Set-membership Method for Discrete Optimal Control

The objective of this paper is twofold. First we propose a new approach for computing Ct0,t f the subset of initial states of a system from which there exists at least one trajectory reaching a target T in a finite time t f from a timet0. This is done considering a discrete time tk and a control vector continuous over a time [tk−1, tk]. Then, using the previously mentioned work and given a cost function, the objective is to estimate an enclosure of the discrete optimal control vector from an initial state of Ct0,t f to the target. Whereas classical methods do not provide any guaranty on the set of state vectors that belong to the Ct0,t f , interval analysis and guaranteed numerical integration allow us to avoid any indetermination. We present an algorithm able to provide guaranteed characterizations of the inner C− t0,t f and an the outer C + t0,t f of Ct0,t f , such that C− t0,t f ⊆ Ct0,t f ⊆ C + t0,t f . In addition to that, the presented algorithm is extended in order enclose the discrete optimal control vector of the system, form an initial state to the target, by a set of discrete trajectories.