First and invariant integrals in stability problems

✍ Scribed by Luigi Salvadori; Francesca Visentin

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 144 KB
• Volume: 55
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/s0362-546x(03)00220-7

✦ Synopsis

We consider again the problem already treated in [Salvadori (Math. Japon. 49 (1) (1999) 1)] of unconditional stability properties of the solution x=0 of a smooth di erential system ẋ=f(t; x), f(t; 0) ≡ 0, for which x = 0 is uniformly asymptotically stable for perturbations lying on an appropriate invariant set . Precisely in [Salvadori (1999)] it is assumed that ={(t; x): F(t; x)= 0}, where F, F(t; 0) ≡ 0, is a smooth ÿrst integral. Previously to our research the problem was analyzed for autonomous systems in [Aeyels (Systems and Control Letters, Vol. 19, North-Holland, Amsterdam, 1992)] and for periodic systems in [Pei er (Rend. Sem. Mat. Univ. Padova, 92 (1994) 165)]. In the present paper we weaken the su cient condition for the stability of the origin given in [Salvadori (1999)] ( -positive deÿnitiveness of F). The su ciency is preserved but the new condition is also necessary for uniform stability and then necessary and su cient for stability in the periodic case. The results follow from the connections which have been found between the stability of the origin and the stability of the set (for perturbations close to the origin). Even the asymptotic stability of x = 0 appears to be connected to corresponding asymptotic stability properties of . If the di erential system is periodic, the origin and have the same stability properties. If the di erential system is autonomous the results are extendable to the stability of a compact invariant set M . In particular it is shown that M and have the same stability properties. This statement still holds if F = 0 is a t-independent invariant integral, with F not necessarily a ÿrst integral. This latter result is illustrated through its application to a bifurcation problem.