Control and Controllability of PDEs with Hysteresis

A controllability problem for a diffusion equation with a complex hysteresis operator is proposed here and consists in finding a control which guarantees that the solution reaches a desired value at a given time. We propose a constructive method based on a two-parameter penalty argument. One small p...

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Bibliographic Details
Published in:Applied mathematics & optimization 2021-08, Vol.84 (1), p.829-847
Main Authors: Gavioli, Chiara, Krejčí, Pavel
Format: Article
Language:English
Subjects:
Applied mathematics
Approximation
Calculus of Variations and Optimal Control
Optimization
Constitutive relationships
Control
Controllability
Hysteresis
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Numerical and Computational Physics
Optimization
Parameters
Partial differential equations
Phase transitions
Simulation
Stability
Systems Theory
Theoretical
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Summary:A controllability problem for a diffusion equation with a complex hysteresis operator is proposed here and consists in finding a control which guarantees that the solution reaches a desired value at a given time. We propose a constructive method based on a two-parameter penalty argument. One small parameter penalizes the distance of the solution at final time from the expected value, the second one is used to approximate the underlying rate independent variational inequalities in the hysteresis term by smooth viscous constitutive relations. We prove that a solution to the controllability problem can be obtained by passing to the limit in a doubly degenerate control system, and the convergence is strong in space and uniform in time.
ISSN:0095-4616
1432-0606
DOI:10.1007/s00245-020-09663-6