Abstract
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.
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1 Introduction and Mathematical Prerequisites
In numerous complicated natural phenomenon, physical constitutive laws, chemical processes, and economic models are often leaded to inequalities rather than the more commonly seen equations. In this context, variational inequalities, as a powerful mathematical tool, have been widely studied. Essentially speaking, variational inequalities emerge from applied models with an underlying convex structure and have been studied extensively since 1960s. Some representative references include on mathematical theories, numerical treatment and application analysis, see e.g., Kinderlehrer-Stampacchia [18], Migórski-Sofonea-Zeng [34], Glowinski-Lions-Trémoliéres [15], Facchinei-Pang [10], Liu-Motreanu-Zeng [26,27,28], Duvaut-Lions [9], Hlaváček-Haslinger-Nečas-Lovíšek [17], Giannessi [14], Han-Sofonea [16], Fukushima [11, 12], Migórski-Khan-Zeng [35, 36].
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 \((X,\Vert \cdot \Vert _X)\) and \((Y,\Vert \cdot \Vert _Y)\) be two reflexive Banach spaces with its dual spaces \((X^*,\Vert \cdot \Vert _{X^*})\) and \((Y^*,\Vert \cdot \Vert _{Y^*})\), respectively. In what follows, we denote by \(\langle \cdot ,\cdot \rangle _X\) (resp., \(\langle \cdot ,\cdot \rangle _Y\)) 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.
Problem 1
Find \((x,y)\in K\times L\) such that
and
To highlight the general form of our problem, we list the following particular cases of Problem 1
-
(i)
if \(\phi \equiv 0\) and \(L=Y\), then Problem 1 becomes to the coupled system, which consists of a mixed variational inequality and a nonlinear equation: find \((x,y)\in K\times Y\) such that
$$\begin{aligned} \begin{array}{lll} \langle G(y,x),v-x\rangle _X+\varphi (v)-\varphi (x)\ge \langle f,v-x\rangle _X \text{ for } \text{ all } v\in K\\ F(x,y)=g. \end{array} \end{aligned}$$ -
(ii)
if \(\phi \equiv 0\), \(\varphi \equiv 0\), \(L=Y\) and \(K=X\), then Problem 1 becomes to the coupled nonlinear equations: find \((x,y)\in K\times Y\) such that
$$\begin{aligned} \begin{array}{lll} G(y,x)=f\\ F(x,y)=g. \end{array} \end{aligned}$$ -
(iii)
when \(\phi \equiv 0\), \(L=Y\) and F is independent of x, then Problem 1 becomes to the parameter control system: find \(x\in K\) and \(y\in U\) such that
$$\begin{aligned} \langle G(y,x),v-x\rangle _X+\varphi (v)-\varphi (x)\ge \langle f,v-x\rangle _X \text{ for } \text{ all } \, v\in K, \end{aligned}$$where \(U:=\{y\in Y\,\mid \,F(y)=g\}\).
-
(iv)
if G is independent of y, \(F\equiv 0\), \(\phi \equiv 0\) and \(g=0\), then Problem 1 reduces the elliptic variational inequality of the first kind studied in [20, 40]: find \(x\in K\) such that
$$\begin{aligned} \langle G(x),v-x\rangle _X+\varphi (v)-\varphi (x)\ge \langle f,v-x\rangle _X \text{ for } \text{ all } \, v\in K. \end{aligned}$$ -
(v)
when G is independent of y, \(F\equiv 0\), \(\phi \equiv 0\), \(g=0\), and \(K=X\), then Problem 1 reduces the elliptic variational inequality of the second kind investigated in [3, 20, 37]: find \(x\in X\) such that
$$\begin{aligned} \langle G(x),v-x\rangle _X+\varphi (v)-\varphi (x)\ge \langle f,v-x\rangle _X \text{ for } \text{ all } v\in X. \end{aligned}$$
The main contribution of this paper is threefold. First, in Sect. 2, we apply Kakutani-Ky Fan fixed point theorem and Minty approach to show the nonemptiness and compactness of solution set to Problem 1, and then use the arguments of monotonicity to establish two uniqueness results for Problem 1. Second, in Sect. 3, we propose a stability result to Problem 1 and consider an optimal control driven by CVI. Finally, 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 "\( \ {\mathop {\longrightarrow }\limits ^{w}} \ \)" and "\(\rightarrow \)" stand for the weak and the strong convergence, respectively. Let \((X,\Vert \cdot \Vert _X)\) be a Banach space with its dual \(X^*\) and denote by \(\langle \cdot ,\cdot \rangle _X\) the duality pairing between \(X^*\) and X. Recall that a function \(f:X\rightarrow {\overline{{\mathbb {R}}}}:={\mathbb {R}}\cup \{+\infty \}\) 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 \((X,\Vert \cdot \Vert _X)\) be a Banach space. Assume that \(\varphi :X \rightarrow {\overline{{\mathbb {R}}}}\) is convex, l.s.c. and \(\varphi \not \equiv +\infty \). Then, \(\varphi \) is bounded below by an affine continuous function, i.e., there exist \(l\in X^*\) and \(c_\varphi \in {\mathbb {R}}\) such that
Remark 3
It is not difficult to see that if \(\varphi :X \rightarrow {\overline{{\mathbb {R}}}}\) is convex, l.s.c. and \(\varphi \not \equiv +\infty \), then we are able to find constants \(\alpha _\varphi ,\beta _\varphi \ge 0\) such that
Let K be a nonempty subset of X, \(\varphi :K\rightarrow {\overline{{\mathbb {R}}}}\) be a proper convex and l.s.c. function, and \(A:K\rightarrow X^*\). We say that A is
-
(i)
monotone, if it holds \( \langle Au-Av,u-v\rangle _X\ge 0\) for all \(u,v\in K\);
-
(ii)
strictly monotone, if it holds \( \langle Au-Av,u-v\rangle _X> 0\) for all \(u,v\in K\) and \(u\ne v\);
-
(iii)
strongly monotone with constant \(m_A>0\), if it holds \( \langle Au-Av,u-v\rangle _X\ge m_A\Vert u-v\Vert _X^2\) for all \(u,v\in K\);
-
(iv)
pseudomonotone, if for any \(u,v\in K\) we have \(\langle Au,v-u\rangle _X\ge 0\), then it entails that\(\langle Av,v-u\rangle _X\ge 0\);
-
(v)
stable pseudomonotone with respect to the set \(W\subset X^*\), if A and \(u\mapsto Au-w\) are pseudomonotone for all \(w\in W\);
-
(vi)
\(\varphi \)-pseudomonotone, if for any \(u,v\in K\) we have \( \langle Au,v-u\rangle _X+\varphi (v)-\varphi (u)\ge 0\), then it entails that \(\langle Av,v-u\rangle _X+\varphi (v)-\varphi (u)\ge 0\);
-
(vii)
stable \(\varphi \)-pseudomonotone with respect to the set \(W\subset X^*\), if A and \(u\mapsto Au-w\) are \(\varphi \)-pseudomonotone for each \(w\in W\).
