Coupled Variational Inequalities: Existence, Stability and Optimal Control

In this paper, we introduce and investigate a new kind of coupled systems, called coupled variational inequalities, which consist of two elliptic mixed variational inequalities on Banach spaces. Under general assumptions, by employing Kakutani-Ky Fan fixed point theorem combined with Minty technique, we prove that the set of solutions for the coupled variational inequality (CVI, for short) under consideration is nonempty and weak compact. Then, two uniqueness theorems are delivered via using the monotonicity arguments, and a stability result for the solutions of CVI is proposed, through the perturbations of duality mappings. Furthermore, an optimal control problem governed by CVI is introduced, and a solvability result for the optimal control problem is established. Finally, to illustrate the applicability of the theoretical results, we study a coupled elliptic mixed boundary value system with nonlocal effect and multivalued boundary conditions, and a feedback control problem involving a least energy condition with respect to the control variable, respectively.

Recently, many scholars noticed that various comprehensive physical phenomenon and engineering applications could be, eventually, modeled by the complicated systems governed by variational inequalities, for example, Nash equilibrium problems of multiple players with shared constraints and dynamic decision processes, contact mechanics problems with adhesion (or wear) effect, and convection diffusion models in porous materials. Among the results we mention: Pang-Stewart [38] in 2008 systematically introduced and studied a class of dynamical systems on finite-dimensional spaces, which is formulated as a combination of ordinary differential equations and time-dependent variational inequalities. They represent powerful mathematical tools with applications to various problems involving both dynamics and constraints arising in mechanical impact processes, electrical circuits with ideal diodes, Coulomb friction for contacting bodies, economical dynamics, dynamic traffic networks. Cojocaru-Matei [6] introduced a Lagrange multiplier system which is composed of a variational inequality of elliptic type and a linear equation, and applied this system to study a boundary value problem involving p-Laplace operator and nonsmooth boundary conditions. By using a surjectivity result for multivalued maps and a fixed point argument for a history-dependent operator, Migórski [31] proved the unique solvability of a system of coupled nonlinear first order history-dependent evolution inclusions in the framework of evolution triples of spaces, and applied these abstract results to a dynamic frictional contact problem in mechanics. For more details on this topic, the reader is referred to Migórski-Zeng [32,33], Liu-Migórski-Zeng [23], Liu-Xu-Lin [22], Liou-Yang-Yao [21], Li-Yang [19], Chen-Wang [4,5], Zeng-Migórski-Liu [44,45], Wang-Huang [42] and the cited references therein. However, it should be pointed out that these results aforementioned cannot be used to study the coupled systems which are made up of two mixed variational inequalities of elliptic type, called coupled variational inequalities. But, coupled variational inequalities could be a useful mathematical tool for investigating numerous coupled mixed boundary value problems, feedback control problems and so forth. Therefore, to fill this gap, the main goal of the present paper is to introduce a new kind of coupled variational inequalities and to deliver the theoretical results concerning existence, uniqueness, stability, optimal control and applications to coupled variational inequalities under consideration.
Before any advancement, let us first introduce the problem that will play the central role in this paper. Let (X , · X ) and (Y , · Y ) be two reflexive Banach spaces with its dual spaces (X * , · X * ) and (Y * , · Y * ), respectively. In what follows, we denote by ·, · X (resp., ·, · Y ) the duality pairing between X * and X (resp., the duality pairing between Y * and Y ). We formulate the following coupled system which consists of two mixed variational inequalities on Banach spaces.
in Sect. 4, we provide novel applications of our abstract results to a coupled elliptic mixed boundary value system with nonlocal effect and multivalued boundary conditions, and a feedback control problem involving a least energy condition with respect to the control variable, respectively. We end the section by recalling a preliminary material to be used in the next sections. More details can be found in [1,7,8,13,30,43].
Throughout the text, the symbols " w −→ " and "→" stand for the weak and the strong convergence, respectively. Let (X , · X ) be a Banach space with its dual X * and denote by ·, · X the duality pairing between X * and X . Recall that a function f : X → R := R ∪ {+∞} is called proper, convex, and lower semicontinuous, if it fulfills the conditions

respectively.
We recall the following important result for the proper convex and l.s.c. functions, see e.g., [2, Proposition 1.10].

