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<p>This paper investigates an optimal control problem governed by a differential hemivariational inequality (DHVI), which arises as the weak formulation of a nonstationary thermistor system with mixed multivalued and nonmonotone boundary conditions. We first establish the nonemptiness and weak compactness of the solution set to the DHVI by employing pseudomonotone operator theory, tools from nonsmooth analysis, and a fixed point approach. We then analyze the associated optimal control problem, focusing on its sensitivity with respect to the initial condition, the electric conductivity and a parameter in a metric space. Using the Kuratowski convergence and techniques from multivalued analysis, we prove the existence of admissible state-control triples. Moreover, we establish two key sensitivity properties of the control problem: the continuity of the value function and the upper semicontinuity of the set of optimal solutions.</p>

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Sensitivity Analysis of an Optimal Control Problem Driven by Differential Hemivariational Inequality for Nonstationary Thermistor Systems

  • 期刊论文

  • 发表日期：

    2026年3月24日

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  • Daomin Cao,
  • Stanisław Migórski,
  • Zijia Peng,
  • Guoqing Zhang

摘要

This paper investigates an optimal control problem governed by a differential hemivariational inequality (DHVI), which arises as the weak formulation of a nonstationary thermistor system with mixed multivalued and nonmonotone boundary conditions. We first establish the nonemptiness and weak compactness of the solution set to the DHVI by employing pseudomonotone operator theory, tools from nonsmooth analysis, and a fixed point approach. We then analyze the associated optimal control problem, focusing on its sensitivity with respect to the initial condition, the electric conductivity and a parameter in a metric space. Using the Kuratowski convergence and techniques from multivalued analysis, we prove the existence of admissible state-control triples. Moreover, we establish two key sensitivity properties of the control problem: the continuity of the value function and the upper semicontinuity of the set of optimal solutions.