Remark 4
The following diagram reveals the essential implications of these monotonicity and generalized monotonicity.
However, it should be mentioned that the inverse direction of each implication relationship in the diagram above does not hold in general (for more details, the reader is welcome to consult the examples, [24, Examples 1 and 2], [25, Example 1], [41, Example 3.1], and [29, Examples 3.4 and 3.5]).
Let Z and Y be topological spaces and \(V\subset Z\) be a nonempty set. In what follows, we denote by \(2^V\) the collection of its subsets. Given a set-valued mapping \(F:Z\rightarrow 2^{Y}\), we use the symbol GrF 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 \(Z\times Y\), if for any sequence \(\{(x_n,y_n)\}\subset Gr F\) is such that
for some \((x,y)\in Z\times Y\), then we have \((x,y)\in Gr F\) (i.e., \(y\in F(x)\)).
Finally, we recall the Kakutani-Ky Fan theorem for a reflexive Banach space, see e.g., [39, Theorem 2.6.7].
Theorem 5
Let Y be a reflexive Banach space and \(D \subseteq Y\) be a nonempty, bounded, closed and convex set. Let \(\Lambda :D \rightarrow 2^D\) be a set-valued map with nonempty, closed and convex values such that its graph is sequentially closed in \(Y_w \times Y_w\) topology. Then, \(\Lambda \) has a fixed point.
2 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 \(S:L\rightarrow 2^K\) and \(T:K\rightarrow 2^L\) defined by
and
respectively.
In order to examine the existence of solutions to Problem 1, we now impose the following assumptions.
\({\underline{H(0)}}\): \(K\subset X\) and \(L\subset Y\) are both nonempty, closed and convex.
\({\underline{H(1)}}\): \(f\in X^*\) and \(g\in Y^*\).
\({\underline{H(\varphi )}}\): \(\varphi :K\rightarrow {\overline{{\mathbb {R}}}}\) is a proper, convex and lower semicontinuous function.
\({\underline{H(G)}}\): \(G:Y\times X\rightarrow X^*\) is such that
-
(i)
for each \(y\in Y\), \(x\mapsto G(y,x)\) is stable \(\varphi \)-pseudomonotone with respect to \(\{f\}\) and satisfies
$$\begin{aligned} \limsup _{\lambda \rightarrow 0}\langle G(y,\lambda v+(1-\lambda )x),v-x\rangle _X\le \langle G(y,x),v-x\rangle _X \end{aligned}$$for all \(y\in Y\) and \(x,v\in X\);
-
(ii)
it holds
$$\begin{aligned} \limsup _{n\rightarrow \infty }\langle G(y_n,v),v-x_n\rangle _X\le \langle G(y,v),v-x\rangle _X, \end{aligned}$$whenever \(v\in X\), \((x,y)\in X\times Y\), \(\{y_n\}\subset Y\) and \(\{x_n\}\subset X\) are such that
$$\begin{aligned} y_n \ {\mathop {\longrightarrow }\limits ^{w}} \ y\text{ in } \, Y \text {and} x_n \ {\mathop {\longrightarrow }\limits ^{w}} \ x \text {in} X \text {as}\,\, n\rightarrow \infty ; \end{aligned}$$ -
(iii)
there exists a function \(r:{\mathbb {R}}_+\times {\mathbb {R}}_+\rightarrow {\mathbb {R}}\) such that
$$\begin{aligned} \langle G(y,x),x\rangle _X\ge r(\Vert x\Vert _X,\Vert y\Vert _Y)\Vert x\Vert _X \text{ for } \text{ all } x\in X \text{ and } y\in Y, \end{aligned}$$and
-
for each nonempty and bounded set \(D\subset {\mathbb {R}}_+\), we have \(r(t,s)\rightarrow +\infty \) as \(t\rightarrow +\infty \) for all \(s\in D\),
-
for any constants \(c_1,c_2\ge 0\), it holds \(r(t,c_1t+c_2)\rightarrow +\infty \) as \(t\rightarrow +\infty \).
-
-
(iv)
there exists a constant \(c_G>0\) such that
$$\begin{aligned} \Vert G(y,x)\Vert _{X^*}\le c_G(1+\Vert x\Vert _X+\Vert y\Vert _Y) \end{aligned}$$for all \((x,y)\in X\times Y\).
\({\underline{H(\phi )}}\): \(\phi :L\rightarrow {\overline{{\mathbb {R}}}}\) is a proper, convex and lower semicontinuous function.
