## Wayne State University Dissertations

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WSU Access

January 2016

Dissertation

Ph.D.

Mathematics

Jose L. Menaldi

#### Abstract

This dissertation considers a stochastic dynamic system which is governed by a multidimensional diffusion process with time dependent coefficients. The control acts additively on the state of the system. The objective is to minimize the expected cumulative cost associated with the position of the system and the amount of control exerted. It is proved that Hamilton-Jacobi-Bellman’s equation of the problem has a solution, which corresponds to the optimal cost of the problem. We also investigate the smoothness of the free boundary arising from the problem.

In the second part of the dissertation, we study the backward parabolic problem for a nonlinear parabolic equation of the form u_t + Au(t) = f (t, u(t)), u(T ) = ϕ, where A is a positive self-adjoint unbounded operator and f is a Lipschitz function. The problem is ill-posed, in the sense that if the solution does exist, it will not depend continuously on the data. To regularize the problem, we use the quasi-reversibility method to establish a modified problem. We present approximated solutions that depend on a small parameter \epsilon > 0 and give error estimates for our regularization. These

results extend some work on the nonlinear backward problem. Some numerical examples are given to justify the theoretical analysis.

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