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We consider a state-constrained optimal control problem of a system of twonon-local partial-differential equations, which is an extension of the oneintroduced in a previous work in mathematical oncology. The aim is to minimizethe tumor size through chemotherapy while avoiding the emergence of resistanceto the drugs. The numerical approach to solve the problem was the combinationof direct methods and continuation on discretization parameters, which happento be insufficient for the more complicated model, where diffusion is added toaccount for mutations. In the present paper, we propose an approach relying onchanging the problem so that it can theoretically be solved thanks to aPontryagin Maximum Principle in infinite dimension. This provides an excellentstarting point for a much more reliable and efficient algorithm combiningdirect methods and continuations. The global idea is new and can be thought ofas an alternative to other numerical optimal control techniques.

Combination of direct methods and homotopy in numerical optimal control: application to the optimization of chemotherapy in cancer