\({\underline{H(F)}}\): \(F:X\times Y\rightarrow Y^*\) is such that
-
(i)
for each \(x\in X\), \(y\mapsto F(x,y)\) is stable \(\phi \)-pseudomonotone with respect to \(\{g\}\) and satisfies
$$\begin{aligned} \limsup _{\lambda \rightarrow 0}\langle F(x,\lambda w+(1-\lambda )y),w-y\rangle _Y\le \langle F(x,y),w-y\rangle _Y \end{aligned}$$for all \(w,y\in Y\) and \(x\in X\);
-
(ii)
it holds
$$\begin{aligned} \limsup _{n\rightarrow \infty }\langle F(x_n,w),w-y_n\rangle _Y\le \langle F(x,w),w-y\rangle _Y, \end{aligned}$$whenever \(w\in Y\), \((x,y)\in X\times Y\), \(\{y_n\}\subset Y\) and \(\{x_n\}\subset X\) are such that
$$\begin{aligned} y_n \ {\mathop {\longrightarrow }\limits ^{w}} \ y \text{ in } Y \text{ and } x_n \ {\mathop {\longrightarrow }\limits ^{w}} \ x \text{ in } X \text{ as } n\rightarrow \infty ; \end{aligned}$$ -
(iii)
there exists a function \(l:{\mathbb {R}}_+\times {\mathbb {R}}_+\rightarrow {\mathbb {R}}\) such that
$$\begin{aligned} \langle F(x,y),y\rangle _Y\ge l(\Vert y\Vert _Y,\Vert x\Vert _X)\Vert y\Vert _Y \text{ for } \text{ all } x\in X \text{ and } y\in Y, \end{aligned}$$and
-
for each nonempty and bounded set \(D\subset {\mathbb {R}}_+\), we have \(l(t,s)\rightarrow +\infty \) as \(t\rightarrow +\infty \) for all \(s\in D\),
-
for any constants \(c_1,c_2\ge 0\), it holds \(l(t,c_1t+c_2)\rightarrow +\infty \) as \(t\rightarrow +\infty \).
-
-
(iv)
there exists a constant \(c_F>0\) such that
$$\begin{aligned} \Vert F(x,y)\Vert _{Y^*}\le c_F(1+\Vert x\Vert _X+\Vert y\Vert _Y) \end{aligned}$$for all \((x,y)\in X\times Y\).
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
-
H(G)(iii)’: there exists a function \(r:{\mathbb {R}}_+\rightarrow {\mathbb {R}}\) with \(r(s)\rightarrow +\infty \) as \(s\rightarrow +\infty \) such that
$$\begin{aligned} \langle G(y,x),x\rangle _X\ge r(\Vert x\Vert _X)\Vert x\Vert _X \text{ for } \text{ all } x\in X \text{ and } y\in Y \end{aligned}$$
(resp. H(F)(iii)’: there exists a function \(l:{\mathbb {R}}_+\rightarrow {\mathbb {R}}\) with \(l(s)\rightarrow +\infty \) as \(s\rightarrow +\infty \) such that
for all \(x\in X\) and \(y\in Y\)).
The first main result of this paper concerning the existence of solutions to Problem 1 is provided as follows.
Theorem 7
Assume that H(G), H(F), H(0), H(1), \(H(\varphi )\) and \(H(\phi )\) are satisfied. Then, the solution set of Problem 1 corresponding to \((f,g)\in X^*\times Y^*\), denoted by \(\Gamma (f,g)\), is nonempty and weakly compact in \(X\times Y\).
To prove this theorem, we need the following lemmas.
Lemma 8
Suppose that H(0), H(1), H(G) and \(H(\varphi )\) are fulfilled. Then, the statements hold
-
(i)
for each \(y\in Y\) fixed, \(x\in K\) is a solution of problem (1.1), if and only if, x solves the following Minty inequality: find \(x\in K\) such that
$$\begin{aligned} \langle G(y,v),v-x\rangle _X+\varphi (v)-\varphi (x)\ge \langle f,v-x\rangle _X \text{ for } \text{ all } \, v\in K; \end{aligned}$$(2.1) -
(ii)
for each \(y\in Y\) fixed, 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 \(S:L\rightarrow 2^K\) is sequentially closed in \(Y_w\times X_w\), i.e., S is sequentially closed from Y endowed with the weak topology into the subsets of X with the weak topology;
-
(iv)
for each \(y\in Y\) fixed, if the mapping \(x\mapsto G(y,x)\) is strictly monotone, then S is a single-valued mapping and weakly continuous.
Proof
The assertions (i) and (ii) are the direct consequences of [24, Theorem 3.3] and [25, Lemma 3.3 and Theorem 3.4].
Next, we show the conclusion (iii). Let \(\{(y_n,x_n)\}\subset Gr S\) be such that
for some \((x,y)\in X\times Y\). Then, for each \(n\in {\mathbb {N}}\), we have \(x_n\in S(y_n)\), i.e.,
for all \(v\in K\). Assertion (i) indicates
for all \(v\in K\). Passing to the upper limit as \(n\rightarrow \infty \) to (2.3), we use hypothesis H(G)(ii) and weak lower semicontinuity of \(\varphi \) (due to the convexity and lower semicontinuity of \(\varphi \)) to find
for all \(v\in K\). Applying assertion (i) again, we conclude that \(x\in S(y)\). Therefore, \((y,x)\in GrS\), namely, the graph of the set-valued mapping \(S:L\rightarrow 2^K\) is sequentially closed in \(Y_w\times X_w\).
Moreover, assume that \(x\mapsto G(y,x)\) is strictly monotone. Let \(x_1,x_2\in K\) be two solutions to problem (1.1). Then, it has
for all \(v\in K\) and \(i=1,2\). 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 \(x\mapsto G(y,x)\) 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. \(\square \)
Likewise, for problem (1.2), we have the following lemma.
