From Chronological Calculus to Exponential Representations of Continuous and Discrete-Time Dynamics: A Lie-Algebraic Approach

In this paper, formal exponential representations of the solutions to nonautonomous nonlinear differential equations are derived. It is shown that the chronological exponential admits an ordinary exponential representation, the exponent being given by an explicitly computable Lie series expansion. The results are then used to describe controlled dynamics, dynamics under sampling and forced discrete-time dynamics. The study emphasizes the role of Lie algebra techniques in nonlinear control theory and specifies structural similarities between nonautonomous differential equations, dynamics under sampling and forced discrete-time dynamics up to hybrid ones.