Proposition 2
Let (X , · X ) be a Banach space. Assume that ϕ : X → R is convex, l.s.c. and ϕ ≡ +∞. Then, ϕ is bounded below by an affine continuous function, i.e., there exist l ∈ X * and c ϕ ∈ R such that Remark 3 It is not difficult to see that if ϕ : X → R is convex, l.s.c. and ϕ ≡ +∞, then we are able to find constants α ϕ , β ϕ ≥ 0 such that Let K be a nonempty subset of X , ϕ : K → R be a proper convex and l.s.c. function, and A : K → X * . We say that A is

Remark 4
The following diagram reveals the essential implications of these monotonicity and generalized monotonicity. Let Z and Y be topological spaces and V ⊂ Z be a nonempty set. In what follows, we denote by 2 V the collection of its subsets. Given a set-valued mapping F : Z → 2 Y , we use the symbol Gr F to stand for the graph of F, i.e., We say that the graph of F is sequentially closed (or F is sequentially closed) in Z ×Y , if for any sequence {(x n , y n )} ⊂ Gr F is such that (x n , y n ) → (x, y) as n → ∞ for some (x, y) ∈ Z × Y , then we have (x, y) ∈ Gr F (i.e., y ∈ F(x)).
Theorem 5 Let Y be a reflexive Banach space and D ⊆ Y be a nonempty, bounded, closed and convex set. Let : D → 2 D be a set-valued map with nonempty, closed and convex values such that its graph is sequentially closed in Y w × Y w topology. Then, has a fixed point.

Existence and Uniqueness
In the section, we are devoted to the study of existence and uniqueness of solution to the abstract coupled variational inequalities, Problem 1. More precisely, under mild assumptions, an existence theorem for the solutions of CVI is established by employing Kakutani-Ky Fan fixed point theorem, Theorem 5, and Minty method. Moreover, we apply the monotonicity arguments to deliver two uniqueness results to Problem 1.
Let us introduce the set-valued mappings S : L → 2 K and T : K → 2 L defined by  (iii) there exists a function r : R + × R + → R such that G(y, x), x X ≥ r ( x X , y Y ) x X for all x ∈ X and y ∈ Y , and • for each nonempty and bounded set D ⊂ R + , we have r (t, s) → +∞ as t → +∞ for all s ∈ D, • for any constants c 1 , c 2 ≥ 0, it holds r (t, c 1 t + c 2 ) → +∞ as t → +∞.
(iv) there exists a constant c G > 0 such that H (φ): φ : L → R is a proper, convex and lower semicontinuous function.
(iii) there exists a function l : and • for each nonempty and bounded set D ⊂ R + , we have l(t, s) → +∞ as t → +∞ for all s ∈ D, • for any constants c 1 , c 2 ≥ 0, it holds l(t, c 1 t + c 2 ) → +∞ as t → +∞.
(iv) there exists a constant c F > 0 such that

Remark 6
Particularly, if function r given in H (G)(iii) (resp. l given in H (F)(iii)) is independent of its second variable, then condition H (G)(iii) (resp. H (F)(iii)) reduces to the following uniformly coercive condtion for all x ∈ X and y ∈ Y ).
The first main result of this paper concerning the existence of solutions to Problem 1 is provided as follows. g), is nonempty and weakly compact in X × Y .

Theorem 7 Assume that H (G), H(F), H(0), H(1), H(ϕ) and H (φ) are satisfied. Then, the solution set of Problem 1 corresponding to
To prove this theorem, we need the following lemmas.

the solution set of problem (1.1), denoted by S(y), is nonempty, bounded, closed and convex; (iii) the graph of the set-valued mapping S
for some (x, y) ∈ X × Y . Then, for each n ∈ N, we have x n ∈ S(y n ), i.e., for all v ∈ K . Passing to the upper limit as n → ∞ to (2.3), we use hypothesis H (G)(ii) and weak lower semicontinuity of ϕ (due to the convexity and lower semicontinuity of ϕ) to find Applying assertion (i) again, we conclude that x ∈ S(y). Therefore, (y, x) ∈ Gr S, namely, the graph of the set-valued mapping S : Inserting v = x 2 and v = x 1 into the inequalities above for i = 1 and i = 2, respectively, we sum up the resulting inequalities to get Hence, the strict monotonicity of x → G(y, x) guarantees that x 1 = x 2 . So, S is a single-valued mapping. But, by virtue of assertion (iii), we can see that S is weakly continuous.
Likewise, for problem (1.2), we have the following lemma.

