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Optimal control of differential quasivariational-hemivariational inequalities with applications

Science China Mathematics 2024年第11期67卷 2587-2606页

作者： Dong-ling Cai Stanislaw Migórski Yi-bin Xiao School of Mathematical Sciences University of Electronic Science and Technology of ChinaChengdu 611731China College of Applied Mathematics Chengdu University of Information TechnologyChengdu 610225China Chair of Optimization and Control Jagiellonian University in KrakowKrakow 30348Poland

In this paper,we study a new class of differential quasivariational-hemivariational inequalities of the elliptic *** problem consists of a system coupling the Cauchy problem for an ordinary differential equation with the variational-hemivariational inequalities,unilateral constraints,and history-dependent ***,based on the Minty formulation and the continuity of the solution map of a parametrized quasivariational-hemivariational inequality,and a fixed point theorem for a history-dependent operator,we prove a result on the ***,we examine optimal control problems for differential quasivariational-hemivariational inequalities,including a time-optimal control problem and a maximum stay control problem,for which we show the existence of *** all the optimal control problems,the system is controlled through a distributed and boundary control,a control in initial conditions,and a control that appears in history-dependent ***,we illustrate the results by considering a nonlinear controlled system for a time-dependent elliptic equation with unilateral constraints.

关键词： differential variational-hemivariational inequality unilateral constraint history-dependent operator optimal control time-optimal control maximum stay problem

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Numerical analysis of history-dependent variational-hemivariational inequalities

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Science China Mathematics 2020年第11期63卷 2207-2232页

作者： Shufen Wang Wei Xu Weimin Han Wenbin Chen School of Mathematical Sciences Fudan UniversityShanghai 200433China Faculty of Science Tongji Zhejiang CollegeJiaxing 314051China Department of Mathematics University of IowaIowa CityIA 52242USA

In this paper,numerical analysis is carried out for a class of history-dependent variationalhemivariational inequalities by arising in contact *** different numerical treatments for temporal discretization are proposed to approximate the continuous ***-point iteration algorithms are employed to implement the implicit scheme and the convergence is proved with a convergence rate independent of the time step-size and mesh grid-size.A special temporal discretization is introduced for the history-dependent operator,leading to numerical schemes for which the unique solvability and error bounds for the temporally discrete systems can be proved without any restriction on the time *** for spatial approximation,the finite element method is applied and an optimal order error estimate for the linear element solutions is provided under appropriate regularity *** examples are presented to illustrate the theoretical results.

关键词： variational-hemivariational inequality Clarke subdifferential history-dependent operator fixedpoint iteration optimal order error estimate contact mechanics

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