On the Role of Well-Posedness in Homotopy Methods for the Stability Analysis of Nonlinear Feedback Systems

We consider the problemintegral quadratic constraintlower hemicontinuousfinely lower hemicontinuousweakly lower hemicontinuouslower hemicontinuousfinely lower hemicontinuousweakly lower hemicontinuousinput/output system of determining the input/output stability of the feedback interconnection of two systems. Dissipativity and graph separation techniques are two related and popular approaches to this problem, and they include well-known passivity and small-gain methods. The use of block diagram transformations with dynamic multipliers can greatly reduce the conservativeness of such approaches, but for the stability of the transformed system to imply that of the original one, these multipliers should admit appropriate factorizations. An alternative approach which circumvents the need to factorize multipliers was provided by Megretski and Rantzer in their seminal 1997 paper on integral quadratic constraints. Their approach is based on homotopy: one constructs a continuous transformation of a trivially stable system into the target system of interest, and by satisfying certain conditions along the homotopy path one guarantees that the target system is also stable. This method assumes that the feedback interconnection is well-posed along the homotopy path, namely, that the feedback equations have solutions for all possible exogenous inputs and that the mapping from these inputs to the solutions is causal. In this chapter we will explore the role of well-posedness in this homotopy method. In so doing we demonstrate that what suffices for the homotopy analysis is a property significantly weaker than well-posedness, one which involves a certain lower hemicontinuity of the feedback interconnection along with a certain controllability of its domain. Moreover, we show that these methods can be applied to general signal spaces, including extended Sobolev spaces, spaces of smooth functions, and spaces of distributions.