Lemma 9
Suppose that H(0), H(1), H(F) and \(H(\phi )\) are fulfilled. Then, the statements hold
-
(i)
for each \(x\in X\) fixed, \(y\in L\) is a solution of problem (1.2), if and only if, y solves the following Minty inequality: find \(y\in L\) such that
$$\begin{aligned} \langle F(x,w),w-y\rangle _Y+\phi (w)-\phi (y)\ge \langle g,w-y\rangle _Y \text{ for } \text{ all } \, w\in L; \end{aligned}$$(2.4) -
(ii)
for each \(x\in X\) 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 \(T:K\rightarrow 2^L\) is sequentially closed in \(X_w\times Y_w\);
-
(iv)
for each \(x\in X\) fixed, if the mapping \(y\mapsto F(x,y)\) 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.
Lemma 10
Assume that H(0), H(1), H(G), H(F), \(H(\varphi )\) and \(H(\phi )\) are fulfilled. If the solution set of Problem 1 is nonempty, thus, \(\Gamma (f,g)\ne \emptyset \), then there exists a constant \(M>0\) such that
for all \((x,y)\in \Gamma (f,g)\).
Proof
Assume that \(\Gamma (f,g)\ne \emptyset \). Let \((x,y)\in \Gamma (f,g)\) be arbitrary and \((x_0,y_0)\in (D(\varphi )\cap K)\times (D(\phi )\cap L)\). Inserting \(v=x_0\) and \(w=y_0\) into (1.1) and (1.2), respectively, we have
and
Recall that \(\varphi \) and \(\phi \) are both proper, convex and l.s.c., it follows from Proposition 2 that there are constants \(\alpha _\varphi ,\alpha _\phi ,\beta _\varphi ,\beta _\phi \ge 0\) satisfying
for all \((v,w)\in X\times Y\) (see Remark 3). Taking account of (2.6)–(2.8), we use hypotheses H(G)(iii)–(iv) and H(F)(iii)–(iv) to obtain
and
Hence,
and
Arguing by contradiction, suppose that \(\Gamma (f,g)\) is unbounded. Then, taking a subsequence if necessary, we are able to find a sequence \(\{(x_n,y_n)\}\subset K\times L\) such that it holds
or
Let us distinguish the following cases:
- ①:
-
(2.11) holds and \(\{y_n\}\) is bounded in Y;
- ②:
-
(2.12) holds and \(\{x_n\}\) is bounded in X;
- ③:
Assume that ① is valid, then we take \(x=x_n\) and \(y=y_n\) to (2.9) for getting
Letting \(n\rightarrow \infty \) 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)
\(\frac{\Vert y_n\Vert _Y}{\Vert x_n\Vert _X}\rightarrow +\infty \) as \(n\rightarrow \infty \);
-
(b)
there exist \(n_0\in {\mathbb {N}}\) and \({\widehat{c}}_0>0\) such that \(\frac{\Vert y_n\Vert _Y}{\Vert x_n\Vert _X}\le {\widehat{c}}_0\) for all \(n\ge n_0\).
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 \(n\rightarrow \infty \) for the inequality above, it gives
Obviously, it is impossible, whereas in terms of the situation (b), it follows from (2.9) that
for \(n\ge n_1\), where \(n_1\ge n_0\) is such that \(\Vert x_{n_1}\Vert _X>1\). This also triggers a contradiction.
To summary, we conclude that \(\Gamma (f,g)\) is bounded in \(X\times Y\). Consequently, we are able to find a constant \(M>0\) such that (2.5) is valid. \(\square \)
Consider the set-valued mapping \(\Lambda :K\times L\rightarrow 2^{K\times L}\) given by
Invoking Lemmas 8 and 9, we can see that \(\Lambda \) is well-defined. The following lemma points out that there exists a bounded, closed and convex set D in \(K\times L\) such that \(\Lambda \) maps D into itself.
Lemma 11
Assume that H(0), H(1), H(G), H(F), \(H(\varphi )\) and \(H(\phi )\) are fulfilled. Then, there exists a constant \({\widehat{M}}>0\) satisfying \(\Lambda (\overline{B(0,{\widehat{M}})})\subset \overline{B(0,{\widehat{M}})}\), where \(\overline{B(0,{\widehat{M}})}\subset X\times Y\) is defined by
Proof
We use the proof by contradiction. Suppose that for each \(n\in {\mathbb {N}}\), it holds \(\Gamma (\overline{B(0,n)})\not \subset \overline{B(0,n)}\). Then, for every \(n\in {\mathbb {N}}\), we are able to find \((x_n,y_n)\in \overline{B(0,n)}\) and \((z_n,w_n)\in \Gamma (x_n,y_n)\) (i.e., \(z_n\in S(y_n)\) and \(w_n\in T(x_n)\)) such that
Therefore, passing to a relabeled subsequence if necessary, we may assume that \(\Vert z_n\Vert _X>n\) for each \(n\in {\mathbb {N}}\) (since the proof for the case that \(\Vert w_n\Vert _Y>n\) for each \(n\in {\mathbb {N}}\) is similar). Using (2.9), it finds
Note that \(\Vert y_n\Vert _Y\le n<\Vert z_n\Vert _X\), so, letting \(n\rightarrow \infty \) to the inequality above, we have
This results in a contradiction. Consequently, there exists a constant \({\widehat{M}}>0\) satisfying \(\Lambda (\overline{B(0,{\widehat{M}})})\subset \overline{B(0,{\widehat{M}})}\). \(\square \)
Proof of Theorem 7
Observe that if \((x^*,y^*)\) is a fixed point of \(\Lambda \) (that is, \((x^*,y^*)\in \Lambda (x^*,y^*)\)), then we have \(x^*\in S(y^*)\) and \(y^*\in T(x^*)\). By the definitions of S and T, it gives
and
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 \(\Lambda \).
