Mathematical Sciences: Operator Theory and Communications Engineering

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Professor Helton's project is research in mathematics that has applications to problems in engineering. The mathematics involves matrix-valued functions on planar regions such as the disc. Among these there is a particularly well-behaved class of functions called analytic. Best-approximation of more or less arbitrary functions by analytic ones, and finding analytic functions that take on prescribed values at prescribed points in the region are problems that turn out to be important in the design of feedback stabilized control systems. By applying methods of functional analysis and operator theory, Helton seeks efficient ways of solving problems like these. Helton's point of view is that commutant lifting in operator theory is nearly equivalent to interpolation theory for matrix- valued functions. He will seek to develop a nonlinear commutant lifting theory, and to link it with the existing theory of nonlinear control systems. He will investigate qualitative and numerical aspects of optimization over spaces of analytic functions; computational experiment and theoretical insight will support one another mutually in this part of the project. A further aspect of the work involves matrix completion, filling in the entries of a partially specified matrix so as to make the result have a particular property such as positivity. Throughout, Helton will keep in view applications to optimal control and worst case design in the frequency domain.