional Euclidean space, x, v be an inner product of elements x, v ∈ R n , (x, v) be a pair of x, v. Assume that F : R n × R n ⇒ R n is a set-valued mapping from R 2n = R n × R n into the set of subsets of R n . The set-valued mapping F is said to be convex if Let us introduce the Hamiltonian function and argmaximum set for a set-valued mapping F as Clearly, the Hamiltonian function H F (·, ·, v * 2 ) is concave for the convex set-valued mapping F .
is called the cone of tangent directions at a point z 0 ∈ A to the set A if from z = (x, u, v) ∈ K A (z 0 ) it follows that z is a tangent vector to the set A at point z 0 ∈ A , i.e., there exists such function γ(λ) ∈ R 3n such that z 0 + λz + γ(λ) ∈ A for sufficiently small λ > 0 and λ −1 γ(λ) → 0 as λ ↓ 0.
It should be pointed out that the cone K A (z 0 ) is not uniquely defined. Since λz, λ ≥ 0 is a vector of tangent direction, if z is the same, then it is clear that such vectors form a cone. In any case we can see that the wider a cone of tangent directions we have the essentially necessary condition for a minimum [1]. Obviously, for a convex mapping F at a point For a convex mapping F a set-valued function defined by . Let intA be the interior of the set A ⊂ R 3n and riA be the relative interior of the set A, i.e. the set of interior points of A with respect to its affine hull Af f A. A function ϕ is called a proper function, if it does not assume the value −∞ and is not identically equal to +∞. Obviously ϕ is proper function if and only if domϕ = ∅ and ϕ(x, y) is finite for (x, y) ∈ domϕ. We note that a convex function is continuous on the relative interior of its domain, it may have discontinuities only on its relative boundary (see, for example, Theorem 1.18 [1]). In this paper, we study the following main secondorder discrete model: where x t ∈ R n , F (·, ·, t) : R 2n ⇒ R n is a time dependent set-valued mapping, ϕ : R 2n → R is continuous function, α 0 , α 1 are fixed vectors and N is fixed natural number, A, B ⊂ R n are nonempty subsets and A ∩ B = ∅. We label this problem as (P D ). A sequence {x t } N t=0 = {x t : t = 0, 1, . . . , N } is called a feasible trajectory for the stated problem (P D ). The problem is to find a solution { x t } N t=0 of the problem (P D ) for the second-order discrete inclusions satisfying (2) − (3) and minimizing the Mayer functional ϕ(x N −1 , x N ). The problem (P D ) is said to be convex, if the F (·, ·, t), ϕ and A, B are convex set-valued mapping, proper convex function, and convex sets respectively.
Definition 2. We say that for the convex problem (P D ) satisfies the regularity condition, if for points x t ∈ R n , one of the following cases is fulfilled: x N ∈ intB, (with the possible exception of one fixed t) and ϕ is continuous at (x N −1 , x N ).
It follows from the regularity condition that, if { x t } N t=0 is the optimal trajectory in the problem (P D ), then the cones of tangent directions K gphF (·,t) (x t ,x t+1 ,x t+2 ) are not separable and consequently the condition of Theorem 3.2 [1] is satisfied. We are interested in the approximate problem for the following evolution differential inclusions with endpoint constraint where F (·, ·, t) : R 2n ⇒ R n is a time dependent set-valued mapping, φ : R 2n → R is continuous function and β 0 , β 1 are fixed vectors and A, B ⊂ R n are nonempty subsets and A ∩ B = ∅.
That problem is called the viability problem. The problem is to find an arc x(·) of the viability problem satisfying (4) almost everywhere on [0, 1] and the initial and endpoint constraints that minimizes the cost functional φ. Here a feasible trajectory x(·) is understood to be an absolutely continuous function on a time interval [0, 1] together with the first-order derivatives for which . Clearly, such a class of functions is a Banach space, endowed with the different equivalent norms. Clearly, the condition (2) in the main discrete model is a discrete analog of second-order differential inclusions (4).
