Mathematical Sciences: Classical Analysis: Complex Analysis,Computation, and Control

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Work on this project combines investigations into mathematical questions of classical complex analysis with applications to control problems and finitely generated algebras of functions of several complex variables. The optimal control problems, arising in engineering, are phrased in terms of vector-valued, bounded, functions defined on certain cylinder-like sets. One is interested in minimizing a certain functional over the class of functions. Sometimes there are no extermal functions, other times they turn out to be analytic. Sometimes there are unique solutions, but not always. The goal of this work is to describe the cylinders which give rise to bounded analytic optimal functions which are smooth on the boundary of the cylinder. A number of positive results in this area have focused the problem to one of taking analytic selections of functions on sets fibered over the unit circle and determining when such functions agree with a given function when restricted to the unit interval. Work on algebra generators evolves from an earlier result of Gleason which suggests that the classes of continuous and holomorphic functions in the two-dimensional domains which vanish at the origin are finitely generated algebras. This is easy to do for the bidisc, although the minimum norm of the generators is not yet known (it is probably unity). This particular question is to be addressed by computational methods first to verify the conjectured minimum. For the general Gleason problem, a new approach will be made by converting it to an equivalent problem in first order partial differential equations. Estimates for the norm of the generators should follow if this plan works out.