Journal Article•

# Globally Asymptotic Stability of a Class of Third Order Nonlinear System

TL;DR: In this article, a better Lyapunov function of third order nonlinear system x,g,f,+h(x)=0 is constructed, and sufficient conditions for the globally asymptotically stability of trivial solution of the nonlinear systems are obtained.

Abstract: The Lyapunov's second method is one of the basic methods to research the stability of nonlinear system.It is critical to construct a suitable Lyapunov function.In this paper,a better Lyapunov function of third order nonlinear system x…+g()+f(,)+h(x)=0 is constructed.The sufficient conditions for the globally asymptotically stability of trivial solution of the nonlinear systems is obtained,and the stronger condition of generally requiring Lyapunov function going to the infinite is removed,the results make further progress of some old results.

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TL;DR: By deflning a Lyapunov functional, the stability and boundedness of solutions to nonlinear third order difierential equations with constant delay were investigated in this paper.

Abstract: By deflning a Lyapunov functional, we investigate the stability and boundedness of solutions to nonlinear third order difierential equation with constant delay, r : x 000 (t) + g(x(t);x 0 (t))x 00 (t) + f(x(t i r);x 0 (t i r)) + h(x(t i r)) = p(t;x(t);x 0 (t);x(t i r);x 0 (t i r);x 00 (t)); when p(t;x(t);x 0 (t);x(t i r);x 0 (t i r);x 00 (t)) = 0 and 6 0; respectively. Our results achieve a stability result which exists in the relevant literature of ordinary nonlinear third order difierential equations without delay to the above functional difierential equation for the stability and boundedness of solutions. An example is introduced to illustrate the importance of the results obtained.

19 citations

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01 Jan 2016

TL;DR: In this article, the stability and boundedness of solutions of nonlinear and non-autonomous differential equations of third order with delay was studied. But the results on the subject are limited.

Abstract: In this paper, we establish some new sufficient conditions which guarantee the stability and boundedness of solutions of certain nonlinear and non autonomous differential equations of third order with delay. By defining appropriate Lyapunov function, we obtain some new results on the subject. By this work, we extend and improve some stability and boundedness results in the literature.

9 citations

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TL;DR: The results improve a stability result in the literature, which was obtained for nonlinear differential equations of third order without delay, to the above differential equation with delay for stability and boundedness of the solutions.

Abstract: We establish some sufficient conditions which guarantee asymptotic stability of the null solution and boundedness of all the solutions of the following nonlinear differential equation of third order with the variable delay, r(t) x ‴ ( t ) + g ( x ′ ( t − r ( t ) ) ) x ″ ( t ) + ψ ( x ′ ( t ) ) + f ( x ′ ( t − r ( t ) ) ) + h ( x ( t − r ( t ) ) ) = p ( t , x ( t ) , x ′ ( t ) , x ( t − r ( t ) ) , x ′ ( t − r ( t ) ) , x ″ ( t ) ) , when p(t, x(t), x′(t), x(t−r(t)), x′(t−r(t)), x′′(t))=0 and ≠0, respectively. By defining an appropriate Lyapunov functional, we prove two new theorems on the stability and boundedness of the solutions of the above equation. We also give an example to illustrate the theoretical analysis in this work. Our results improve a stability result in the literature, which was obtained for nonlinear differential equations of third order without delay, to the above differential equation with delay for stability and boundedness of the solutions.

1 citations