Simultaneous distributed-boundary optimal control problems driven by nonlinear complementarity systems

Abstract

The primary goal of this paper is to study a nonlinear complementarity system (NCS, for short) with a nonlinear and nonhomogeneous partial differential operator and mixed boundary conditions, and a simultaneous distributed-boundary optimal control problem governed by (NCS), respectively. First, we formulate the weak formulation of (NCS) to a mixed variational inequality with double obstacle constraints (MVI, for short), and prove the existence and uniqueness of solution to (MVI). Then, a power penalty method is applied to (NCS) for introducing an approximating mixed variational inequality without constraints (AMVI, for short). After that, a convergence result that the unique solution of (MVI) can be approached by the unique solution of (AMVI) when a penalty parameter tends to infinity, is established. Moreover, we explore the solvability of the simultaneous distributed-boundary optimal control problem described by (MVI), and consider a family of approximating optimal control problems driven by (AMVI). Finally, we provide a result on asymptotic behavior of optimal controls, system states and minimal values to approximating optimal control problems.

A New Viscosity Implicit Approximation Method for Solving Variational Inequalities over the Common Fixed Points of Nonexpansive Mappings in Symmetric Hilbert Space

“…Hartman and Stampacchia [1] proposed and established the initial theory of variational inequality in 1964. Since then, scholars have carried out extensive research on variational inequality that covers a wide range of disciplines, including optimization, optimal control, mechanics, and finance (see, e.g., [2][3][4][5]). In the theory of variational inequalities, an important and interesting problem is determination of the approximate solutions of variational inequalities by creating a feasible and effective iterative algorithm.…”