UDC531.36 A. A. Martynyuk* (Inst. Mech. Nat. Acad. Sci. Ukraine, Kiev), I. P. Stavronlakis (Univ. Ioannina, Greece) STABILITY ANALYSIS OF LINEAR IMPULSIVE DIFFERENTIAL SYSTEMS UNDER STRUCTURAL PERTURBATION AHAdlI3 CTIi:IKOCTI dIIHIi:IHHX J1H| IMIIYdlbCHHX CHCTEM I3 CTPYKTYPHHMH 3BYPEHHRMH The stability and asymptotic stability of the solutions of large-scale linear impulsive systems under structural perturbationsare investigated. Sufficientconditionsfor stabilityand instability are formulated in terms of the fixed signs of special matrices, j]ocJfi~tmyloTLcz crittKiCTb Ta acHMrrro'mtma eritlKiCTb pO3n'Jl3Kie BeattKOMacmTa6Hoi ~iHitlHOi iM- ny~bcHoiCHCTeMH npH eTpyKTypHHX 36ypellH.qX./]oera-mi yMOBH eTiltKOeTi "ra HCeTi~IKOCTi cqbopMy- /lbOBaHi Ha OCHOBi ~laKOBtl3HatleHOeTi crlellia.rlbltHX MaTpHLU,. 1, Introduction. Many of the processes in engineering and technology deal with an overcoming of the "threshold" phenomena. This is expressed in particular, in accomulation by the process of some property with the consequent sudden change of the state. The modelling of such a process when the ordinary differential equations are employed is difficult to some extend and an attempt to encorporate new classes of the systems of equations seems natural. The impulsive systems with structural perturbations belong to a class of this type. This paper concentrates on the investigation of stability and instability of large scale linear impulsive systems under structural perturbations by the Lyapunov's direct method in terms of matrix-valued functions. Sufficient conditions for various types of stability and instability are established. A numerical example showing the application of some general results is given. 2. Preliminaries. According to [1 -3] we consider the linear large scale impulsive system decomposed into s subsystems dx i s = + X S U t jffil $ ax~ = Jkixi + ~ Jkijx~, j=l j*i t --- xk(x), (1) i=1,2 ..... s, k=l,2 ..... where xie R n', X~=l ni = n, x = (x~,x~i .... xrs) r e R n, A i, J~, Aij, Jkij are Constant matrices of the correspondent dimensions, the set ~, and matrices S, Si, Sij are defined in Appendix 1, the values xt(x), k = 1, 2 ..... are ordered by %k(x) < < xk+ i (x) and such that %t(X) ~ + .0 as k --> + -0. We shall assume, for simplicity, that the system (1) satisfies all required conditions so that all solutions x(t) = x(t, t o, x0) of (1) exist for alI t > t0. For the system (1) we construct a matrix-valued function * This work was done while the author was visiting the Department of Mathematics, University of Ioannina. in the framworkof the NATO ScienceFellowshipsProgrammethrough the Greek Ministry of National Economy. O A. A. MARTYNYUK.I. P. STAVROULAKtS, 1999 784 155N 0041-6053. Yr, p. uam, ~. pn., 1999, m. 51,1~ 6