On Some Theorems of Liapunov's Asymptotic Stability
Yu Bohua
In this paper six theorems, giving some generalization of Chow Shuer's and Matrosov's results, are proved. Consider the differential equation dx/dt=X(x,t)(x,t)∈G [X(0,t)=0],(1) where x and X are real nvectors, t denotes time (a real variable), and X is defined on We assume X smooth enough to ensure existence, uniqueness and continuous dependence of the solutions of the initial value. We obtain the following results: If there exists a continuous function v(x,t) which is positivedefinite in the domain G with v(0,t)=0 and where i) θ(t) is an integrable function defined in the interval t≥0, and satisfies the condition uniformly in t_0≥0; ii) the function C(v) is continuous, C(v)0 for v0 and C(0)=0; iii) the function v~*=v~*(x,t) is negativedefinite or v~*≡0 in the domain G, then the zero solution of (1) is asymptotically stable in the sense of Liapunov and is equiasymptotically stable. If v(x,t) is positivedefinite with infinitely small upper bound and satisfies the conditions i), ii), iii), here i)' the function θ(t) is defined in the interval t≥0, integrable and in any t_0≥0, for tt_0 then the zero solution of (1) is uniformly stable. If v(x,t) is positivedefinite with infinitely small upper bound and satisfies the conditions i), ii), iii), then the zero solution of (1) is uniformly asymptotically stable. If the differential equation (1) and the function v(x,t) are definited in E~n×E_+~1, where E_+~1:0≤t∧+ ∞, E~n is Euclidean space, and if v(x,t) is positivedefinite and infinitely large and the conditions i),ii), iii) are held, then the zero solution of (1) is globally asymptotically stable. In addition to above conditions, if v(x,t) exists an infinitely small upper bound, then the zero solution of (1) is globally uniformly asymptotically stable.


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