Indeed, it follows from Lemmas 8, 9 and 11 that \(\Lambda :\overline{B(0,{\widehat{M}})}\rightarrow 2^{\overline{B(0,{\widehat{M}})}}\) has nonempty, closed and convex values and the graph of \(\Lambda \) is sequentially closed in \((X\times Y)_w\times (X\times Y)_w\). So, all conditions of Theorem 5 are verified. Using this theorem, we conclude that there exists \((x^*,y^*)\in K\times L\) such that \((x^*,y^*)\in \Lambda (x^*,y^*)\). Therefore, \((x^*,y^*)\) is a solution to Problem 1, that is, \(\Gamma (f,g)\ne \emptyset \).
From Lemma 10, we can see that \(\Gamma (f,g)\) is bounded in \(X\times Y\). Next, we shall show that \(\Gamma (f,g)\) is weakly closed. Let \(\{(x_n,y_n)\}\subset \Gamma (f,g)\) be such that
for some \((x,y)\in K\times L\). It is not difficult to see that for each \(n\in {\mathbb {N}}\), it holds \((x_n,y_n)\in \Lambda (x_n,y_n)\). Keeping in mind that \(\Lambda \) is sequentially closed from \((X\times Y)_w\) to \((X\times Y)_w\) (see Lemmas 8 and 9), we, therefore, imply that \((x,y)\in \Lambda (x,y)\). This means that \((x,y)\in \Gamma (f,g)\). Consequently, from the boundedness of \(\Gamma (f,g)\), we conclude that \(\Gamma (f,g)\) is weakly compact. \(\square \)
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.
Theorem 12
Assume that H(G), H(F), H(0), H(1), \(H(\varphi )\) and \(H(\phi )\) are satisfied. If, in addition, the following inequality holds,
for all \((x_1,y_1),(x_2,y_2)\in X\times Y\) with \((x_1,y_1)\ne (x_2,y_2)\), then Problem 1 has a unique solution.
Proof
Theorem 7 ensures that \(\Gamma (f,g)\ne \emptyset \). We now show the uniqueness of Problem 1. Let \((x_1,y_1),(x_2,y_2)\in \Gamma (f,g)\). Then, we have
and
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. \(\square \)
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), \(H(\varphi )\) and \(H(\phi )\) are satisfied. If, in addition, the following conditions hold
-
for each \(y\in Y\), the function \(x\mapsto G(y,x)\) is strongly monotone with constant \(m_G>0\), and for each \(x\in X\) the function \(y\mapsto G(y,x)\) is Lipschitz continuous with constant \(L_G>0\),
-
for each \(x\in X\), the function \(y\mapsto F(x,y)\) is strongly monotone with constant \(m_F>0\), and for each \(y\in Y\) the function \(x\mapsto F(x,y)\) is Lipschitz continuous with constant \(L_F>0\),
-
\(\displaystyle \frac{L_GL_F}{m_Gm_F}<1\),
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 \(\frac{L_GL_F}{m_Gm_F}<1\) derives that \(x_1=x_2\) and \(y_1=y_2\). Consequently, Problem 1 has a unique solution. \(\square \)
3 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 \(J_X:X\rightarrow X^*\) and \(J_Y:Y\rightarrow Y^*\) be the duality mappings of the spaces X and Y, respectively, namely:
Let real sequences \(\{\varepsilon _n\}\) and \(\{\delta _n\}\) be such that
For each \(n\in {\mathbb {N}}\), consider the following perturbated problem corresponding to Problem 1.
Problem 14
Find \((x_n,y_n)\in K\times L\) such that
and
We make the following assumptions.
\({\underline{H(2)}}\): \(x\mapsto G(y,x)\) and \(y\mapsto F(x,y)\) are monotone, and satisfy
for all \(w,y\in Y\) and \(v,x\in X\).
\({\underline{H(3)}}\): \(x\mapsto G(y,x)\) and \(y\mapsto F(x,y)\) are strongly monotone with constants \(m_G>0\) and \(m_F>0\), respectively, and satisfy
for all \(w,y\in Y\) and \(v,x\in X\).
The following theorem delivers the existence and convergence of solutions to Problem 14.
Theorem 15
Assume that H(G)(ii)–(iv), H(F)(ii)–(iv), H(0), H(1), \(H(\varphi )\) and \(H(\phi )\) hold. Then, we have
-
(i)
if, in addition, H(2) holds, then for each \(n\in {\mathbb {N}}\), Problem 14 has at least a solution \((x_n,y_n)\in K\times L\);
-
(ii)
if, in addition, H(2) 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
$$\begin{aligned} (x_n,y_n) \ {\mathop {\longrightarrow }\limits ^{w}} \ (x,y) \text{ in } X\times Y \text{ as } n\rightarrow \infty , \end{aligned}$$(3.4)where \((x,y)\in K\times L\) 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
$$\begin{aligned} (x_n,y_n)\rightarrow (x,y) \text{ in } X\times Y \text{ as } n\rightarrow \infty , \end{aligned}$$(3.5)where \((x,y)\in K\times L\) is a solution of Problem 1.
Proof
(i) Set \(G_n(y,x)=G(y,x)+\varepsilon _nJ_X(x)\) and \(F_n(x,y)=F(x,y)+\delta _nJ_Y(y)\) for all \((x,y)\in X\times Y\). 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 \(y\in Y\), \(x\mapsto G_n(y,x)\) satisfies H(G)(i). By using the facts, \(\Vert J_X(x)\Vert _X=\Vert x\Vert _X\) and \(\langle J_X(x),x\rangle _X=\Vert x\Vert _X^2\) for all \(x\in X\), 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
and
It could be carried out by using the same arguments as in the proof of Lemma 10 that \(\{(x_n,y_n)\}\) is bounded in \(X\times Y\).
