Systems & Control Letters 34 (1998) 93–100 Comments on integral variants of ISS 1 Eduardo D. Sontag ∗ Department of Mathematics, Rutgers University, Piscataway, NJ 08854-8019, USA Received 20 June 1997; received in revised form 15 November 1997 Abstract This note discusses two integral variants of the input-to-state stability (ISS) property, which represent nonlinear generalizations of L 2 stability, in much the same way that ISS generalizes L ∞ stability. Both variants are equivalent to ISS for linear systems. For general nonlinear systems, it is shown that one of the new properties is strictly weaker than ISS, while the other one is equivalent to it. For bilinear systems, a complete characterization is provided of the weaker property. An interesting fact about functions of type KL is proved as well. c  1998 Elsevier Science B.V. All rights reserved. Keywords: Input-to-state stability; ISS; Nonlinear stability; System gains 1. Introduction We deal here with controlled systems of the general form ˙ x = f(x; u); (1) where f : R n × R m → R n is continuous, and locally Lipschitz on x for bounded u, and inputs u(·): [0; ∞) → R m are assumed to be locally essentially bounded. The paper [7] introduced the notion of “input to state stability” (ISS), which roughly states that “no matter what is the initial state, if the inputs are uniformly small, then the state must eventually be small”. Some results, applications, and further developments can be found in, e.g. [4, 5, 9–11] as well as many other recent papers. One frequently remarked shortcoming of the ISS property is that provides no useful bounds in the situation in which inputs u(·) are unbounded but have in some sense * Fax: +1 908 445 5530; e-mail: sontag@control.rutgers.edu. 1 Supported in part by US Air Force Grant F49620-95-1-0101. nite energy. This note deals with two variants of ISS which take into account such “energy” information, and shows that one of them is, in fact, equivalent to ISS, while the other one is strictly weaker. Notations: For a vector z in a space R q , |z| de- notes Euclidean norm. If z : I → R q is a measurable function dened on an interval containing [0;t ], ‖z t ‖ denotes the (essential) supremum of {|z(s)|;s ∈ [0;t ]}; for t = ∞, we write just ‖z‖. In order to motivate the denitions, we rst review the classical case of linear systems ˙ x = Ax + Bu: One often denes input= output stability of such a sys- tem in various ways, depending on the norms being used for state and input trajectories. The most com- mon choices are L 2 and L ∞ norms. These can be used in various combinations, one of which (“L ∞ to L 2 ”) is less interesting, being far too restrictive. The three possibilities that remain are dened by requiring the existence of constants c and , with ¿0, so that, for each input u(·) and each initial state , the solution 0167-6911/98/$19.00 c  1998 Elsevier Science B.V. All rights reserved PII S0167-6911(98)00003-6