Necessary optimality conditions for control of strongly monotone variational inequalities (Q2712218)

The author considers the following optimal control of the strongly monotone variational inequality: NEWLINE\[NEWLINE\min J(y,u)\;\text{s.t. } y\in K,\;u\in U_{\text{ad}} \tag{OCVI}NEWLINE\]NEWLINE NEWLINE\[NEWLINE\langle F(y,u), y'- y\rangle\geq 0\quad\forall y'\in K,NEWLINE\]NEWLINE where \(K\) and \(U_{\text{ad}}\) are closed and convex subsets of Asplund spaces \(V\) and \(U\), respectively, \(U_{\text{ad}}\) is contained in a finite-codimensional closed subspace of \(U\) with the nonempty interior with respect to this suubspace, \(J: V\times U_{\text{ad}}\to \mathbb{R}\) is Lipschitz near the solution \((\overline y,\overline u)\) of (OCVI), \(F: V\times U_{\text{ad}}\to V^*\) is strictly differentiable at \((\overline y,\overline u)\) and locally strongly monotone in \(y\) uniformly in \(u\).NEWLINENEWLINENEWLINEThis problem is a generalization of those proposed in [\textit{J. L. Lions}, CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics. VI, 92 p. (1972; Zbl 0275.49001)] and studied in [\textit{V. Barbu}, ``Optimal control of variational inequalities'' (1984; Zbl 0574.49005)].NEWLINENEWLINENEWLINEExploiting the notion of a limiting subgradient and the coderivative of a set-valued map, the author proves a necessary condition for a pair \((\overline y,\overline u)\) to be a solution of (OCVI).NEWLINENEWLINEFor the entire collection see [Zbl 0958.00050].