Abstract
In this paper, we develop dissipation inequalities for a class of well-posed systems described by partial differential equations (PDEs). We study passivity, reachability, induced input–output norm boundedness, and input-to-state stability (ISS). We consider both cases of in-domain and boundary inputs and outputs. We study the interconnection of PDE–PDE systems and formulate small gain conditions for stability. For PDEs polynomial in dependent and independent variables, we demonstrate that sum-of-squares (SOS) programming can be used to compute certificates for each property. Therefore, the solution to the proposed dissipation inequalities can be obtained via semi-definite programming. The results are illustrated with examples.
Highlights
A powerful tool in the study of robustness and input-tostate/output properties of dynamical systems is dissipation inequalities (Hill & Moylan, 1976; Willems, 1972)
This paper aims at developing dissipation inequalities for systems defined by partial differential equations (PDEs)
In Valmorbida, Ahmadi, and Papachristodoulou, we proposed a methodology to solve integral inequalities involving functions specified by a set of boundary conditions using SOS optimization cast as semidefinite programs (SDPs)
Summary
A powerful tool in the study of robustness and input-tostate/output properties of dynamical systems is dissipation inequalities (Hill & Moylan, 1976; Willems, 1972). The Kalman–Yakubovic–Popov lemma (Kalman, 1963) presents necessary and sufficient conditions to construct quadratic storage functions certifying the passivity dissipation inequality of linear ODE systems. In Valmorbida, Ahmadi, and Papachristodoulou (in press), we proposed a methodology to solve integral inequalities involving functions specified by a set of boundary conditions using SOS optimization cast as semidefinite programs (SDPs). We generalize the results in Ahmadi et al (2014), wherein dissipation inequalities for different input-state/output properties of PDEs in the space of square integrable functions with in-domain inputs and outputs were studied, and we present a framework for input-state/output analysis of a class of well-posed PDEs. Each input-state/output property, namely passivity, reachability, induced input–output norms and ISS, is defined in the appropriate Sobolev norms.
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