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Stability and stabilization of implicit systems

This paper is concerned with stability and stabilization of implicit systems. Stability criteria are provided in terms of the Kronecker form, Lyapunov equation and inequality and conditions on extended rank and invertibility of the system pencil. Then stabilization of implicit systems via interconnection is considered based on the notion of initial-freedom preservation as a formulation of regularity of interconnection. To analyze stabilization with initial-freedom preservation, we define complete-stabilizability and zero-detectability. It is shown that stabilizability via interconnection with initial-freedom preservation is equivalent to complete-stabilizability and zero-detectability. Thus standard results on output feedback stabilization of state-space and descriptor systems are generalized to stabilization of implicit systems via interconnection.