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Well-Posedness of History/State-Dependent Implicit Sweeping Processes

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Abstract

This paper is devoted to the study of a new class of implicit state-dependent sweeping processes with history-dependent operators. Based on the methods of convex analysis, we prove the equivalence of the history/state dependent implicit sweeping process and a nonlinear differential equation, which, through a fixed point argument for history-dependent operators, enables us to prove the existence, uniqueness, and continuous dependence of the solution in a very general framework. Moreover, we present some new convergence results with respect to perturbations in the data, including perturbations of the associated moving sets. Finally, the theoretical results are applied to prove the well-posedness of a history-dependent quasi-static contact problem.

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Acknowledgements

The authors are grateful to the referee for careful reading of the paper and valuable suggestions and comments. S. Zeng received funding from the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Skłodowska-Curie grant agreement No. 823731—CONMECH. It is also supported by National Science Center of Poland under Preludium Project No. 2017/25/N/ST1/00611. E. Vilches was funded by ANID Chile under grant Fondecyt de Iniciación No. 11180098.

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Authors and Affiliations

  1. Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing, Yulin Normal University, Yulin, 537000, P.R. China

    Shengda Zeng

  2. Jagiellonian University in Krakow, Chair of Optimization and Control, ul. Lojasiewicza 6, 30348, Krakow, Poland

    Shengda Zeng

  3. Universidad de O’Higgins, Rancagua, Chile

    Emilio Vilches

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  1. Shengda Zeng
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  2. Emilio Vilches
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Correspondence to Emilio Vilches.

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Communicated by Boris S. Mordukhovich.

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Zeng, S., Vilches, E. Well-Posedness of History/State-Dependent Implicit Sweeping Processes. J Optim Theory Appl 186, 960–984 (2020). https://doi.org/10.1007/s10957-020-01730-w

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