Input-output criteria for stability and expansiveness of dynamical systems

The aim of the present paper is to provide new criteria for the asymptotic behaviors of dynamical systems, approaching simultaneously four directions: nonuniform, uniform stability and respectively nonuniform, uniform expansiveness, treating both discrete and continuous-time case and using computational type methods. In order to characterize the asymptotic properties of discrete variational systems, we consider two classes of weighted ℓ∞-spaces and we define two types of nonuniform input-output properties, by employing spaces from these classes in the admissible pair. Our study continues the line initiated in Dragičević, Sasu and Sasu [J. Differential Equations 268 (2020), 4786-4829] and [J. Dynam. Differential Equations, doi.org/10.1007/s10884-020-09918-4], giving a completely new perspective regarding the connections between admissibility techniques for stability and expansiveness. We obtain necessary and sufficient conditions for nonuniform, uniform exponential stability and also for nonuniform, uniform exponential expansiveness, providing a unified approach and emphasizing the key differences when passing from nonuniform to uniform behavior. Next, we present two categories of applications of the discrete-time results. First, we deduce criteria for stability and expansiveness in terms of input-output type properties along period orbits. After that, we obtain characterizations for uniform exponential stability and uniform exponential expansiveness of skew-product semiflows in terms of the solvability of an associated integral control system between well-chosen weighted spaces of continuous functions, pointing out new computational methods of recovering the continuous information from the discrete-time behavior in the framework of nonuniform input-output techniques. Our results are applicable to general classes of variational systems.