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We investigate the long-time dynamics and optimal control problem of adiffuse interface model that describes the growth of a tumor in presence of anutrient and surrounded by host tissues. The state system consists of aCahn-Hilliard type equation for the tumor cell fraction and areaction-diffusion equation for the nutrient. The possible medication thatserves to eliminate tumor cells is in terms of drugs and is introduced into thesystem through the nutrient. In this setting, the control variable acts as anexternal source in the nutrient equation. First, we consider the problem of`long-time treatment' under a suitable given source and prove the convergenceof any global solution to a single equilibrium as $t\to+\infty$. Then weconsider the `finite-time treatment' of a tumor, which corresponds to anoptimal control problem. Here we also allow the objective cost functional todepend on a free time variable, which represents the unknown treatment time tobe optimized. We prove the existence of an optimal control and obtain firstorder necessary optimality conditions for both the drug concentration and thetreatment time. One of the main aim of the control problem is to realize in thebest possible way a desired final distribution of the tumor cells, which isexpressed by the target function $\phi_\Omega$. By establishing the Lyapunovstability of certain equilibria of the state system (without external source),we see that $\phi_{\Omega}$ can be taken as a stable configuration, so that thetumor will not grow again once the finite-time treatment is completed.

Long-time Dynamics and Optimal Control of a Diffuse Interface Model for Tumor Growth