On Characterizations of Input-to-State Stability with Respect to Compact Sets Eduardo D. Sontag * and Yuan Wang ** * Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA ** Department of Mathematics, Florida Atlantic University, Boca Raton, FL 33431, USA Abstract. Previous characterizations of iss-stability are shown to generalize without change to the case of stability with respect to sets. Some results on iss-stabilizability are mentioned as well. Key Words. Set Stability; input-to-state stability; Lyapunov functions; control Lyapunov func- tions; practical stability 1. Introduction System stability with respect to input perturba- tions is one of the central issues to be studied in control. During the past few years, the property called “input to state stability” (iss) has been pro- posed (originally in (Sontag, 1989a)) as a founda- tion for the study of such problems, and has been subsequently employed by many authors in areas ranging from robust control to highly nonlinear small-gain theorems (see for instance (Jiang et al., 1994; Tsinias, 1989)). In the paper (Sontag and Wang, 1995), the authors established the equiv- alence among several natural characterizations of the iss property, stated in terms of dissipation in- equalities, robustness margins, and Lyapunov-like functions. The iss property was originally stated for stabil- ity with respect to a given equilibrium state of interest. On the other hand, in many applications it is of interest to study stability with respect to an invariant set A, where A does not necessarily consist of a single point. Examples of such appli- cations include problems of robust control and the various notions usually encompassed by the term “practical stability.” This motivated the study of the “set” version of the iss property, originally in (Sontag and Lin, 1992), and developed with ap- plications to the study of parameterized families of systems in (Lin et al., 1995). Given the interest in set-iss, it is an obvious question to ask whether the equivalent characterizations given in (Sontag and Wang, 1995) for the special case A = equilib- rium extend to the more general set case. It is the main purpose of this paper to point out that, at * Supported in part by US Air Force Grant F49620-95-1- 0101 ** Supported in part by NSF Grants DMS-9457826 and DMS-9403924 least for the case of compact invariant sets A, the results in (Sontag and Wang, 1995) indeed can be generalized with little or no change; in particu- lar, iss with respect to a given A is equivalent to the existence of an “iss-Lyapunov function” rela- tive to A. (Most of the proofs valid in the equi- librium case extend easily to the set case.) The property of there existing some invariant com- pact set A so that there is iss with respect to A will be called the “compact-iss” property; various remarks about this notion are included, includ- ing the equivalence with what is sometimes called “practical stability.” Finally, we also sketch some applications of the iss notion to feedback design and disturbance attenuation. 2. Set Input to State Stability We deal with systems of the following general form: ˙ x = f (x, u) , (Σ) where f : IR n × IR m → IR n is a locally Lipschitz map, and we interpret x and u as functions of time t ∈ IR, with values x(t) ∈ IR n and u(t) ∈ IR m , for some positive integers n and m. (Generaliza- tions to systems evolving on manifolds, and/or re- stricted control value sets, are also of interest, but in this paper, all spaces are Euclidean.) From now on, one such system is assumed given. A control or input is a measurable locally essen- tially bounded function u : IR ≥0 → IR m . We use the notation kuk to indicate the L m ∞ -norm of u, and |·| for Euclidean norm in IR n and IR m . For each ξ ∈ IR n and each u ∈ L m ∞ , we denote by x(t, ξ, u) the trajectory of the system Σ with initial state x(0) = ξ and the input u. (This solution is a priori only defined on some maxi- mal interval [0,T ξ, u ), with T ξ, u ≤ +∞, but the main definition will include the requirement that T ξ, u =+∞.) We recall in an appendix the no-