Taking to a relabeled subsequence if necessary, we may assume that
for some \((x,y)\in K\times L\). Applying the monotonicity of \(x\mapsto G(y,x)\) and \(y\mapsto F(x,y)\), we have
and
Passing to the upper limit as \(n\rightarrow \infty \) and using hypotheses H(G)(ii) and H(F)(ii) imply
and
where we have used the boundedness of \(\{(x_n,y_n)\}\) in \(X\times Y\). Employing Minty technique, we conclude that \((x,y)\in K\times L\) is a solution of Problem 1, i.e., \((x,y)\in \Gamma (f,g)\).
(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 \(n\rightarrow \infty \) to the inequality above and using hypothesis H(G)(ii), we get
This means that \(x_n\rightarrow x\) in X as \(n\rightarrow \infty \). Analogically, it has \(y_n\rightarrow y\) in Y as \(n\rightarrow \infty \).
\(\square \)
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 \(x_0\in Z_1\) and \(y_0\in Z_2\), 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:
Problem 16
Find \((f^*,g^*)\in U\times V\) such that
where the cost function \(I:U\times V\rightarrow {\mathbb {R}}\) is defined by
Here, \(\Gamma (f,g)\) is the solution set of Problem 1 associated with \((f,g)\in X^*\times Y^*\), and \(\rho >0\), \(\theta >0\) are two regularized parameters.
For the function h, we assume that it reads the following conditions.
\({\underline{H(h)}}\): \(h:U\times V\rightarrow {\mathbb {R}}\) is such that
-
(i)
h is bounded from below;
-
(ii)
h is coercive on \(U\times V\), namely it holds
$$\begin{aligned} \lim _{(f,g)\in U\times V,\Vert f\Vert _U+\Vert g\Vert _V\rightarrow \infty }h(f,g)\rightarrow +\infty ; \end{aligned}$$ -
(iii)
h is weakly lower semicontinuous on \(U\times V\), i.e., \(\liminf _{n\rightarrow \infty }h(f_n,g_n)\ge h(f,g)\), whenever \(\{(f_n,g_n)\}\subset U\times V\) and \((f,g)\in U\times V\) are such that \((f_n,g_n) \ {\mathop {\longrightarrow }\limits ^{w}} \ (f,g)\) in \(U\times V\) as \(n\rightarrow \infty \).
We examine the following existence result for Problem 16.
Theorem 17
Assume that H(G)(ii)–(iv), H(F)(ii)–(iv), H(0), H(1), \(H(\varphi )\) and \(H(\phi )\) hold. If, in addition, H(h) and H(2) are fulfilled, then Problem 16 admits an optimal control pair.
Proof
For every \((f,g)\in U\times V\) fixed, the closedness of \(\Gamma (f,g)\) (see Theorem 7) guarantees that there exists \(({\hat{x}},{\hat{y}})\in \Gamma (f,g)\) such that
i.e., \(\inf _{(x,y)\in \Gamma (f,g)}\left( \frac{\rho }{2}\Vert x-x_0\Vert _{Z_1}^2+\frac{\theta }{2}\Vert y-y_0\Vert _{Z_2}^2\right) \) is attainable.
It follows from the definition of I and hypothesis H(h)(i) that there exists a minimizing sequence \(\{(f_n,g_n)\}\subset U\times V\) such that
We assert that the sequence \(\{(f_n,g_n)\}\) is bounded in \(U\times V\). Arguing by contradiction, we suppose that
The latter together with hypothesis H(h)(ii) deduces
This leads to a contradiction, so, \(\{(f_n,g_n)\}\) is bounded in \(U\times V\). Passing to a relabeled subsequence if necessary, we may assume that
for some \((f^*,g^*)\in U\times V\).
Let sequence \(\{(x_n,y_n)\}\subset K\times L\) be such that (3.9) holds by taking \({\hat{x}}=x_n\), \({\hat{y}}=y_n\), and \((f,g)=(f_n,g_n)\). Next, we are going to show that \(\{(x_n,y_n)\}\subset K\times L\) is uniformly bounded in \(X\times Y\). A direct computation finds
and
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 \(\{(x_n,y_n)\}\subset K\times L\) is uniformly bounded in \(X\times Y\). Without loss of generality, we may suppose that
for some \((x^*,y^*)\in K\times L\). Employing Minty approach derives
and
The compactness of the embedding from (U, V) into \((X^*,Y^*)\) and (3.11) indicate that \((f_n,g_n)\rightarrow (f^*,g^*)\) in \(X^*\times Y^*\) as \(n\rightarrow \infty \). Passing to the upper limit as \(n\rightarrow \infty \) for inequalities (3.15)–(3.16), we have
and
where we have applied the conditions H(F)(ii) and H(G)(ii). Using Minty trick again, it leads to \((x^*,y^*)\in \Gamma (f^*,g^*)\).
But, the weak lower semicontinuity of \(\Vert \cdot \Vert _{Z_1}\) and \(\Vert \cdot \Vert _{Z_2}\) implies
Recall that h is weakly lower semicontinuous on \(U\times V\), it yields
Taking account of (3.17) and (3.18), we have
This combined with (3.10) concludes that
namely \((f^*,g^*)\) is an optimal control of Problem 16. \(\square \)
4 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.
4.1 A Coupled Mixed Boundary Value System
Given a bounded domain \(\Omega \) in \({\mathbb {R}}^N\) \((N\ge 2)\) such that its boundary \(\Gamma =\partial \Omega \) is locally Lipschitz and
with \(\Gamma _i\cap \Gamma _j=\emptyset \) for \(i,j=1,2,3\), \(i\ne j\), and \(\Gamma _p\cap \Gamma _q=\emptyset \) for \(p,q=a,b,c\), \(p\ne q\), and meas\((\Gamma _1)>0\) and meas\((\Gamma _a)>0\). In what follows, we denote by \(\nu \) the outward unit normal to the boundary \(\Gamma \). The classical form of the coupled mixed boundary value system is given as follows.
