New Characterizations of Input-to-State Stability - Automatic Control, IEEE Transactions on

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 trajectories 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