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