# Sliding mode control for a phase field system related to tumor growth

## TL;DR

This paper develops a sliding mode control approach for a tumor growth model combining phase field and reaction-diffusion equations, ensuring the tumor phase reaches a desired state in finite time.

## Contribution

It introduces a novel SMC law for a coupled tumor growth system and proves finite-time convergence to a prescribed phase state.

## Key findings

- System reaches the sliding manifold in finite time
- Control law stabilizes tumor phase at target value
- Well-posedness of the controlled system established

## Abstract

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 $\phi^*$ in finite time.