Mathematical Sciences: Infinite Dimensional Nonlinear Programming Problems

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This award supports mathematical research in the areas of partial differential equations and functional differential equations. The work focuses on the applications of infinite dimensional nonlinear programming theory to optimal control problems for systems governed by equations such as the Navier-Stokes equations, reaction-diffusion equations and nonlinear wave equations. The object is to obtain existence results in the realm of relaxed controls, necessary conditions for optimal controls of the type of Pontryagin's maximum principle, convergence results for numerically obtained sequences of suboptimal controls and feedback solutions by means of Hamilton-Jacoby equations. Work will be done both in the case of distributed and boundary control for problems that include sate constraints. Regularization of problems containing singular arcs and the role of the Hamiltonian in free arrival time problems will also be examined. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations.