Anzeige
Erschienen in:
2013 | OriginalPaper | Buchkapitel
12. Invariance and monotonicity
verfasst von : Francis Clarke
Erschienen in: Functional Analysis, Calculus of Variations and Optimal Control
Verlag: Springer London
Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.
Wählen Sie Textabschnitte aus um mit Künstlicher Intelligenz passenden Patente zu finden. powered by
Markieren Sie Textabschnitte, um KI-gestützt weitere passende Inhalte zu finden. powered by
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 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.
$$x \,' (t)\: \in \:F( x(t)) , \;\;t\in\, [\,a ,b\,]{~\,\text{a.e.}}$$