Mathematical Sciences: Viscosity Solutions and Control Problems

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This project concerns several problems involving viscosity solutions and optimal control problems. The first problem involves the extension of viscosity solution results to systems with quadratic and non-local coupling terms. These coupling terms can represent interaction terms in competition and cooperative systems. Monotone iteration schemes will be developed to solve such systems. The second problem involves viscosity solutions for second order elliptic systems associated with two player games. In an illustrative example, one player controls the switching of a diffusion process and the other player controls the stopping of the process. One wants to show existence results for the system and prove that the viscosity solution gives the value of the saddle point of the game. The third problem is a direct study of optimality systems in which the state is governed by first order PDEs. In minimizing a cost functional involving a target function, the goal is to characterize the optimal control and solve the optimality system (state equation with optimal control and the adjoint equation).