The peaking Phenomenon and Singular Perturbations : An Extension of Tikhonov's Theorem

Abstract

We study the asymptotic behaviour, when the parameter $\gamma$ tends to infinity, of a class of singularly perturbed triangular systems $\dot x=f(x,y)$, $\dot y=G(y,\gamma)$. The first equation may be considered as a control system recieving the inputs from the states of the second equation. With zero input, the origin of the first equation is globally asymptotically stable. We assume that all solutions of the second equation tend to zero arbitrarily fast when $\gamma$ tends to infinity. Some states of the second equation may peak to very large values, before they rapidly decay to zero. Such peaking states can destabilize the first equation. The paper introduces the concept of \em instantaneous stability, to measure the fast decay to zero of the solutions of the second equation, and the concept of uniform infinitesimal boundedness to measure the effects of peaking on the first equation. Whe show that all the solutions of the triangular system tend to zero when $\gamma$ and $t$ tend to infinity. Our results are a generalization of the classical Tikhonov's theorem of singular perturbation theory, concerning the asymptotic behaviour of the solutions in the particular case where the second equation is of the form $\dot y=\gamma G(y)$. Our results are formulated in both classical mathematics and nonstandard analysis