Optimal control for obstacle problems involving time-dependent variational inequalities with Liouville–Caputo fractional derivative

Advances in Difference Equations

Apassara Suechoei¹  Parinya Sa Ngiamsunthorn²  Poom Kumam³ 

[1] Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Band Mod, Thrung Khru, 10140, Bangkok, Thailand;Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Band Mod, Thrung Khru, 10140, Bangkok, Thailand;Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Science Laboratory Building, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha Uthit Road, Bang Mod, Thung Khru, 10140, Bangkok, Thailand;KMUTT Fixed Point Research Laboratory, KMUTT-Fixed Point Theory and Applications Research Group, SCL 802 Fixed Point Laboratory, Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha Uthit Road, Bang Mod, Thung Khru, 10140, Bangkok, Thailand;Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Science Laboratory Building, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha Uthit Road, Bang Mod, Thung Khru, 10140, Bangkok, Thailand;

关键词: Existence of solutions;    Fractional calculus;    Obstacle problems;    Optimal control;    35R11;    49J20;    49J40;   

DOI  :  10.1186/s13662-021-03453-2

来源: Springer

【 摘 要 】

We consider an optimal control problem for a time-dependent obstacle variational inequality involving fractional Liouville–Caputo derivative. The obstacle is considered as the control, and the corresponding solution to the obstacle problem is regarded as the state. Our aim is to find the optimal control with the properties that the state is closed to a given target profile and the obstacle is not excessively large in terms of its norm. We prove existence results and establish necessary conditions of obstacle problems via the approximated time fractional-order partial differential equations and their adjoint problems. The result in this paper is a generalization of the obstacle problem for a parabolic variational inequalities as the Liouville–Caputo fractional derivatives were used instead of the classical derivatives.

【 授权许可】

CC BY