On the Continuity of Asymptotically Stable Compact Sets for Simulations of Hybrid Systems

2006 ◽  
Author(s):  
Ricardo G. Sanfelice ◽  
Andrew R. Teel
Keyword(s):  
Hybrid Systems ◽  
Asymptotically Stable ◽  
Compact Sets

Asymptotic stability, an in-depth treatment

2012 ◽  
Author(s):  
Rafal Goebel ◽  
Ricardo G. Sanfelice ◽  
Andrew R. Teel
Keyword(s):  
Asymptotic Stability ◽  
Lyapunov Function ◽  
Hybrid Systems ◽  
Basin Of Attraction ◽  
Asymptotically Stable ◽  
Compact Sets ◽  
Stable Sets ◽  
Bounded Set ◽  
Well Posed ◽  
The Basin Of Attraction

This chapter defines local pre-asymptotic stability for a compact (closed and bounded) set and studies its properties for systems that are nominally well-posed or well-posed. While Chapter 3 dealt with global (and uniform) pre-asymptotic stability, the more general local pre-asymptotic stability is studied in this chapter, although for the more restrictive case of compact sets. Properties of the basin of attraction and uniformity of convergence are here analyzed, and some general examples of locally pre-asymptotically stable sets are given. The chapter reveals that, for nominally well-posed hybrid systems, pre-asymptotic stability turns out to be equivalent to uniform pre-asymptotic stability. For well-posed systems, pre-asymptotic stability turns out to be equivalent to uniform, robust pre-asymptotic stability and implies the existence of a Smooth Lyapunov function.

scholarly journals Asymptotically stable attracting sets in the Navier-Stokes equations

1986 ◽  
Vol 34 (1) ◽  
pp. 37-52 ◽  
Author(s):  
P. E. Kloeden
Keyword(s):  
Stokes Equations ◽  
Galerkin Approximation ◽  
Navier Stokes ◽  
Asymptotically Stable ◽  
Compact Sets ◽  
Navier Stokes Equations ◽  
Attracting Set ◽  
Galerkin Approximations ◽  
Steady Solutions ◽  
The Stability

The planar Navier-Stokes equations with periodic boundary conditions are shown to have a nearby asymptotically stable attracting set whenever a Galerkin approximation involving the eigenfunctions of the Stokes operator has such an attracting set, provided the approximation has sufficiently many terms and its attracting set is sufficiently strongly stable. Lyapunov functions are used to characterize the stability of these attracting sets, which are compact sets of arbitrary geometric shape. This generalizes earlier results of Constantin, Foias and Temam and of the author for asymptotically stable steady solutions in the Navier-Stokes equations and such Galerkin approximations.

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