32,487 research outputs found

    Input-to-state stability of infinite-dimensional control systems

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    We develop tools for investigation of input-to-state stability (ISS) of infinite-dimensional control systems. We show that for certain classes of admissible inputs the existence of an ISS-Lyapunov function implies the input-to-state stability of a system. Then for the case of systems described by abstract equations in Banach spaces we develop two methods of construction of local and global ISS-Lyapunov functions. We prove a linearization principle that allows a construction of a local ISS-Lyapunov function for a system which linear approximation is ISS. In order to study interconnections of nonlinear infinite-dimensional systems, we generalize the small-gain theorem to the case of infinite-dimensional systems and provide a way to construct an ISS-Lyapunov function for an entire interconnection, if ISS-Lyapunov functions for subsystems are known and the small-gain condition is satisfied. We illustrate the theory on examples of linear and semilinear reaction-diffusion equations.Comment: 33 page

    The converse of the passivity and small-gain theorems for input-output maps

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    We prove the following converse of the passivity theorem. Consider a causal system given by a sum of a linear time-invariant and a passive linear time-varying input-output map. Then, in order to guarantee stability (in the sense of finite L2-gain) of the feedback interconnection of the system with an arbitrary nonlinear output strictly passive system, the given system must itself be output strictly passive. The proof is based on the S-procedure lossless theorem. We discuss the importance of this result for the control of systems interacting with an output strictly passive, but otherwise completely unknown, environment. Similarly, we prove the necessity of the small-gain condition for closed-loop stability of certain time-varying systems, extending the well-known necessity result in linear robust control.Comment: 15 pages, 3 figure
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