Abstract
This paper deals with an optimal control problem for a haptotaxis model which describes two types of cancer cell invading the surrounding healthy tissue. Cancer cells can be treated by therapeutic agents (radiotherapy or chemotherapy) considered as the control variable. Our primary goal is to characterize an optimal control which balances the therapeutic benefits with its side effects. Firstly, the global bounded weak solution of state system in spatial dimensions $$1\le N\le 5$$ are obtained by using the Banach fixed-point theorem and a new iteration criterion of $$L^p$$ estimates as well as the adapted Moser-type iteration technique of $$L^\infty $$ estimates. Subsequently, the existence of optimal pair is proved by applying the technique of minimizing sequence. Furthermore, we derive the first-order necessary optimality condition by means of the Lipschitz continuity of the control-to-state mapping and the methods of convex perturbation. Finally, we present numerically the spatio-temporal dynamics of species, optimal control and optimal regimen of therapeutic agents administrated to illustrate the concrete realization of the theoretical results obtained in this work, and to validate some clinical findings.
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