Lemma 9 Suppose that H (0), H(1), H(F) and H (φ) are fulfilled. Then, the statements hold
(i) for each x ∈ X fixed, y ∈ L is a solution of problem (1.2), if and only if, y solves the following Minty inequality: find y ∈ L such that for each x ∈ X fixed, the solution set of problem (1.2), denoted by T (x), is nonempty, bounded, closed and convex; (iii) the graph of the set-valued mapping T :

y) is strictly monotone, then T is a single-valued mapping and weakly continuous.
Furthermore, we provide a priori estimates for the solutions of Problem 1.
Assume that ① is valid, then we take x = x n and y = y n to (2.9) for getting Letting n → ∞ for the inequality above and using (2.11) as well as H (G)(iii) turn out This generates a contradiction. Similarly, for the case ②, we could use (2.10) to get a contradiction as well. However, suppose ③ occurs, we, further, consider the following two situations: (a) y n Y x n X → +∞ as n → ∞; (b) there exist n 0 ∈ N and c 0 > 0 such that y n Y x n X ≤ c 0 for all n ≥ n 0 .
If item (a) is true, then we put x = x n and y = y n into (2.10) to yield Passing to the limit as n → ∞ for the inequality above, it gives Obviously, it is impossible, whereas in terms of the situation (b), it follows from (2.9) that for n ≥ n 1 , where n 1 ≥ n 0 is such that x n 1 X > 1. This also triggers a contradiction.
To summary, we conclude that ( f , g) is bounded in X × Y . Consequently, we are able to find a constant M > 0 such that (2.5) is valid.
Consider the set-valued mapping : K × L → 2 K ×L given by (2.13) Invoking Lemmas 8 and 9, we can see that is well-defined. The following lemma points out that there exists a bounded, closed and convex set D in K × L such that maps D into itself.

Lemma 11 Assume that H (0), H(1), H(G), H(F), H(ϕ) and H
Proof We use the proof by contradiction. Suppose that for each n ∈ N, it holds (B(0, n)) ⊂ B(0, n). Then, for every n ∈ N, we are able to find (x n , y n ) ∈ B(0, n) and (z n , w n ) ∈ (x n , y n ) (i.e., z n ∈ S(y n ) and w n ∈ T (x n )) such that z n X > n or w n Y > n. (2.14) Therefore, passing to a relabeled subsequence if necessary, we may assume that z n X > n for each n ∈ N (since the proof for the case that w n Y > n for each n ∈ N is similar). Using (2.9), it finds Note that y n Y ≤ n < z n X , so, letting n → ∞ to the inequality above, we have This results in a contradiction. Consequently, there exists a constant M > 0 satisfying (B(0, M)) ⊂ B(0, M).

Proof of Theorem 7.
Observe that if (x * , y * ) is a fixed point of (that is, (x * , y * ) ∈ (x * , y * )), then we have x * ∈ S(y * ) and y * ∈ T (x * ). By the definitions of S and T , it gives Then, it is obvious that (x * , y * ) is also a solution to Problem 1. Based on this important fact, we are going to apply Kakutani-Ky Fan fixed point theorem, Theorem 5, for examining the existence of a fixed point of .
Indeed, it follows from Lemmas 8, 9 and 11 that : B(0, M) → 2 B(0, M) has nonempty, closed and convex values and the graph of is sequentially closed in (X ×Y ) w ×(X ×Y ) w . So, all conditions of Theorem 5 are verified. Using this theorem, we conclude that there exists (x * , y * ) ∈ K × L such that (x * , y * ) ∈ (x * , y * ). Therefore, (x * , y * ) is a solution to Problem 1, that is, ( f , g) = ∅.
From Lemma 10, we can see that ( f , g) is bounded in X × Y . Next, we shall show that ( f , g) is weakly closed. Let {(x n , y n )} ⊂ ( f , g) be such that for some (x, y) ∈ K × L. It is not difficult to see that for each n ∈ N, it holds (x n , y n ) ∈ (x n , y n ). Keeping in mind that is sequentially closed from (X × Y ) w to (X × Y ) w (see Lemmas 8 and 9), we, therefore, imply that (x, y) ∈ (x, y). This means that (x, y) ∈ ( f , g). Consequently, from the boundedness of ( f , g), we conclude that ( f , g) is weakly compact.
Theorem 7 has revealed the nonemptiness and weak compactness of the solution set of Problem 1. Naturally, a problem arises: whether can we prove the uniqueness to Problem 1 under necessary assumptions? The following theorems give a positive answer for the issue. If, in addition, the following inequality holds, , then Problem 1 has a unique solution.
Proof Theorem 7 ensures that ( f , g) = ∅. We now show the uniqueness of Prob- A simple calculation gives This combined with the condition (2.16) implies that x 1 = x 2 and y 1 = y 2 . Therefore, Problem 1 has a unique solution.
The following theorem also provides a uniqueness result for Problem 1 by using an alternative condition to (2.16).

