Mathematical Sciences: Nonlinear Partial Differential Equations in Optimal Control and Probability

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The principal investigator will study several problems in optimal deterministic or stochastic control and probability, using the theory of nonlinear partial differential equations. The project consists of three parts. The optimal control of a singularly controlled Brownian motion will be studied using the results for free boundary problems. This approach has yielded regularity results when the dimension is equal to two and it is the first multi-dimensional result which yields the construction of the optimal process. It now appears that it can be generalized to higher dimensions. The second part of this project is devoted to the perturbation theory of infinite dimensional Hamilton-Jacobi equations. As a case study, the principal investigator will study an asymptotic problem related to simple exclusion processes. Finally, an application of the finite dimensional perturbation theory to a problem of production planning is described. The model to be studied has failure-prone machines, and the goal is to construct approximate optimal policies by exploiting the hierarchical structure of the model.