Optimal Distributed Control of an Extended Model of Tumor Growth with Logarithmic Potential
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Optimal Distributed Control of an Extended Model of Tumor Growth with Logarithmic Potential Andrea Signori1
© Springer Science+Business Media, LLC, part of Springer Nature 2018
Abstract This paper is intended to tackle the control problem associated with an extended phase field system of Cahn–Hilliard type that is related to a tumor growth model. This system has been investigated in previous contributions from the viewpoint of well-posedness and asymptotic analyses. Here, we aim to extend the mathematical studies around this system by introducing a control variable and handling the corresponding control problem. We try to keep the potential as general as possible, focusing our investigation towards singular potentials, such as the logarithmic one. We establish the existence of optimal control, the Lipschitz continuity of the control-to-state mapping and even its Fréchet differentiability in suitable Banach spaces. Moreover, we derive the first-order necessary conditions that an optimal control has to satisfy. Keywords Distributed optimal control · Tumor growth · Phase field model · Cahn–Hilliard equation · Optimal control · Necessary optimality conditions · Adjoint system Mathematics Subject Classification 35K61 · 35Q92 · 49J20 · 49K20 · 92C50
1 Introduction In this paper, we deal with a distributed optimal control problem for a system of partial differential equations whose physical context is that of tumor growth dynamics. Our aim is to devote this section to explain the general purpose of the work and we postpone all the technicalities for the forthcoming sections. In the next one, we will state precisely the problem and have the care to present in detail our notation and the mathematical framework in which set the problem. Here, let us only mention that with
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Andrea Signori [email protected] Dipartimento di Matematica e Applicazioni, Università di Milano–Bicocca, via Cozzi 55, 20125 Milan, Italy
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Applied Mathematics & Optimization
⊂ R3 we denote the set where the evolution takes place and, for a given final time T > 0, we fix Q := × (0, T ) and := × (0, T ). The distributed control problem, referred as (CP), consists of minimizing the socalled cost functional J(ϕ, σ, u) =
b2 b3 b1 ϕ − ϕ Q 2L 2 (Q) + ϕ(T ) − ϕ 2L 2 () + σ − σ Q 2L 2 (Q) 2 2 2 b4 b0 2 2 + σ (T ) − σ L 2 () + u L 2 (Q) , (1.1) 2 2
subject to the control constraints u ∈ Uad := {u ∈ L∞ (Q) : u ∗ ≤ u ≤ u ∗ a.e. in Q},
(1.2)
and to the state system α∂t μ + ∂t ϕ − μ = P(ϕ)(σ − μ) in Q μ = β∂t ϕ − ϕ + F (ϕ) in Q
(1.3) (1.4)
∂t σ − σ = −P(ϕ)(σ − μ) + u in Q ∂n μ = ∂n ϕ = ∂n σ = 0 on
(1.5) (1.6)
μ(0) = μ0 , ϕ(0) = ϕ0 , σ (0) = σ0 in .
(1.7)
Let us give just some overall indications on the involved quantities of the above equations. The symbols b0 , b1 , b2 , b3 , b4 represent nonnegative constants, not all zero, while ϕ Q , ϕ , σ Q , σ , u ∗ , and u ∗ denote given functions. As regards these latter, the first four model some targets, while the last two fix the box in which the cont
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