Abstract
In this paper a generalization of Lyapunov's equation for the stability of linear dynamical systems to globally asymptotically stable nonlinear systems is presented by embedding the system in a linear infinite-dimensional one on a tensor space by using Carleman linearization. This linear representation allows the definition and solution of a Lyapunov equation as in the usual linear case. The converse result is also discussed using the fact that globally asymptotically stable non linear systems are essentially linear.
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