Existence and necessary optimality conditions for nonlinear inverse problems governed by quasi-hemivariational inequalities with parameter-dependent constraints

This paper investigates nonlinear inverse problems governed by a class of evolutionary quasi-hemivariational inequalities featuring constraints that depend simultaneously on parameters and unknowns. We begin by establishing the well-posedness of the parametric forward problem within the framework of evolution triples, providing existence, uniqueness, and a priori estimates for the state system. The associated inverse problem is subsequently formulated as a Tikhonov-regularized output-least-squares optimization problem, where the existence of at least one optimal parameter is proved utilizing the direct method of the calculus of variations. A primary analytical contribution of this work is the derivation of necessary optimality conditions in the presence of both nonsmoothness and state constraints with potentially empty interiors. To overcome these inherent challenges, we employ a rigorous double-approximation strategy combining smoothing and penalization techniques. This methodology leads to the derivation of an adjoint system and a first-order variational inequality associated with a singular Lagrange multiplier. Finally, to demonstrate the utility and applicability of our abstract theoretical framework, we apply our results to a nonlinear parabolic quasi-obstacle problem. Within this context, the abstract framework is verified in appropriate Sobolev spaces, and explicit optimality conditions are successfully derived.