Weak exponential stability of linear time-varying differential behaviors

We develop a new approach to exponential stability of linear time-varying (LTV) differential behaviors that is analogous to that in our paper on exponential stability of discrete LTV behaviors (H. Bourlès et al., 2015 [5]). Stability theory for differential state space systems with smooth coefficients is an important subject in the literature. For differential LTV behaviors with arbitrary smooth coefficients there is no reasonable stability theory. Therefore we restrict the smooth varying coefficients to functions that are defined by means of locally convergent Puiseux series. All rational functions are of this type. We introduce a new kind of behaviors and prove a module-behavior duality for these. We define a new notion of weak exponential stability (w.e.s.) of a behavior B and its associated finitely generated (f.g.) module M and show that the w.e.s. modules and behaviors are closed under isomorphisms, subobjects, factor objects and extensions. The standard uniform exponential stability of state space equations is not preserved under behavior isomorphisms and unsuitable for a behavioral theory. In the main result we assume a nonzero f.g. torsion module M and its associated autonomous behavior B. Such a module may be regular or irregular singular according to the Galois theory of differential equations. If it is nonzero and regular singular it is never w.e.s. For irregular singular M we characterize w.e.s. of most B algebraically and constructively.