Optimal Control of Heterogeneous Systems with Endogenous Domain of Heterogeneity

The paper deals with optimal control of heterogeneous systems, that is, families of controlled ODEs parameterized by a parameter running over a domain called domain of heterogeneity. The main novelty in the paper is that the domain of heterogeneity is endogenous: it may depend on the control and on the state of the system. This extension is crucial for several economic applications and turns out to rise interesting mathematical problems. A necessary optimality condition is derived, where one of the adjoint variables satisfies a differential inclusion (instead of equation) and the maximization of the Hamiltonian takes the form of ??min-max??. As a consequence, a Pontryagin-type maximum principle is obtained under certain regularity conditions for the optimal control. A formula for the derivative of the objective function with respect to the control from L _?? is presented together with a sufficient condition for its existence. A stylized economic example is investigated analytically and numerically.