Problem 18
Find functions \(x:\Omega \rightarrow {\mathbb {R}}\) and \(y:\Omega \rightarrow {\mathbb {R}}\) such that
and
Next, let us introduce two subspaces X and Y of \(H^1(\Omega )\), which are defined by
It is not difficult to prove that X endowed with the inner product and the corresponding norm
for all \(x,v\in X\), is a separable Hilbert space. Also, Y is a separable Hilbert space with the norm and inner product by
for all \(y,w\in Y\), respectively. Denote by \(\gamma _1:X\rightarrow L^2(\Omega )\) (resp., \(\gamma _2:Y\rightarrow L^2(\Omega )\)) the embedding operator from X to \(L^2(\Omega )\) (resp., the embedding operator from Y to \(L^2(\Omega )\)).
We make the following assumptions.
\({\underline{H(s)}}\): \(s:\Omega \times {\mathbb {R}}\rightarrow {\mathbb {R}}\) is a Carathéodory function such that \(\theta \mapsto s(z,\theta ) \) is monotone for a.e. \(z\in \Omega \) and there exist \(c_s\in L^2(\Omega )_+\) and \(d_s>0\) satisfying
\({\underline{H(t)}}\): \(t:\Omega \times {\mathbb {R}}\rightarrow {\mathbb {R}}\) is a Carathéodory function such that \(\theta \mapsto t(z,\theta ) \) is monotone for a.e. \(z\in \Omega \) and there exist \(c_t\in L^2(\Omega )_+\) and \(d_t>0\) satisfying
\({\underline{H(4)}}\): \(f_0,g_0\in L^2(\Omega )\), \(f_1\in L^2(\Gamma _2)\), \(g_1\in L^2(\Gamma _b)\), \(k_1\in L^2(\Gamma _3)_+\) and \(k_2\in L^2(\Gamma _c)_+\).
\({\underline{H(m_1)}}\): \(m_1:{\mathbb {R}}_+\rightarrow {\mathbb {R}}_+\) is a continuous function such that there exist constants \(0<c_{m_1}<d_{m_1}<+\infty \) satisfying
\({\underline{H(m_2)}}\): \(m_2:{\mathbb {R}}_+\rightarrow {\mathbb {R}}_+\) is a continuous function such that there exist constants \(0<c_{m_2}<d_{m_2}<+\infty \) satisfying
Let \((x,y)\in X\times Y\) be such that (4.1)–(4.4) hold. For any \(v\in X\) fixed, we multiply (4.1) by \(v-x\) and use Green’s formula to find
Note that
it follows from the boundary conditions (4.2) and (4.3) that
By virtue of the definition of convex subgradient, boundary condition (4.4) can be rewritten to the following inclusion form
where the term \(\partial _c|\theta |\) is the convex subdifferential operator of the modulus function \({\mathbb {R}}\ni \theta \mapsto |\theta |\in {\mathbb {R}}_+\). Therefore, it has
Combining (4.10)–(4.11), we have
for all \(v\in X\). Similarly, it also gets
for all \(w\in Y\).
Let us consider the functions \(G:Y\times X\rightarrow X^*\) and \(F:X\times Y\rightarrow Y^*\) defined by
and
for all \(x,v\in X\) and \(y,w\in Y\).
Taking account of (4.12) and (4.13), we get the variational formulation of Problem 18 as follows.
Problem 19
Find functions \(x\in X\) and \(y\in Y\) such that
for all \(v\in X\), and
for all \(w\in Y\), where the functions \(\varphi :X\rightarrow {\mathbb {R}}\) and \(\phi :Y\rightarrow {\mathbb {R}}\) are defined by
for all \(v\in X\) and \(w\in Y\), 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 \(y\in Y\) fixed, let \(x_1,x_2\in X\) be arbitrary. Note that
it follows from hypotheses H(s) and \(H(m_1)\) that \(x\mapsto G(y,x)\) is monotone. Besides, the continuity of s implies that for each \(y\in Y\) the function \(x\mapsto G(y,x)\) is continuous too. So, condition H(G)(i) is valid. Let sequences \(\{x_n\}\subset X\) and \(\{y_n\}\subset Y\) be such that
for some \((x,y)\in X\times Y\). Recall that the embedding from Y into \(L^2(\Omega )\) is compact, we have \(y_n\rightarrow y\) and \(x_n\rightarrow x\) in \(L^2(\Omega )\) as \(n\rightarrow \infty \). Then, we have
This combined with the continuity of s, the convergence \(x_n\rightarrow x\) in \(L^2(\Omega )\) and Lebesgue dominated convergence theorem implies
namely H(G)(ii) is available.
For any \(x\in X\), using H(s) and \(H(m_1)\) deduces
where we have used Hölder inequality and \({{\widetilde{c}}}_1>0\) is such that
Let us consider the function \(r:{\mathbb {R}}_+\rightarrow {\mathbb {R}}\) 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 \(x\in X\) and \(y\in Y\) fixed, we apply hypotheses \(H(m_1)\) and H(s) to find
Employing Hölder inequality finds
It is not difficult to see that condition H(G)(iv) is valid with the constant \(c_G>0\) defined by
Additionally, we can verify that F fulfills hypotheses H(F) with the function \(l:{\mathbb {R}}_+\rightarrow {\mathbb {R}}\)
and constant \(c_F>0\)
where \({{\widetilde{c}}}_2>0\) is such that
For each \(v\in X\) and \(w\in Y\), we utilize Hölder inequality, hypotheses H(4), and the continuity of the embeddings of X to \(L^2(\Gamma _3)\) and Y to \(L^2(\Gamma _c)\) for getting
But, from the definitions of \(\varphi \) and \(\phi \), we can see that \(\varphi \) and \(\phi \) 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 \(X\times Y\). \(\square \)
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 \(\{\varepsilon _n\}\) and \(\{\delta _n\}\) satisfy (3.1). We, next, consider the following regularized problem corresponding to Problem 19.