Let us formulate the conditions of optimality for the discrete problem (P D ) before we begin the discussion on the approximation problem with second-order inclusions.

Necessary and sufficient conditions of optimality for discrete problem
We consider the discrete problem (P D ) in this section. First, let us introduce a vector u = (x 0 , x 1 , . . . , x N ) ∈ R n(N +1) and define the following convex sets in the space R n(N +1) ∈ gphF (·, ·, t) , t = 0, 1, . . . , N − 2, The discrete problem (P D ) is now reduced to solving a convex programming problem by setting f (u) = ϕ(x N −1 , x N ). In fact, it is easy to see that the problem (P D ) is equivalent to the following one: C is the convex set. This transformation allows us to prove rigorously that if { x t } N t=0 is the optimal solution to the problem (P D ), then u is the solution to the problem (5). It should be noted that under the regularity condition Definition 2 the dual cone associated with intersection of cones of tangent directions is equal to the algebraic sum of their dual cones. Thus, from Theorem 1.11 [1], we have Moreover, the results taken from [1] provide necessary conditions of optimality for the convex mathematical programming (5). We can prove necessary conditions of optimality for problem (5) with geometric constraints based on results concerning convex mathematical programming. Thus, by continuity of ϕ at points of some feasible solution { x t } N t=0 it follows from Theorem 3.4 [1], there exist vectors u * (t) ∈ K * Mt (ũ), t = 0, 1, . . . , N − 2, w * (t) ∈ K * Pt (ũ), t = 0, . . . , N and v * (0) ∈ K * Q 0 (ũ), v * (1) ∈ K * Q 1 (ũ), v * (N ) ∈ K * Q (ũ) and the number µ ∈ {0, 1}, not all equal to zero, such that whereû * ∈ ∂ u f (ũ). From definition of the function f it is easy to see that the vectorû * ∈ ∂ u f (ũ) has a formû * = 0, . . . , 0, First of all it is not hard to compute the dual cones K * Pt ( u), Then we should compute the dual cone K * Mt ( u) in the following lemma.
Proof. From the dual cone definition, x t+1 , x t+2 ) and from the arbitrariness of components x k , k = t, t + 1, t + 2 of vectors u it follows that x * k = 0, k = t, t + 1, t + 2. Therefore the inequality Now, we are ready to give the conditions of optimality for the discrete problem (P D ).
Theorem 1. Let F (·, ·, t) be a convex set-valued mapping and ϕ be proper convex functional and continuous at the points of some feasible trajectory. Then for optimality of the trajectory {x t } N t=0 in the Mayer problem (1) − (3) with second-order discrete inclusions, initial and endpoint constraints, it is necessary that there exist a number µ ∈ {0, 1} and vectors x * t , ξ * t , η * t , t = 0, . . . , N not all equal zero satisfying the Euler-Lagrange discrete inclusions and transversality conditions: In addition, under the regularity condition these conditions are sufficient for optimality of the trajectory {x t } N t=0 .

The conditions of optimality for discrete approximation problem
In what follows, we give an idea of the construction of the discrete-approximation problem for the viability problem (4) with second-order differential inclusions by introducing the first and secondorder difference operators: where δ is a step on the t-axis and x(t) ≡ x δ (t) is a grid functions on a uniform grid on [0, 1]. We define the following discrete-approximation problem associated with the continuous problem (4) We apply the results of Theorem 1 to the problem (14) − (16). To do this we rewrite the discreteapproximation inclusion (15) in the relevant form by introducing the following auxiliary set-valued mapping Then we rewrite the problem (14)−(16) as follows : x(0) = β 0 , ∆x(0) = β 1 ,  ) it is necessary that there exist vectors x * (t), ξ * (t), η * (t) and a number µ = µ δ ∈ {0, 1}, not all zero, such that (22) We should express the LAM G * in the above relationship (20) in terms of LAM F * , which plays a central role in our developments in the next results.
