{"title":"Optimal control of a parabolic equation with memory","authors":"E. Casas, J. Yong","doi":"10.1051/cocv/2023013","DOIUrl":null,"url":null,"abstract":"An optimal control problem for a semilinear parabolic partial differential equation with memory is considered. The well-posedness as well as the first and the second order differentiability of the state equation is established by means of Schauder fixed point theorem and the implicity function theorem. For the corresponding optimal control problem with the quadratic cost functional, the existence of optimal control is proved. The first and the second order necessary conditions are presented, including the investigation of the adjoint equations which are linear parabolic equations with a measure as a coefficient of the operator. Finally, the sufficiency of the second order optimality condition for the local optimal control is proved.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"11 1","pages":""},"PeriodicalIF":1.3000,"publicationDate":"2023-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Esaim-Control Optimisation and Calculus of Variations","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1051/cocv/2023013","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"AUTOMATION & CONTROL SYSTEMS","Score":null,"Total":0}
引用次数: 2
Abstract
An optimal control problem for a semilinear parabolic partial differential equation with memory is considered. The well-posedness as well as the first and the second order differentiability of the state equation is established by means of Schauder fixed point theorem and the implicity function theorem. For the corresponding optimal control problem with the quadratic cost functional, the existence of optimal control is proved. The first and the second order necessary conditions are presented, including the investigation of the adjoint equations which are linear parabolic equations with a measure as a coefficient of the operator. Finally, the sufficiency of the second order optimality condition for the local optimal control is proved.
期刊介绍：
ESAIM: COCV strives to publish rapidly and efficiently papers and surveys in the areas of Control, Optimisation and Calculus of Variations.
Articles may be theoretical, computational, or both, and they will cover contemporary subjects with impact in forefront technology, biosciences, materials science, computer vision, continuum physics, decision sciences and other allied disciplines.
Targeted topics include:
in control: modeling, controllability, optimal control, stabilization, control design, hybrid control, robustness analysis, numerical and computational methods for control, stochastic or deterministic, continuous or discrete control systems, finite-dimensional or infinite-dimensional control systems, geometric control, quantum control, game theory;
in optimisation: mathematical programming, large scale systems, stochastic optimisation, combinatorial optimisation, shape optimisation, convex or nonsmooth optimisation, inverse problems, interior point methods, duality methods, numerical methods, convergence and complexity, global optimisation, optimisation and dynamical systems, optimal transport, machine learning, image or signal analysis;
in calculus of variations: variational methods for differential equations and Hamiltonian systems, variational inequalities; semicontinuity and convergence, existence and regularity of minimizers and critical points of functionals, relaxation; geometric problems and the use and development of geometric measure theory tools; problems involving randomness; viscosity solutions; numerical methods; homogenization, multiscale and singular perturbation problems.
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