On the perturbation of bimodal systems

Abstract

Given a bimodal system de¯ned by the equations ½ x_ (t) = A1x(t) + Bu(t) if ctx(t) · 0 x_ (t) = A2x(t) + Bu(t) if ctx(t) ¸ 0 (1) where B 2Mn;m and Ai 2Mn, i = 1; 2, are such that A1;A2 coincide on the hyper- plane V =Kerct. We consider in the set of matrices de¯ning the above systems the simultaneous feedback equivalence de¯ned by ([A1;B]; [A2;B]) » ([A0 1;B0]; [A0 2;B0]) if [A0 i B0] = S¡1[Ai B] · S 0 R T ¸ i = 1; 2 with S(V) = V This equivalent relation corresponds to the action of a Lie group. Under this action we obtain, in the case m · 1, the semiuniversal deformation, following Arnold's technique. Then the problem of structural stability is studied.