On Characterizations of the Input-to-State Stability Property Eduardo D. Sontag * Department of Mathematics Rutgers University New Brunswick, NJ 08903 sontag@hilbert.rutgers.edu Yuan Wang † Department of Mathematics Florida Atlantic University Boca Raton, FL 33431 ywang@polya.math.fau.edu Abstract We show that the well-known Lyapunov sufficient condition for “input-to-state stability” isalso necessary, settling positively an open question raised by several authors during the past few years. Additional characterizations of the ISS property, including one in terms of nonlinear stability margins, are also provided. 1 Introduction In practice, control systems are very often affected by noise, expressed for instance as pertur- bations on controls and errors on observations. Thus, it is desirable for a system not only to be stable, but also to display so-called “input/state” stability properties. Intuitively, this means that the behavior of the system should remain bounded when its inputs are bounded, and should tend to equilibrium when inputs tend to zero. These notions are closely related to the topic of stability under perturbations (total stability), studied in the classical dynamical systems literature. In the late 1980s, one of the coauthors introduced a particular precise definition of in- put/state stability , and established a few basic results; see for instance [4], [6], and [5]. These results then were applied in different areas, including observer design and new small-gain the- orems; see for instance [7], [8], [1], and [3]. One of the main observations used in [4] as well as in subsequent papers has been the fact that a system is input/state stable if it admits an “ISS-Lyapunov function.” This motivates checking the ISS property by investigating ISS-Lyapunov functions for the given system. In this work, we show that it is also necessary for a input/state stable system to admit an ISS-Lyapunov function. The proof is based on a reduction to a question about systems with disturbances and relies on a new Converse Lyapunov Theorem for such systems proved in [2]. In the process of proving the main result, a number of other natural characterizations became available, including one in terms of nonlinear stability margins. The main theorem of this paper will state the equivalence of all these new notions, which will probably be of interest in themselves. * Supported in part by US Air Force Grant AFOSR-91-0346 † Supported in part by NSF Grant DMS-9108250 Keywords: Nonlinear stability, Input/state stability, Lyapunov function techniques AMS(MOS) subject classifications: 93D05, 93D09, 93D20, 93D25, 34D20 1