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Sliding mode control for a phase field system related to tumor growth. (English) Zbl 1420.35434

Summary: In the present contribution we study the sliding mode control (SMC) problem for a diffuse interface tumor growth model coupling a viscous Cahn-Hilliard type equation for the phase variable with a reaction-diffusion equation for the nutrient. First, we prove the well-posedness and some regularity results for the state system modified by the state-feedback control law. Then, we show that the chosen SMC law forces the system to reach within finite time the sliding manifold (that we chose in order that the tumor phase remains constant in time). The feedback control law is added in the Cahn-Hilliard type equation and leads the phase onto a prescribed target \(\varphi^*\) in finite time.

MSC:

35Q92 PDEs in connection with biology, chemistry and other natural sciences
35K25 Higher-order parabolic equations
35K61 Nonlinear initial, boundary and initial-boundary value problems for nonlinear parabolic equations
93B52 Feedback control
92C50 Medical applications (general)
97M60 Biology, chemistry, medicine (aspects of mathematics education)
92C37 Cell biology
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