Mathematical Sciences: Commutant Lifting Methods in Operator Theory and Robust Control Theory

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Abstract Gu The proposed work is in two directions. The first direction lies at the interface of operator theory and system and control theory. The purpose of this part of the project is to extend to a more general class of operators the commutant lifting theory along with its connection with interpolation problems, Hankel and Toeplitz operators and robust control theory. The project will also include implementable algorithms for the computation of the norms of the type operators which appear in the solutions of interpolation problems and robust control problems by using commutant lifting methods. The qualitative and quantitative results for those operators are relevant to the design of controllers which stabilize the systems with parameter or modeling uncertainty while at the same time meet certain performance specifications. Specifically, the investigator will continue his study of a general operator extension problem which includes the causal commutant lifting approach for nonlinear systems. He will develop algorithms for the mixed sensitivity minimization problem of a class of multivariable distributed systems which is one of the fundamental problems in robust control theory. Another direction is related to the study of commutators of Toeplitz operators on certain important spaces of analytic functions including vector-valued Hardy spaces of the unit disk and Hardy spaces of the polydisk. These commutators have their origin in mathematical physics and non-commutative analysis. The research in this direction should be of interest to mathematical analysts as well as to scientists in many fields of applied sciences.