Bracketing backward reach sets of a dynamical system

Abstract

In this paper, we present a new method for bracketing (i.e., characterizing from inside and from outside) backward reach set of the target region T of a continuous time dynamical system. The principle of the method is to formalize the problem as a constraint network, where the variables are the trajectories (or paths) of the system. The resolution is made possible by using mazes which is a set of paths that contain all solutions of the problem. As a result, we will be able to derive a method able to compute a backward reach set for a huge class of systems without any knowledge of a parametric Lyapunov function and without assuming any linearity for our system. The method will be illustrated on several examples.

Interval centred form for proving stability of non-linear discrete-time systems

“…Moreover, to our knowledge, none of the existing methods is able to give an approximation for the neighbourhoods δ and ε used in Lyapunov's definition. Now, finding values for δ and ε is needed in practice, for instance to initialize algorithms which approximate basins of attraction [13,21,26] or reachable sets [15]. This paper proposes an original approach to prove Lyapunov stability of a non-linear discrete system, but also to find values for δ and ε.…”

Kleene Algebra to Compute Invariant Sets of Dynamical Systems

“…Remark 1. In our context, the lattice L will correspond to the set of all subsets of R n and L M to the set of all machine sets (or mazes [39,40]). In Figure 1, the red polygon could represent the set of all positive invariant sets included in the set represented by e. The figure could also illustrate that given a set A, there exists a smallest invariant set that contains A, and there exists a largest element in P, which is included in A.…”