On the asymptotic behavior of solutions

in the critical case of two pairs of purely imaginary roots

K.Peiffer

Inst. de Math Pure et Appl., Chemin du cyclotron 2, 1348 Louvain-la-Neuve, Belgium

A.Ya.Savchenko, A.L.Zuyev

Inst. Prikl. Mat. Mekh., Ul. Rozi-Luxemburg 74, 340114 Donetsk-114, Ukraine

 

The asymptotic behavior of a differential system is investigated in the critical case of two pairs of purely imaginary roots. It is supposed that all other roots have negative real part and that stability is obtained using forms of finite order. The asymptotic behavior of such a system can sometimes be deduced from an associated simpler system. This has been proved by G.V.Kamenkov and is applied here to problems of nonlinear optimal passive stabilization.

Let us consider the system

,љљљ .

We suppose that the solution љis not asymptotically stable in the sense of Lyapunov. The problem of passive stabilization by means of defreezing parameters leads us to introduce an additional state variable , in such a way that the origin of the following extended system is asymptotically stable:

љљљљљљљљљљљљљљљљљљљљљљ љљљљљљљљљљљљљљљљљљљљљљљљљ(1)

.

where

љ

In practice, matrix A and coefficients љshould be determined in some "admissible" way which is dictated by the modelled phenomena.

The critical case of one pair of purely imaginary roots was cosidered in the earlier paper. It has been shown that if some polynomial of fourth degree is a Lyapunov function, then the critial variables behave like љas .

The present paper deals with the case where the linear part of (1) has two pairs of purely imaginary roots, the other ones having negative real part.