Well-Posedness, Optimal Control, and Sensitivity Analysis for a Class of Differential Variational-Hemivariational Inequalities
Shengda Zeng, Stanisław Migórski, Zhenhai Liu
Abstract
<p>The objective of the paper is to investigate a dynamical system called a differential variational-hemivariational inequality (DVHVI) which couples an abstract variational-hemivariational inequality of elliptic type and a nonlinear evolution inclusion problem in a Banach space. Under appropriate assumptions, the nonemptiness and compactness of the solution set for DVHVI are established<br>by using the Fan–Knaster–Kuratowski–Mazurkiewicz principle, the Minty approach, and the methods of nonsmooth analysis. Then, we explore properties of solution mapping for DVHVI which involve the relative compactness, continuity, and convergence in the Kuratowski sense. Employing these properties, we prove existence of a solution to the optimal control problem driven by a DVHVI. Next, well-posedness results for DVHVI are obtained, including the existence, uniqueness, and stability of the solution. Furthermore, we study sensitivity of a perturbed problem with multiparameters corresponding to DVHVI. Finally, a comprehensive parabolic-elliptic system with obstacle effect is considered as an illustrative application</p>