Problem 21
Find \((x_n,y_n)\in X\times Y\) such that
for all \(v\in X\), and
for all \(v\in Y\).
We invoke Theorem 15 directly to obtain the following existence and convergence results.
Theorem 22
Suppose that H(s), H(t), H(4), \(H(m_1)\), \(H(m_2)\) and (3.1) are fulfilled. Then, we have
-
(i)
for each \(n\in {\mathbb {N}}\), Problem 21 has at least a solution \((x_n,y_n)\in X\times Y\);
-
(ii)
for any sequence of solutions \(\{(x_n,y_n)\}\) of Problem 21, there exists a subsequence of \(\{(x_n,y_n)\}\), still denoted by the same way, such that
$$\begin{aligned} (x_n,y_n)\rightarrow (x,y) \text{ in } X\times Y \text{ as } n\rightarrow \infty , \end{aligned}$$where \((x,y)\in X\times Y\) is a solution of Problem 19.
Let \(x_0,y_0\in L^2(\Omega )\). We end the subsection to consider the following optimal control problem.
Problem 23
Find \((f^*,g^*)\in L^2(\Omega )\times L^2(\Omega )\) such that
where the cost function \(I:U\times V\rightarrow {\mathbb {R}}\) is defined by
Here, \(\Gamma (f,g)\) is the solution set of Problem 19 associated with \((f,g)\in X^*\times Y^*\), \(\rho >0\), \(\theta >0\) are two regularized parameters, and \(x_0,y_0\in L^2(\Omega )\) are two given target profiles.
Theorem 24
Suppose that H(s), H(t), H(4), \(H(m_1)\) and \(H(m_2)\) are fulfilled. Then Problem 23 admits an optimal control pair.
Proof
Let \(U=V=Z_1=Z_2=L^2(\Omega )\). It is obvious that the embeddings of U to \(X^*\) and V to \(Y^*\) are both continuous and compact. Set \(h:U\times V\rightarrow {\mathbb {R}}\), \(h(f,g)=\Vert f\Vert _{L^2{(\Omega )}}+\Vert g\Vert _{L^2(\Omega )}\). It is obvious that \(h(f,g)\ge 0\) for all \((f,g)\in U\times V\), h is coercive on \(U\times V\), and h is weakly lower semicontinuous on \(U\times V\). We are now in a position to utilize Theorem 17 to conclude that Problem 23 admits an optimal control pair. \(\square \)
4.2 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 \(\Omega \) in \({\mathbb {R}}^N\) \((N\ge 2)\) be a bounded domain such that its boundary \(\Gamma =\partial \Omega \) is locally Lipschitz and is divided into three measurable and disjoint parts \(\Gamma _1\), \(\Gamma _2\), and \(\Gamma _3\) with \(meas(\Gamma _1)>0\). Let X be the Hilbert space defined in (4.9). The elliptic feedback control system is formulated as follows.
Problem 25
Find functions \(x:\Omega \rightarrow {\mathbb {R}}\) and \(y:\Omega \rightarrow {\mathbb {R}}\) such that
where the control variable \(y\in H^1_0(\Omega )\) satisfies the following least energy condition
and P(x, y) is defined by
\({\underline{H(\psi )}}\): \(\psi :{\mathbb {R}}\rightarrow {\mathbb {R}}\) is a convex and lower semicontinuous function.
Using a standard procedure, we obtain the variational formulation of Problem 25 as follows:
Problem 26
Find functions \(x\in X\) and \(y\in Y:=H^1_0(\Omega )\) such that
for all \(v\in X\), and
However, it is not difficult to show that Problem 26 is equivalent to the following one.
Problem 27
Find functions \(x\in X\) and \(y\in Y\) such that
for all \(v\in X\), and
for all \(w\in H^1_0(\Omega )\).
Arguing as in the proof of Theorem 20, we have the following existence result for Problem 27.
Theorem 28
Assume that H(s), \(H(\psi )\), \(f_1\in L^2(\Gamma _2)\), and \(h_0\in L^2(\Omega )\) hold. Then, Problem 27 admits a solution.
5 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 quasi-variational inequalities.
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Acknowledgements
This project has received funding from the NNSF of China Grant Nos. 12001478, 12026255, 12026256, 12001072, 11971084, 11991024 and 11991020, and the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Skłodowska-Curie grant agreement No. 823731 CONMECH, National Science Center of Poland under Preludium Project No. 2017/25/N/ST1 /00611, and the Startup Project of Doctor Scientific Research of Yulin Normal University No. G2020ZK07. It is also supported by Natural Science Foundation of Guangxi Grant Nos. 2021GXNSFFA196004 and 2020GXNSFBA297137, and the Ministry of Science and Higher Education of Republic of Poland under Grants Nos. 4004/GGPJII/H2020/2018/0 and 440328/PnH2/2019. The forth author is supported by the China Postdoctoral Science Foundation Project (No. 2019M653332), and the Chongqing Natural Science Foundation Project (No. cstc2019jcyj-bshX0092).
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Communicated by Massimo Pappalardo.
Dedicated to Professor Franco Giannessi on the occasion of his 85th birthday.
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Liu, J., Yang, X., Zeng, S. et al. Coupled Variational Inequalities: Existence, Stability and Optimal Control. J Optim Theory Appl 193, 877–909 (2022). https://doi.org/10.1007/s10957-021-01995-9
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DOI: https://doi.org/10.1007/s10957-021-01995-9
Keywords
- Coupled variational inequality
- Existence
- Uniqueness
- Optimal control
- Mixed boundary value system
- Feedback control problem