On Barriers in State and Input Constrained Nonlinear Systems

Abstract

In this paper, the problem of state and input constrained control is addressed, with multidimensional constraints. We obtain a local description of the boundary of the admissible subset of the state space where the state and input constraints can be satisfied \emph{for all times}. This boundary is made of two disjoint parts: the subset of the state constraint boundary on which there are trajectories pointing towards the interior of the admissible set or tangentially to it; and a barrier, namely a semipermeable surface which is constructed via a minimum-like principle.Comment: 36 pages, 8 figures, submitte

Epidemic management with admissible and robust invariant sets

“…• To our knowledge, this is the first paper that deals with the problem of maintaining a hard infection cap for the SIR and SEIR epidemic models (under the assumption of both perfect and imperfect modelling), via a set-theoretic approach: the theory of barriers, [19,20].…”

“…We now summarise the relevant theory from [19,20], which we will apply to describe the sets of the two epidemic models. Consider a state and input constrained nonlinear system:…”

“…Definition 3.1. The admissible set also known as the viability kernel in Viability theory, [22], see [19], of the system (10) and (11), denoted by A, is the set of initial states for which there exists a u 2 U such that the corresponding solution satisfies the constraints (11) for all future…”

“…In this paper we build on the recent work concerning the application of set-based methods to epidemiology. We present a set-based analysis of two well-known compartmental epidemic models: the SIR and SEIR models, see for example [17,18], subjected to constraints on their inputs (social distancing, quarantine rate and/or vaccination rate) and state (a hard cap on the proportion of infectives), utilising the theory of barriers, [19,20]. These models often serve as good initial candidates to model new diseases, later being elaborated on to be more accurate.…”

Construction of invariant polytopic sets with specified complexity

“…The large number of vertices is due to the fact that the maximal controlled invariant set S max of the continuous-time system, although convex, is not a polytope. A local description of the boundary of the maximal controlled invariant set with curves for general continuous-time systems has been recently established in (De Dona and Levine 2013). The algorithmic procedure described in Example 4.3 was applied directly to the continuoustime system, with the obvious modification of changing the optimization constraints of the optimization problem solved to relations (33)-(38).…”