Optimality conditions for an extended tumor growth model with double obstacle potential via deep quench approach

Abstract

In this work, we investigate a distributed optimal control problem for an extended phase field system of Cahn--Hilliard type which physical context is that of tumor growth dynamics. In a previous contribution, the author has already studied the corresponding problem for the logarithmic potential. Here, we try to extend the analysis by taking into account a non-smooth singular nonlinearity, namely the double obstacle potential. Due to its non-smoothness behavior, the standard procedure to characterize the necessary conditions for the optimality cannot be performed. Therefore, we follow a different strategy which in the literature is known as the "deep quench" approach in order to obtain some optimality conditions that have to be interpreted in a more general framework. We establish the existence of optimal controls and some first-order optimality conditions for the system are derived by employing suitable approximation schemes.

References in corpus (4)

• Long-time Dynamics and Optimal Control of a Diffuse Interface Model for Tumor Growth
• Optimal Distributed Control of a Cahn-Hilliard-Darcy System with Mass Sources
• Optimal distributed control of an extended model of tumor growth with logarithmic potential
• Optimal velocity control of a convective Cahn-Hilliard system with double obstacles and dynamic boundary conditions: a `deep quench' approach

Cited by in corpus (10)

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• Optimal control of a phase field system modelling tumor growth with chemotaxis and singular potentials
• Penalisation of Long Treatment Time and Optimal Control of a Tumour Growth Model of Cahn-Hilliard
• Strong well-posedness and inverse identification problem of a non-local phase field tumor model with degenerate mobilities
• Vanishing parameter for an optimal control problem modeling tumor growth
• Existence of weak solutions to multiphase Cahn-Hilliard-Darcy and Cahn-Hilliard-Brinkman models for stratified tumor growth with chemotaxis and general source terms
• Optimal control problems with sparsity for phase field tumor growth models involving variational inequalities
• Second-order analysis of an optimal control problem in a phase field tumor growth model with singular potentials and chemotaxis