First and invariant integrals in stability problems Original Research Article

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 differential system View the MathML source, f(t,0)≡0, for which x=0 is uniformly asymptotically stable for perturbations lying on an appropriate invariant set Φ. Precisely in [Salvadori (1999)] it is assumed that View the MathML source, where F, F(t,0)≡0, is a smooth first 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 [Peiffer (Rend. Sem. Mat. Univ. Padova, 92 (1994) 165)]. In the present paper we weaken the sufficient condition for the stability of the origin given in [Salvadori (1999)] (Φ-positive definitiveness of F). The sufficiency is preserved but the new condition is also necessary for uniform stability and then necessary and sufficient 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 Φ (for perturbations close to the origin). Even the asymptotic stability of x=0 appears to be connected to corresponding asymptotic stability properties of Φ. If the differential system is periodic, the origin and Φ have the same stability properties. If the differential system is autonomous the results are extendable to the stability of a compact invariant set M. In particular it is shown that M and Φ have the same stability properties. This statement still holds if F=0 is a t-independent invariant integral, with F not necessarily a first integral. This latter result is illustrated through its application to a bifurcation problem.