比较定理 the comparison theorem.docx

比较定理 the comparison theoremA Note on Comparison Theorems for Nonnegative MatricesInRiemannian geometryit is a traditional name for a number of theorems that compare various metrics and provide various estimates in Riemannian geometry. Rauch comparison theoremrelates thesectional curvatureof aRiemannian manifoldto the rate at which itsgeodesicsspread apart. Toponogovs theorem Myerss theorem Hessian comparison theorem Laplacian comparison theorem MorseSchoenberg comparison theorem Berger comparison theorem,RauchBerger comparison theorem, M. Berger, An Extension of Rauchs Metric Comparison Theorem and some Applications, llinois J. Math., vol. 6 (1962) 700712 BergerKazdan comparison theorem2 Warner comparison theoremforlengthsofN-Jacobi fields(Nbeing a submanifold of a complete Riemannian manifold) F.W. Warner, Extensions of the Rauch Comparison Theorem to Submanifolds (Trans. Amer. Math. Soc., vol. 122, 1966, pp. 341356). BishopGromov inequality, conditional on a lower bound for theRicci curvatures(R.L. Bishop & R. Crittenden,Geometry of manifolds) Lichnerowicz comparison theorem Eigenvalue comparison theorem Chengs eigenvalue comparison theoremSee also:Comparison triangleLyapunov comparison theorem qualitative analysis of large scale dynamical systemsChapter IV Systems Described by Stochastic Differential Equations【1】Lyapunov stability and practical stability of nonlinear delay stochastic systems: a unified approachTheorem 3.1【1】. Let (AI) and (A21 hold. Assume that for the solution process x(t)=w(to.#,)(t) of (2.11, EIY(t,x(t)l exists on thto-7. Then dA(t) E I Y t , t , # t l r t , # t , trto (3.7) whenever ElV( to+s.#,(s) 1 I,(s), SE J (3.8) Stability of linear differential equations with distributed delayDefinition 1 (comparison principle) A dynamic system ( A ) is said to be a Comparison System of a dynamic system ( B ) with regard to the property P (for example, stability of its zero solution), i f the verification of property P for system ( A ) implies the same property for system ( B ) .Remark 1 Note that in the linear cases such an overvaluing system constitutes a comparison system with regard to (asymptotic) stability. Indeed, if V ( x ( t ; to; 4 ) ) 5 z ( t ; to; %to),& 2 to, then any property relative to stability verified b y z(t; to; %to) is verified b y x(t; to; 4).Improved Razumikhin-Type Theorem for Input-to-State Stability of Nonlinear Time-Delay SystemsTheorem 1. Assume that there exists a continuously differentiable function V : R+ Rn R+, functions 1, 2 K,functions , K, and a continuous function : R+ R such that(i) 1(|x|) V (t, x) 2(|x|), (t, x) R+ Rn;(ii) V (t, (0) (t)V (t, (0) for V (t, (0) max(|u|), (V (t+, (), (t, (0) R+Rn, and u Rm;(iii) R t0+(s)ds , and there exists a positive constant such that, for a sufciently large t, R tt0(s)ds (t t0), where +(s) = (s) 0and (s) = (s) 0; and(iv) r C(r) K.Then, System (1) is uniformly ISS and (r) = 11(2C(r),where C = eR 0+(s)ds .Small-Gain Conditions and the Comparison Principle Partial robust connective stability analysis for a class of interconnected power systems1Large-Scale Dynamic Systems: Stability and Structurecomparison theorem stochasticAsymptotic stability of stochastic impulsive system via comparison approach定积分上的比较定理性质5 设f为a，b上的可积函数。若则 （4） 证 由于在a，b上，因此f的任一积分和都为非负。由f在a，b上可积，则有 推论（积分不等式性）(比较定理)若f与g为a，b上的两个可积函数，且a，b，则有 （5） 证 令Fa，b，由性质2知道F在a，b上可积，且由性质5推