Optimal control and pattern formation for a haptotaxis model of solid tumor invasion

This paper deal with an optimal control problem for a haptotaxis model of solid tumor invasion by considering the multiple treatments of cancer (a combination of radiotherapy and chemotherapy). Firstly, we obtain the existence and uniqueness of weak solution for the controlled system with spatial dimensions N = 1, 2, 3 by applying the Leray-Schauder fixed point theorem and developing adapted a priori estimates. Subsequently, the existence of optimal pair are proved by means of the technique of minimizing sequence. Furthermore, we verify the Lipschitz continuity of control-to-state mapping based on some a priori estimates, hence derive the first-order necessary optimality condition and establish the optimality systems. Finally, the ringlike diffusion and aggregation patterns and the dynamics of tumor invasion as well as the optimal control strategies are presented numerically, which demonstrate that the optimal treatment strategies are capable of breaking the pattern formation, and preventing the tumor invading and metastasizing, even eliminating the tumor possibly. The results of this work improved and extended previous results partially. (C) 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.