IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 41, NO. 9, SEPTEMBER 1996 1283 New Characterizations of Input-to-State Stability Eduardo D. Sontag, Fellow, IEEE, and Yuan Wang Abstruct- We present new characterizations of the input-to- state stability property. As a consequence of these results, we show the equivalence between the ISS property and several (apparent) variations proposed in the literature. I. INTRODUCTION HIS paper studies stability questions for systems of the general form E: i = f(z, U) (1) with states z(t) evolving in Euclidean space EL” and controls U(.) taking values u(t) E W & Et” for some positive integers w and m (in all the main results, W = R”). The questions to be addressed all concern the study of the size of each solution z(t)-its asymptotic behavior as well as maximum value-as a function of the initial condition z(O) and the magnitude of the control U(.). One of the most important issues in the study of control systems is that of understanding the dependence of state tra- jectories on the magnitude of inputs. This is especially relevant when the inputs in question represent disturbances acting on a system. For linear systems, this leads to the consideration of gains and the operator-theoretic approach, including the formulation of H““ control. For not-necessarily linear systems, there is no complete agreement as yet regarding what are the most useful formulations of system stability with respect to input perturbations. One candidate for such a formulation is the property called “input-to-state stability” (ISS), introduced in [12]. Various authors, e.g., [4]-[6], [lo], and [17], have subsequently employed this property in studies ranging from robust control and highly nonlinear small-gain theorems to the design of observers and the study of parameterization issues; for expositions see [14] and most especially [7] and [XI. The ISS property is defined in terms of a decay estimate of solutions and is known (cf. [15]) to be equivalent to the validity of a dissipation inequality T holding along all possible trajectories (this is reviewed below) for an appropriate “energy storage” function V and comparison Manuscript received July 21, 1995; revised April 29, 1996. Recommended by Associate Editor, A. M. Bloch. This research was supported in part by the US Air Force under Grant F49620-95-1-0101 and also in part by the NSF under Grants DMS-9457826 and DMS-9403924. E. D. Sontag is with the Department of Mathematics, SYCON-Rutgers Center for Systems and Control, Rutgers University, New Brunswick, NJ 08903 USA (e-mail: sontag@control.rutgers.edu). Y. Wang is with Department of Mathematics, Florida Atlantic University, Boca Raton, FL 33431 USA. Publisher Item Identifier S 001 8-9286(96)06777-3. functions o, Q. (A dual notion of “output-to-state stability” (OSS) can also be introduced and leads to the study of nonlinear detectability; see [ 161.) In some cases, notably in [2], [6], and [18], authors have suggested apparent variations of the ISS property which are more natural when solving particular control problems. The main objective of this paper is to point out that such variations are in fact theoretically equivalent to the original ISS defini- tion. (This does not in any way diminish the interest of these other authors’ contributions; on the contrary, the alternative characterizations are of great interest, especially since the actual estimates obtained may be more useful in one form than another. For instance, the “small-gain theorems” given in [2] and [6] depend, in their applicability, on having the ISS property expressed in a particular form. This paper merely states that from a theoretical point of view, the properties are equivalent. For an analogy, the notion of “convergence” in R” is independent of the particular norm used-e.g., all Lp norms are equivalent-but many problems are more naturally expressed in one norm than another.) One of the main conclusions of this paper is that the ISS property is equivalent to the conjunction of the following two properties: i) asymptotic stability of the equilibrium z = 0 of the unforced system (that is, of the system defined by (1) with U = 0) and ii) every trajectory of (1) asymptotically approaches a ball around the origin whose radius is a function of the supremum norm of the control being applied. We prove this characterization along with many others. Since it is not harder to do so, the results are proved in slightly more generality, for notions relative to an arbitrary compact attractor rather than the equilibrium z = 0. A. Basic Definitions and Notations Euclidean norm in R” or Rm is denoted simply as 1 . 1. More generally, we will study notions relative to nonempty subsets A of R”; for such a set A, 1tl.A = d(E, A) = inf {d(q, E), 11 E A} denotes the point-to-set distance from E R” to A. (So for the special case A = {0}, IEl{o> = It\.) We also let, for each E > 0 and each set A Most of the results to be given are new even for A = {0}, so the reader may wish to assume this and interpret 1El.A simply as the norm of E. (We prefer to deal with arbitrary A because of potential applications to systems with parameters as well as the “practical stability” results given in Section VI.) 00 18-9286/96$05.00 0 1996 IEEE