Abstract
This paper concerns a distributed optimal control problem for a tumor growth model of Cahn–Hilliard type including chemotaxis with possibly singular potentials, where the control and state variables are nonlinearly coupled. First, we discuss the weak well-posedness of the system under very general assumptions for the potentials, which may be singular and nonsmooth. Then, we establish the strong well-posedness of the system in a reduced setting, which however admits the logarithmic potential: this analysis will lay the foundation for the study of the corresponding optimal control problem. Concerning the optimization problem, we address the existence of minimizers and establish both first-order necessary and second-order sufficient conditions for optimality. The mathematically challenging second-order analysis is completely performed here, after showing that the solution mapping is twice continuously differentiable between suitable Banach spaces via the implicit function theorem. Then, we completely identify the second-order Fréchet derivative of the control-to-state operator and carry out a thorough and detailed investigation about the related properties.
Highlights
Lots of disclosures have been obtained in the past decades concerning tumor growth modeling: see, e.g., the pioneering works [13, 14, 48]
We assume the growth and proliferation of the tumor to be driven by the absorption and consumption of Keywords and phrases: Optimal control, tumor growth models, singular potentials, optimality conditions, second-order analysis
We investigate a distributed optimal control problem for a tumor growth model of viscous Cahn–Hilliard type with source term including chemotaxis effects and possibly singular potentials
Summary
Lots of disclosures have been obtained in the past decades concerning tumor growth modeling: see, e.g., the pioneering works [13, 14, 48]. In order to better emulate in-vivo tumor growth, it is possible to include in similar models the effects generated by the fluid flow development by postulating a Darcy’s law or a Stokes–Brinkman’s law. In this direction, we refer to [15, 19, 22, 24,25,26,27,28, 30, 48], and we mention [31], where elastic effects are included. We turn our attention to the strong well-posedness of (1.2)–(1.6) in the cases of the regular Freg and logarithmic Flog potentials This is done, while the corresponding optimal control problem is investigated . Precise constants we could refer to are treated in a different way
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