Ellipsoidal Techniques for Reachability Under State Constraints

Abstract

The paper presents a scheme to calculate approximations of reach sets and tubes for linear control systems with time-varying coefficients, bounds on the controls, and constraints on the state. The scheme provides tight external approximations by ellipsoidalvalued tubes. These tubes touch the reach tubes from the outside at each point of their boundary so that the surface of the reach tube is totally covered by curves that belong to the approximating tubes. The result is an exact parametric representation of reach tubes through families of external ellipsoidal tubes. The parameters that characterize the approximating elliposids are solutions of ordinary differential equations with coefficients given partly in explicit analytical form and partly through the solution of a recursive optimization problem. The scheme combines the calculation of external approximations of infinite sums and intersections of ellipsoids, and suggests an approach to calculate reach sets of hybrid systems.

State-Constrained Stochastic Optimal Control Problems via Reachability Approach

“…By the "levelset approach" we aim at describing the backward reachable set as the zero-level set of a suitable function. The idea was first introduced by Osher and Sethian [30] in the context of front propagation, and then extended to reachability problems, see [25,18,7,2,8] and the references therein.…”

A discontinuous Galerkin scheme for front propagation with obstacles

“…Then using a Gronwall Lemma we deduce that (w, w) ≤ e Ct (w(0), w(0)) = 0, thus w ≡ 0 and u 2 ≡ u 1 . Hence we have shown the equivalence between the "min" formulation of the obstacle problem (15) (or (7)), and the variational formulation (16).…”

“…We consider the problem to reach a target C. It is known from [16] (see also [8]), that if we considerφ 0 ≥ 0 such thatφ…”

Minimal Time Problems With Moving Targets and Obstacles

“…The case when K is an open set in R d is investigated in [15]. We also refer to [14] for a short discussion linking the reachable sets under state constraints to HJB equations. The treatment in this reference assumed a C 1 value function.…”