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

Lei Duan; Li-jun Zhu; Zhenhua He; Lei Duan; Li-jun Zhu; Zhenhua He

doi:10.3934/era.2026118

Electronic Research Archive

2026, Volume 34, Issue 4: 2548-2571. doi: 10.3934/era.2026118

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Research article

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

1.
School of Mathematics and Information Science, North Minzu University, Yinchuan 750021, China
2.
The Key Laboratory of Intelligent Information and Big Data Processing of NingXia Province, North Minzu University, Yinchuan 750021, China
3.
College of Innovation and Entrepreneurship, Xinhua College of Ningxia University, Yinchuan 750021, China

Received: 09 January 2026 Revised: 15 February 2026 Accepted: 26 February 2026 Published: 24 March 2026

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.
- hemivariational inequality,
- Clarke generalized gradient,
- asymptotic analysis,
- optimal control,
- nonsmooth analysis
Citation: Lei Duan, Li-jun Zhu, Zhenhua He. Asymptotic convergence of solutions to elliptic hemivariational inequalities and optimal control problem[J]. Electronic Research Archive, 2026, 34(4): 2548-2571. doi: 10.3934/era.2026118

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Abstract

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.

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