Theorem 13 Assume that H (G), H(F), H(0), H(1), H(ϕ) and H (φ) are satisfied.
If, in addition, the following conditions hold • for each y ∈ Y , the function x → G(y, x) is strongly monotone with constant m G > 0, and for each x ∈ X the function y → G(y, x) is Lipschitz continuous with constant L G > 0, • for each x ∈ X , the function y → F(x, y) is strongly monotone with constant m F > 0, and for each y ∈ Y the function x → F(x, y) is Lipschitz continuous with constant L F > 0, then Problem 1 has a unique solution.
Proof Let (x 1 , y 1 ) and (x 2 , y 2 ) be two solutions to Problem 1. Then, it has Hence, we have Analogously, it gets The last two inequalities imply But, the inequality L G L F m G m F < 1 derives that x 1 = x 2 and y 1 = y 2 . Consequently, Problem 1 has a unique solution.

Stability and Optimal Control for Coupled Variational Inequalities
In the present section, we move our attention to explore the stability and optimal control for coupled variational inequalities. More precisely, we, first, introduce a family regularized problems corresponding to Problem 1 which are perturbated by duality mappings. Then, a stability result, which shows that any sequence of solutions to regularized problems has at least a subsequence to converge to some solution of the original problem, Problem 1, is obtained. Furthermore, we consider an optimal control problem driven by CVI, and prove the solvability of the optimal control problem.
Recall that X and Y are two reflexive Banach spaces, so, they can be renormed such that X and Y become strictly convex. So, without loss of generality, we may assume that X and Y are strictly convex. Let J X : X → X * and J Y : Y → Y * be the duality mappings of the spaces X and Y , respectively, namely: Let real sequences {ε n } and {δ n } be such that ε n > 0, δ n > 0, ε n → 0 and δ n → 0.
(3.1) For each n ∈ N, consider the following perturbated problem corresponding to Problem 1.

Problem 14
Find (x n , y n ) ∈ K × L such that

2)
and We make the following assumptions. H (2): x → G(y, x) and y → F(x, y) are monotone, and satisfy lim sup x → G(y, x) and y → F(x, y) are strongly monotone with constants m G > 0 and m F > 0, respectively, and satisfy lim sup The following theorem delivers the existence and convergence of solutions to Problem 14. where (

