Invariance and monotonicity

Abstract

A venerable notion from the classical theory of dynamical systems is that of flow invariance. When the basic model consists of an autonomous ordinary differential equation x ′(t)=f(x(t)) and a set S, then flow invariance of the pair (S,f) is the property that for every initial point α∈ S, the solution x(⋅) satisfying x(0)=α remains in S : x(t)∈ S for all t ⩾ 0. In this chapter, we study highly useful generalizations of this concept to situations wherein the differential equation is replaced by an inclusion. A trajectory x of the multifunction F, on a given interval [ a,b ], refers to a function $x:[\,a ,b\,]\to\,{\mathbb{R}}^{ n}$ which satisfies the differential inclusion

$$x \,' (t)\: \in \:F( x(t)) , \;\;t\in\, [\,a ,b\,]{~\,\text{a.e.}}$$

When F(x) is a singleton {f(x)} for each x, the differential inclusion reduces to an ordinary differential equation. Otherwise, we would generally expect there to be multiple trajectories from the same initial condition. In such a context, the invariance question bifurcates: do we require some of, or all of, the trajectories to remain in S? This will be the difference between weak and strong invariance.