Asymptotic convergence of solutions to elliptic hemivariational inequalities and optimal control problem

This paper investigates a class of elliptic hemivariational inequalities (HVIs) arising from steady-state heat conduction problems that involve linear and nonlinear mixed boundary effects and explores their applications to optimal control problems. First, we use nonsmooth analysis and pseudomonotone operator theory to establish the existence of solutions of the HVIs for the model. Under a relaxed monotonicity condition, we obtain uniqueness of the solution. Second, we prove that the solutions of the HVIs converge strongly to the solution of the corresponding Dirichlet boundary problem as the coefficient $\lambda$ tends to infinity. Finally, a convergence analysis is provided for the solutions of the optimal control problem governed by HVIs. The results provide a theoretical framework for the study of elliptic problems with nonmonotone and multivalued boundary conditions, and they contribute to the mathematical foundation of related control and engineering applications.