$\mathcal{KL}$ and Lyapunov Approaches for Discrete-time Peak Computation Problems

In this paper, we propose a method to solve discrete-time peak computation problems (DPCPs for short). DPCPs are optimization problems that consist of maximizing a function over the reachable values set of a discrete-time dynamical system. The optimal value of a DPCP can be rewritten as the supremum of the sequence of optimal values. Previous results provide general techniques for computing the supremum of a real sequence from a well-chosen pair of a strictly increasing continuous function on [0,1] and a positive scalar in (0,1). In this paper, we exploit the specific structure of the optimal value of the DPCP to construct such a pair from classical tools from stability theory: $\mathcal{KL}$ certificate and Lyapunov functions.