Solvability and sliding mode control for the viscous Cahn–Hilliard system with a possibly singular potential

Colli Pierluigi; Gilardi Gianni; Marinoschi Gabriela

doi:10.3934/mcrf.2020051

Article Contents

2021, Volume 11, Issue 4: 905-934. Doi: 10.3934/mcrf.2020051 open access

This issue Previous Article Stability of N-D transmission problem in viscoelasticity with localized Kelvin-Voigt damping under different types of geometric conditions Next Article Strict dissipativity analysis for classes of optimal control problems involving probability density functions

Solvability and sliding mode control for the viscous Cahn–Hilliard system with a possibly singular potential

1.
Dipartimento di Matematica ``F. Casorati'', Università di Pavia, via Ferrata 5, 27100 Pavia, Italy
2.
``Gheorghe Mihoc-Caius Iacob'', Institute of Mathematical Statistics and Applied Mathematics, of the Romanian Academy (ISMMA), Calea 13 Septembrie 13, 050711 Bucharest, Romania

Received: March 31, 2020

Revised: August 31, 2020

Early access: December 2020

Published: December 2021

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Abstract

In the present contribution we study a viscous Cahn–Hilliard system where a further leading term in the expression for the chemical potential $ \mu $ is present. This term consists of a subdifferential operator $ S $ in $ L^2(\Omega) $ (where $ \Omega $ is the domain where the evolution takes place) acting on the difference of the phase variable $ \varphi $ and a given state $ {\varphi^*} $, which is prescribed and may depend on space and time. We prove existence and continuous dependence results in case of both homogeneous Neumann and Dirichlet boundary conditions for the chemical potential $ \mu $. Next, by assuming that $ S = \rho\;{\rm{sign}} $, a multiple of the $ \;{\rm{sign}} $ operator, and for smoother data, we first show regularity results. Then, in the case of Dirichlet boundary conditions for $ \mu $ and under suitable conditions on $ \rho $ and $ \Omega $, we also prove the sliding mode property, that is, that $ \varphi $ is forced to join the evolution of $ {\varphi^*} $ in some time $ T^* $ lower than the given final time $ T $. We point out that all our results hold true for a very general and possibly singular multi-well potential acting on $ \varphi $.

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Mathematics Subject Classification: Primary: 35K52; Secondary: 58J35, 80A22, 93B52, 93C20.

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