Application Of Lozinskii Measure In The Stability Of Ordinary Differential Equation Systems

In this paper ,we discuss the stability problems of ordinary differential equation systems:And P (t) is an n order matrix.R(t, x) is constant when t≥t 0,â•‘x║≤M, and satisfies If R(t, x)≡0, then (1) changes into x′= P(t)x (2) Usually (2) is the homogeneous linear equations of (1) .Using the following symbolics:1. Supposes R is a real number set.2. C is a complex matrix set, and R+=[0,+∞). 3. Supposes X, Y are two Banach spaces.4. Uses C(X, Y), C′(X, Y) and Llog(X, Y) to represent continual, continuously differentiable, and the operators set of locally integal on X and map X on Y.5. Uses C*′(X, Y) to represent an continuous-differential operator set , particulary it is a closed subset with zero measure except for X.6.To any n×n order matrix P, useλi (P), (i=1,2,…, n) to represent the characteristic values of matrix P, these characteristic values satisfy Reλ1(P)≥Reλ2(P)≥…≥Reλn(P)Research the problem about the stability of the ordinary differential equation system by matrix Lozinskii measure.First, (refers to [1-6]) give the definition of stability (definition 2.1) and (according to [6-14] definition and nature) the definition and the important nature of matrix Lozinskii measure (definition 2.2) .Then gives and proves lemma 1.1 Lemma 1.1 Supposes X(t) is a n×n order matrix solution of (2) , E, F∈C(*′[t 0,∞],Cn×n)are two non-singular matrices ,to all t≥t₀,By lemma , we get corollary 2.1.1 and corollary 2.1.2, then get the stability criterion of linear ordinaary differential systems (2) :Theorem 1.1 (â…°) If there is a non-singular matrix the equation (2) is unstable.(â…±) If there is a non-singular matrix to all t≥t₀≥0, the equation (2) is stable. the equation (2) is asymptotic stability. (â…³)If there is a non-singular matrix to all t≥t₀≥0, the equation (2) is continuous stability. (â…´)If there is a non-singular matrix to K ,α> 0,t≥t₀≥0, the equation (2) is continuous and asymptotic stability. Among them K,αare constants, K ( t 0) is a function about t₀.Theorem 1.2 (â…°) One necessary and sufficient condition to keep equation (2) stable is that there must be a non-singular matrix E∈C(*′R+,Cn×n), make E ?1 limited when t≥0,and(â…±) One necessary and sufficient condition to keep equation (2) as an asymptotic stability system is that there must be a non-singular matrix E∈C(*′R+,Cn×n), make E^-1 limited when t≥0,andGenerally, to the nonlinear ordinary differential systems(1), if we can satisfy that to be added with the item ERX ? 1 E?1, we can also get the similar conclusion in all the lemmas, the deductions, the theorem integrands, then we get the stability criterion of nonlinear ordinary differential systems (theorem 2.2.1 and theorem 2.2.2).