Frequency Conditions in Controller Design for Dynamical Systems in Hilbert Spaces

Abstract

The present paper develops the results of [1], where a generalization of H∞-optimal control theory to the case of dynamical systems in Hilbert spaces and, in particular, systems governed by delay differential equations was considered. It was mentioned in Remark 4 in [1] that the main difficulty encountered in the design of an H∞-optimal dynamic controller is in finding a positive definite solution of the corresponding operator equation or inequality. In practice, one sidesteps this difficulty by passing to a finite-dimensional approximation with the subsequent use of numerical methods. It would be more logical not to use solutions of any operator equations or inequalities in the controller equations, as was done in [2] for the finite-dimensional version of the problem. The aim of the present paper is to generalize this “inverse” approach [3, 4] in control law design to systems with infinite-dimensional state space. In this case, it becomes unnecessary to find solutions of operator equations or inequalities, and “frequency” inequalities for positive matrix functions used in this approach become inequalities for matrix functions of a more general class. Note that such a generalization was once carried out in absolute stability theory [5].