Lyapunov theory done categorically
(2026)
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- (en) It has been observed that in a physical system, energy is either conserved or dissipated. Under these conditions, the minimum-energy configuration is a stable equilibrium of the system. Alexandre Lyapunov extended this observation to any dynamical system by broadening the notion of energy, which is replaced by a Lyapunov function. Moreover, the existence of such a function is a necessary and sufficient condition to prove that a configuration is a stable equilibrium. These results form the basis of Lyapunov theory. In practice, Lyapunov theory is fragmented, because each class of dynamical system (continuous-time, discrete-time, hybrid, varying-geometry, stochastic, etc.) requires, for technical reasons, its own definition of stability, its own definition of Lyapunov function, and its own proof of the equivalence between the two, and so on. In this thesis, we use a mathematical tool (category theory) to abstract away the technical arguments and unify all these proofs. Thus, a single definition of stability is needed, which applies in different contexts and specializes into the usual definitions of stability for continuous-time/discrete-time/hybrid/etc. systems. Following this categorical definition, we state and prove all the necessary steps to perform Lyapunov second method in an abstract context, and recover usual theorems when looking at specific contexts.
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Mattenet, S. (2026). Lyapunov theory done categorically. https://hdl.handle.net/2078.5/280838