9704199 Lenhart Optimal control of partial differential equations, ordinary differential equations, and variational inequalities will be investigated. The first area of research is the control of hybrid systems of partial differential equations and ordinary differential equations, with applications in bioremediation and population modeling. The second area is optimal control of the obstacle in variational inequalities, with elliptic or fourth order operators (involving biharmonic operators). The third area is the investigation of optimal chemotherapy strategies in HIV models. First, the existence of an optimal control must be shown for the models and objective functionals appropriate to each application area. Secondly, this optimal control must be characterized through necessary conditions from the derivation of an optimality system, which is the state system coupled with an adjoint system. Due to the variety of systems treated, existence, uniqueness, and characterization of the optimal control requires new and different a priori estimates and convergence arguments. For some of the applications, numerical simulations will be completed. This research involves theory and applications of optimal control of partial differential equations, ordinary differential equations, and variational inequalities. The first application area is control of a bioremediation model for encouraging the remediation of contaminants by metabolism of bacteria. The level of the nutrient inputs is controlled to optimally run a bioreactor. The approach will be lab tested on a gas phase bioreactor with pseudomonas putida bacteria degrading paraxylene. Other population models will be investigated. The second application area is the control of variational inequalities, which model the designing of the shape of the membrane by controlling an obstacle put under the membrane. These models arise in flow through porous media, hydrodynamic lubrication, elasto-plastic analysis, continuous casting, and electrochemical machining. The third application area is the design of optimal chemotherapies for the treatment of HIV infection. The effect of a "cocktail of various drugs" can be simulated for an immunology model involving CD4+T white blood cells and the HIV virus. Also a model involving the time elapsed since the cells became infected with HIV will be studied. Numerical simulations for the optimal control and resulting controlled quantities (like CD4+T cells), will be completed using realistic data.
