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We consider a system of two coupled integro-differential equations modellingpopulations of healthy and cancer cells under therapy. Both populations arestructured by a phenotypic variable, representing their level of resistance tothe treatment. We analyse the asymptotic behaviour of the model under constantinfusion of drugs. By designing an appropriate Lyapunov function, we prove thatboth densities converge to Dirac masses. We then define an optimal controlproblem, by considering all possible infusion protocols and minimising thenumber of cancer cells over a prescribed time frame. We provide a quasi-optimalstrategy and prove that it solves this problem for large final times. For thismodelling framework, we illustrate our results with numerical simulations, andcompare our optimal strategy with periodic treatment schedules.

Asymptotic analysis and optimal control of an integro-differential system modelling healthy and cancer cells exposed to chemotherapy