Usually, some equivalence theorems are needed for any development in problems given by differential inclusions. Let us first prove two propositions concerning the Hamiltonian functions of the setvalued mappings F and G, and the sets of subdifferential of the Hamiltonian functions H G and H F . Lemma 2. Let F and G be formula-specified convex set-valued mappings (17). Then there is the following relation between the Hamiltonian H G and H F functions: Proof. We get the lemma proof immediately as follows, keeping in mind the definition of the Hamiltonian functions of the set-valued mappings G, F Lemma 3. The following relation holds for subdifferentials of the Hamiltonian functions H G and H F : is a 2n × 2n matrix partitioned into submatrices, I, −I δ , −I δ and n × n zero matrix, where I is an n × n identity matrix and Θ * is transposes of Θ.
In the construction of LAM for the original discrete approximation problem (14)-(16), the following theorem which is definitely of independent interest plays a crucial role.
Theorem 2. If G is a convex set-valued mapping defined by (17), then the following statements for the LAM s are equivalent: is the argmaximum set for mapping G.
Proof. By useful Frobenius formula, the following inverse of 2 × 2 block matrices holds if A, B are invertible where ∆ = B − CA −1 D is the Schur complement of A. Therefore, denoting A = δ 2 I, B = δI, C = 0, D = −δI, we compute that Furthermore, by Lemma 3 it is easy to see that are equivalent. Then from (24) and (25), it means Since 2 ) and by using the fact that Here take into account that v 2 ∈ G(x, v 1 ; v * 2 ) and x−2v 1 +v 2 δ 2 ∈ F (x, v 1 −x δ ; v * 2 ) ensure that the LAM s are nonempty at a given points.
Proof. By the definition of subdifferential sets we can write This latter relationship gives that When (27) and (28) are compared, we deduce that x * = x * − y * δ , y * = y * δ or, in other words, The proof of theorem is completed.
Theorem 3. Let F be a convex set-valued mapping and φ 0 be proper convex functional and continuous at the points of some feasible trajectory. Then for optimality of the trajectory {x(t)} in the discrete approximation problem, it is necessary that there exist a number µ = µ δ ∈ {0, 1} and vectors x * (t), η * (t), v * (t) which are not all equal zero, satisfying the approximate Euler-Lagrange and transversality inclusions: ∈ F * (x * (t + 2δ); (x(t), ∆x(t), ∆ 2x (t), t)), And, under the regularity condition these conditions are also sufficient for optimality of the trajectory {x(t)}.
Remark 1. For future directions, we note that the results obtained in this section could be useful for deriving optimality conditions for the secondorder viability problem (4) given by differential inclusions with endpoint constraint. At least we emphasize that the key to our success is to pass formally to the limit in the conditions of Theorem 3. Consequently, by setting µ = 1 and passing formally to the limit as δ → 0, the sufficient conditions of optimality can be formulated for the continuous problem (4). Moreover, by using the functional analysis approach in the convex problem, the necessity of these conditions for optimality can be justified. Corollary 1. Let F be a convex set-valued mapping and φ be proper convex functional and continuous at the points of some feasible trajectory. Then for optimality of the trajectoryx(t) in the problem (4), it is sufficient that there exist a number µ and vectors x * (t), η * (t), v * (t) which are not all equal zero, satisfying the second-order Euler-Lagrange type adjoint A (x(t)), t ∈ [0, 1) , and the transversality conditions at the endpoint t = 1 consist of the following Note that the transformation to the continuous problem (4) is in any case, a separate subject of discussion and is therefore omitted. Note also that this construction of optimality conditions in the presented paper for second-order discrete and discrete approximation inclusions can be useful for the optimization of an arbitrary order discrete and ordinary differential inclusions.
Theorem 4. If Ψ and ϕ are continuously differential functions and the cone K gphF (x, v 1 , v 2 ) is non-empty for all ( x t , x t+1 , x t+2 ) then the existence of {x * t , ξ * t , η * t } satisfying the adjoint discrete inclusions (34)-(36) is necessary and sufficient for the optimality of the trajectory { x t } N t=0 of problem (32).