Mathematical Sciences: Optimal Control Theory of Input-Output Systems - Control Theory of Partial Differential Equations

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This work will focus on the control theory of partial differential equations. Two primary thrusts will be undertaken. The first line of investigation deals with optimal control of input-output systems described by partial differential equations, functional differential equations and associated applications. This work will expand on recent work of the principal investigator which extended the classical Pontryagin maximum principle to general nonlinear systems in infinite dimensions. Applications of the resulting convergence principles have only been exploited in classical cases to date. Work will now be directed to at least two new areas: functional differential equations and to shape optimization where the control is a time-dependent set. Here the general maximum principle is especially suited since the state equation is a partial differential equation in the variable domain and does not fit any of the available maximum principles. Work is also planned on the problem of characterizing the set of attainable states for both the heat equation and the wave equation, primarily on special geometric domains at first. In this active area of research, boundary conditions through which the control action is exercised are not specified. These questions often reduce to problems in Fourier analysis in which estimates on the norms of sequences are required. Fundamental questions focus on determining attainable states by means of boundary control, a state being represented by a function in the space variables which one wishes to reach in prescribed time from an initial shape.