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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
It is of interest to know under what conditions the existence of a Lyapunov function φ is necessary, as well as sufficient, for system stability; we shall not pursue the issue of such converse Lyapunov theorems, however.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag London
About this chapter
Cite this chapter
Clarke, F. (2013). Invariance and monotonicity. In: Functional Analysis, Calculus of Variations and Optimal Control. Graduate Texts in Mathematics, vol 264. Springer, London. https://doi.org/10.1007/978-1-4471-4820-3_12
Download citation
DOI: https://doi.org/10.1007/978-1-4471-4820-3_12
Published:
Publisher Name: Springer, London
Print ISBN: 978-1-4471-4819-7
Online ISBN: 978-1-4471-4820-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)