x, y) ∈ K × L is a solution of Problem 1; (iii) if, in addition, H (3) holds, then for any sequence of solutions {(x n , y n )} of Problem 14, there exists a subsequence of {(x n , y n )}, still denoted by the same way, such that
where (x, y) ∈ K × L is a solution of Problem 1.
We shall verify that G n and F n satisfy hypotheses H (G) and H (F), respectively. Note that J X is demicontinuous and we use hypotheses H (2) to find that for each y ∈ Y , x → G n (y, x) satisfies H (G)(i). By using the facts, J X (x) X = x X and J X (x), x X = x 2 X for all x ∈ X , it is not difficult to prove that G n enjoys the conditions H (G)(ii)-(iv). Analogously, F n satisfies hypotheses H (F). Therefore, we use Theorem 7 to conclude that Problem 14 admits a solution.
(ii) Let {(x n , y n )} be an arbitrary sequence of solutions of Problem 14. Then, a careful computation gives It could be carried out by using the same arguments as in the proof of Lemma 10 that Taking to a relabeled subsequence if necessary, we may assume that for some (x, y) ∈ K × L. Applying the monotonicity of x → G(y, x) and y →  F(x, y), we have Passing to the upper limit as n → ∞ and using hypotheses H (G)(ii) and H (F)(ii) imply where we have used the boundedness of {(x n , y n )} in X × Y . Employing Minty technique, we conclude that (x, y) ∈ K × L is a solution of Problem 1, i.e., (x, y) ∈ ( f , g).
(iii) It follows from assertion (ii) that for any sequence of solutions {(x n , y n )} of Problem 14, there exists a subsequence of {(x n , y n )}, still denoted by the same way, such that (3.4) is valid. We assert that {(x n , y n )} converges strongly to (x, y). It is not difficult to obtain that Passing to the upper limit as n → ∞ to the inequality above and using hypothesis This means that x n → x in X as n → ∞. Analogically, it has y n → y in Y as n → ∞.
Let Z 1 , Z 2 be two Banach spaces such that the embeddings from X into Z 1 and from Y in Z 2 are both continuous. Given two target profiles x 0 ∈ Z 1 and y 0 ∈ Z 2 , let U and V be subspaces of X * and Y * , respectively, such that the embeddings from U to X * and V to Y * are compact. Next, we focus our attention on the investigation of the following optimal control problem: where the cost function I : U × V → R is defined by Here, ( f , g) is the solution set of Problem 1 associated with ( f , g) ∈ X * × Y * , and ρ > 0, θ > 0 are two regularized parameters.
For the function h, we assume that it reads the following conditions.
We examine the following existence result for Problem 16.
Proof For every ( f , g) ∈ U × V fixed, the closedness of ( f , g) (see Theorem 7) guarantees that there exists (x,ŷ) ∈ ( f , g) such that i.e., inf (x,y)∈ ( f ,g) We assert that the sequence {( f n , g n )} is bounded in U × V . Arguing by contradiction, we suppose that f n U + g n V → +∞ as n → ∞.
The latter together with hypothesis This leads to a contradiction, so, Passing to a relabeled subsequence if necessary, we may assume that for some ( f * , g * ) ∈ U × V . Let sequence {(x n , y n )} ⊂ K × L be such that (3.9) holds by takingx = x n , y = y n , and ( f , g) = ( f n , g n ). Next, we are going to show that and l( y n Y , x n X ) (3.13) Since the embeddings from U to X * and from V to Y * are both continuous, so, we can apply the same arguments as in the proof of Lemma 10 to obtain that {(x n , y n )} ⊂ K ×L is uniformly bounded in X × Y . Without loss of generality, we may suppose that for some (x * , y * ) ∈ K × L. Employing Minty approach derives (3.15) and F(x n , w), w − y n Y + φ(w) − φ(y n ) ≥ g n , w − y n Y for all w ∈ L. (3.16) The compactness of the embedding from (U , V ) into (X * , Y * ) and (3.11) indicate that ( f n , g n ) → ( f * , g * ) in X * × Y * as n → ∞. Passing to the upper limit as n → ∞ for inequalities (3.15)-(3.16), we have where we have applied the conditions H (F)(ii) and H (G)(ii). Using Minty trick again, it leads to (x * , y * ) ∈ ( f * , g * ).
But, the weak lower semicontinuity of · Z 1 and · Z 2 implies (3.17) Recall that h is weakly lower semicontinuous on U × V , it yields This combined with (3.10) concludes that namely ( f * , g * ) is an optimal control of Problem 16.

Applications
The goal of the section is to illustrate the applicability of the theoretical results established in Sections 3 and 4 to the study of two elliptic partial differential systems. The first application is a coupled elliptic mixed boundary value system with nonlocal effect and a multivalued boundary condition which is described by subgradient for a convex superpotential. But, the second application is a feedback control problem involving a least energy condition with respect to the control variable.

A Coupled Mixed Boundary Value System
Given a bounded domain in R N (N ≥ 2) such that its boundary = ∂ is locally Lipschitz and and p ∩ q = ∅ for p, q = a, b, c, p = q, and meas( 1 ) > 0 and meas( a ) > 0. In what follows, we denote by ν the outward unit normal to the boundary . The classical form of the coupled mixed boundary value system is given as follows.

Problem 18
Find functions x : → R and y : → R such that

4)
and for all v ∈ X . Similarly, it also gets for all w ∈ Y . Let us consider the functions G : Y × X → X * and F : and for all x, v ∈ X and y, w ∈ Y . Taking account of (4.12) and (4.13), we get the variational formulation of Problem 18 as follows.

Problem 19
Find functions x ∈ X and y ∈ Y such that for all v ∈ X , and for all w ∈ Y , where the functions ϕ : X → R and φ : Y → R are defined by for all v ∈ X and w ∈ Y , respectively.
We are now in a position to give the existence theorem for Problem 19.

