Mathematical Sciences: Systems of PDEs: Viscosity Solutions and Optimal Control Applications

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This is a multi-faceted investigation whose primary focus is the application of partial differential equations and their optimal control to problems in mechanics,population dynamics, mathematical ecology and mathematical physiology. Among the problems to be investigated are bilinear control of Kirchhoff plate equations;viscosity solutions of systems of systems of degenerate parabolic partial differential equations that arise in population dynamics; optimal control of a system of ordinary differential equations which models the interaction of the immune system with the human immunodeficiency virus (HIV) with the goal of devising an optimal scheduling procedure for chemotherapy treatments; and optimal control of a model for bioremediation of hazardous waste using the input pumping levels of nutrients as the control variable. This project is jointly funded by the Applied Mathematics Program and the Classical Analysis Program. This is a multi-faceted investigation whose primary focus is the application of partial differential equations and their optimal control to problems in mechanics, population dynamics, mathematical ecology and mathematical physiology. Among the specific problems to be investigated are (1) the bilinear optimal control of plate equations, with applications to control of elastic structures; (2) an analysis of systems of partial differential systems which model population dynamics, with applications to competitive and cooperative scenarios in social, political, military, and environmental situations; (3) optimal control of chemotherapy treatment scheduling in a model which describes the interaction of the immune system with the human immunodeficiency virus (HIV) with the goal of reducing viral replication rates; and (4) optimal control of hazardous waste disposal through the process of bioremediation in which the control variable is the delivery pattern of the essential nutrients that optimizes the efficiency of the bioremediation process. Th is project is jointly funded by the Applied Mathematics Program and the Classical Analysis Program.