Stability Of Functional Differential Equations Research Articles

Liapunov-Razumikhin conditions for asymptotic stability in functional differential equations of Volterra type

In [l], B.S. Razumikhin gave conditions for the stability and asymptotic stability of the zero state of systems of ordinary differential equations involving a fixed finite time delay. His conditions make use of Liapunov functions defined on the finite-dimensional state space rather than on the space of functions continuous on the interval of delay; for methods involving functions of the latter type, cf., for example, Yoshizawa 121. In a recent paper [3], the author has used Liapunov functions of Razumikhin type to give conditions sufficient for the stability of the zero state of a system of ordinary differential equations involving an interval of delay which becomes unbounded as t + + co; an example of such a system is an integrodifferential equation of Volterra type:

On the Uniform Asymptotic Stability of Functional Differential Equations of the Neutral Type

On the Uniform Asymptotic Stability of Functional Differential Equations of the Neutral Type

On the uniform asymptotic stability of functional differential equations of the neutral type

Consider the functional equations of neutral type (1) ( d / d t ) D ( t , x t ) = f ( t , x t ) (d/dt)D(t,{x_t}) = f(t,{x_t}) and (2) ( d / d t ) [ D ( t , x t ) − G ( t , x t ) ] = f ( t , x t ) + F ( t , x t ) (d/dt)[D(t,{x_t}) - G(t,{x_t})] = f(t,{x_t}) + F(t,{x_t}) where D , f D,f are bounded linear operators from C [ a , b ] C[a,b] into R n {R^n} or C n {C^n} for each fixed t t in [ 0 , ∞ ) , F = F 1 + F 2 , G = G 1 + G 2 , | F 1 ( t , ϕ ) | ≦ v ( t ) | ϕ | , | G 1 ( t , ϕ ) | ≦ π ( t ) | ϕ | , π ( t ) [0,\infty ),F = {F_1} + {F_2},G = {G_1} + {G_2},|{F_1}(t,\phi )| \leqq v(t)|\phi |,|{G_1}(t,\phi )| \leqq \pi (t)|\phi |,\pi (t) , bounded and for any ε > 0 \varepsilon > 0 , there exists δ ( ε ) > 0 \delta (\varepsilon ) > 0 such that | F 2 ( t , ϕ ) | ≦ ε | ϕ | , | G 2 ( t , ϕ ) | ≦ ε | ϕ | , t ≧ 0 , | ϕ | > δ ( ε ) |{F_2}(t,\phi )| \leqq \varepsilon |\phi |,|{G_2}(t,\phi )| \leqq \varepsilon |\phi |,t \geqq 0,|\phi | > \delta (\varepsilon ) . The authors prove that if (1) is uniformly asymptotically stable, then there is a ζ 0 , 0 > ζ 0 > 1 {\zeta _0},0 > {\zeta _0} > 1 , such that for any p > 0 , 0 > ζ > ζ 0 p > 0,0 > \zeta > {\zeta _0} , there are constants v 0 > 0 , M 0 > 0 , s 0 > 0 {v_0} > 0,{M_0} > 0,{s_0} > 0 , such that if π ( t ) > M 0 , t ≧ s 0 , ( 1 / p ) ∫ t t + p v ( s ) d s > ζ v 0 , t > 0 \pi (t) > {M_0},t \geqq {s_0},(1/p)\int _t^{t + p} {v(s)ds > \zeta {v_0}} ,t > 0 , then the solution x = 0 x = 0 of (2) is uniformly asymptotically stable. The result generalizes previous results which consider only terms of the form F 1 , G 1 {F_1},{G_1} or F 2 , G 2 {F_{2,}}{G_2} but not both simultaneously, and the stronger hypothesis lim t → ∞ π ( t ) = 0 {\lim _{t \to \infty }}\pi (t) = 0 .

Stability of functional differential equations of neutral type

Abstract : A functional differential equation of neutral type is a differential system in which the rate of change of the system depends not only upon the past history but also the derivative of the past history of the system. For example, the system (1.1) x dot (t) + A x dot (t - 1) = f(t,x(t),x(t - 1)) is a functional differential or differential difference equation of neutral type. It is the purpose of this paper to give sufficient conditions for the stability and instability of solutions of a large class of equations (1.1) in terms of functions similar to those occurring in the application of the second method of Liapunov to ordinary and functional differential equations of retarded type. The basic restriction on the class of systems is that the derivatives occur linearly with coefficients depending only upon t and that the 'difference' operator associated with the equation is stable. (Author)