Theorem 20 Assume that H (s), H(t), H(4), H(m 1 ) and H (m 2 ) hold. Then, Problem 19 admits a solution.
Proof We apply Theorem 7 to prove the desired conclusion. For any y ∈ Y fixed, let x 1 , x 2 ∈ X be arbitrary. Note that This combined with the continuity of s, the convergence x n → x in L 2 ( ) and Lebesgue dominated convergence theorem implies For any x ∈ X , using H (s) and H (m 1 ) deduces where we have used Hölder inequality and c 1 > 0 is such that Let us consider the function r : R + → R defined by It is obvious that r fulfills the conditions of H (G)(iii). This means that G satisfies hypothesis H (G)(iii).
For any x ∈ X and y ∈ Y fixed, we apply hypotheses H (m 1 ) and H (s) to find where c 2 > 0 is such that For each v ∈ X and w ∈ Y , we utilize Hölder inequality, hypotheses H (4), and the continuity of the embeddings of X to L 2 ( 3 ) and Y to L 2 ( c ) for getting But, from the definitions of ϕ and φ, we can see that ϕ and φ are both continuous and convex.
Set K = X and L = Y . Therefore, all conditions of Theorem 7 are verified. Employing this theorem, we conclude that the solution set of Problem 19 is nonempty and weakly compact in X × Y .
Since X and Y are Hilbert spaces, so, the duality mappings of X and Y are the identity operators I X in X and I Y in Y , respectively. Let sequences {ε n } and {δ n } satisfy (3.1). We, next, consider the following regularized problem corresponding to Problem 19.
for all v ∈ X , and F(x n , y n ) + δ n I Y (y n ), w − y n Y + φ(w) − φ(y n ) ≥ g 0 , w − y n Y (4.20) for all v ∈ Y .
We invoke Theorem 15 directly to obtain the following existence and convergence results.
where (x, y) ∈ X × Y is a solution of Problem 19.
Let x 0 , y 0 ∈ L 2 ( ). We end the subsection to consider the following optimal control problem.
where the cost function I : U × V → R is defined by Here, ( f , g) is the solution set of Problem 19 associated with ( f , g) ∈ X * × Y * , ρ > 0, θ > 0 are two regularized parameters, and x 0 , y 0 ∈ L 2 ( ) are two given target profiles.
Proof Let U = V = Z 1 = Z 2 = L 2 ( ). It is obvious that the embeddings of U to X * and V to Y * are both continuous and compact. Set h : U × V → R, h( f , g) = f L 2 ( ) + g L 2 ( ) . It is obvious that h( f , g) ≥ 0 for all ( f , g) ∈ U × V , h is coercive on U × V , and h is weakly lower semicontinuous on U × V . We are now in a position to utilize Theorem 17 to conclude that Problem 23 admits an optimal control pair.

An Elliptic Feedback Control System
This subsection is devoted to the investigation of an elliptic mixed boundary value system with distributed control in which the distributed control is described by a least energy equation which explicitly relies on the status variable.
Let in R N (N ≥ 2) be a bounded domain such that its boundary = ∂ is locally Lipschitz and is divided into three measurable and disjoint parts 1 , 2 , and 3 with meas( 1 ) > 0. Let X be the Hilbert space defined in (4.9). The elliptic feedback control system is formulated as follows.

Problem 26
Find functions x ∈ X and y ∈ Y := H 1 0 ( ) such that for all v ∈ X , and P(x, y) ≤ P(x, w) for all w ∈ H 1 0 ( ).
However, it is not difficult to show that Problem 26 is equivalent to the following one.

Conclusion
In this paper, we have introduced and studied a new kind of coupled variational inequalities on Banach spaces. Using Kakutani-Ky Fan fixed point theorem combined with Minty method and the arguments of monotonicity, we delivered the results concerning existence and uniqueness of solution to CVI. Then, we established a stability result for CVI and considered an optimal control problem driven by CVI. Moreover, these theoretical results were applied to explore two complicated elliptic partial differential systems: a coupled elliptic mixed boundary value system with nonlocal effect and a multivalued boundary condition, and a feedback control problem involving a least energy condition with respect to the control variable.
In fact, problems of this type are encountered in transport optimization, Nash equilibrium problem of multiple players, contact mechanics problems, and related fields. In the future, we plan to apply the theoretical results established in the current paper to an Nash equilibrium problem of multiple players, and investigate coupled quasivariational inequalities.