Measure Optimal Controls for Models Inspired by Biology

The paper is concerned with optimal control problems for a parabolic system, coupled with zero Neumann boundary conditions and with nonlinear source terms. Inspired by applications in biology and medicine, the system aims to describe two species in competition in the same spatial region and is supplemented with measure valued distributed controls, acting as source terms. Introducing general cost functionals, one can study optimal control problems. We prove the existence of solutions for the parabolic equations with measure valued controls, together with suitable stability estimates. Moreover, the existence of optimal solutions in a distributional sense is also established.