Liapunov functions for autonomous systems of second order

Anderson, Larry R.; Leighton, Walter

doi:10.1016/0022-247x(68)90145-5

Cited by 12 publications

(21 citation statements)

References 8 publications

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“…With kx= k2> 0, the function (2.3) becomes kt K2/2, where V = I(y) + P(x) is the function given in [2], In this case the estimate of the stability region given by (2.3) is the same as the estimate given by V. There are, however, many general cases in which constants A.! and X2 may be chosen so as to improve the estimate given by V. One such general case will be explored thoroughly in the following section.…”

Section: Vg(x Y) = [ 6{x S)f(x S) Ds -[ 0[s /I(s)]g[s /I(s)] Dsmentioning

confidence: 91%

“…In the special case of a Lienard equation we give general conditions under which one can choose constants ll5 x2 in (1.2) so as to improve the estimate of the region of asymptotic stability over well-known estimates that may be obtained by using 6 with -k2 . Finally, we apply the method to an equation that appears extensively in the literature [2,3,4,5] and obtain improved estimates.…”

Section: Vg(x Y) = [ 6{x S)f(x S) Ds -[ 0[s /I(s)]g[s /I(s)] Dsmentioning

confidence: 99%

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Weight functions for a class of Liapunov functions in the plane

Anderson¹,

Ducich²

1981

Quart. Appl. Math.

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Abstract. In this paper we give a class of weight functions which generate Liapunov functions for a general second-order differential system. In the special case of a Lienard equation we give conditions under which these weight functions may be chosen so as to improve certain known estimates of regions of asymptotic stability. The procedure is applied to a well-known equation and new estimates are obtained.

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Section: Vg(x Y) = [ 6{x S)f(x S) Ds -[ 0[s /I(s)]g[s /I(s)] Dsmentioning

confidence: 91%

Section: Vg(x Y) = [ 6{x S)f(x S) Ds -[ 0[s /I(s)]g[s /I(s)] Dsmentioning

confidence: 99%

Weight functions for a class of Liapunov functions in the plane

Anderson¹,

Ducich²

1981

Quart. Appl. Math.

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“…As in [4] and [5], we construct a new energy function, adapted to system (2.17), by using the Anderson and Leighton formula for autonomous systems, see [1]. Let…”

Section: (113)mentioning

confidence: 99%

The p-Laplace Heat Equation With a Source Term: Self-similar Solutions Revisited

Bidaut‐Véron

2006

Advanced Nonlinear Studies

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We study the self-similar solutions of any sign of the equationwhere p, q > 1. We extend the results of Haraux-Weissler obtained for p = 2 to the case q > p − 1 > 0. In particular we study the existence of slow or fast decaying solutions. For given t > 0, the fast solutions u(t, .) have a compact support in R N when p > 2, and |x| p/(2−p) u(t, x) is bounded at infinity when p < 2. We describe the behaviour for large |x| of all the solutions. According to the position of q with respect to the first critical exponent p − 1 + p/N and the critical Sobolev exponent q * , we study the existence of positive solutions, or the number of the zeros of u(t, .). We prove that any solution u(t, .) is oscillatory when p < 2 and q is closed to 1.

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“…Then Vw is a Liapunov function for the system (1.1), where a = (<f>°)(<£°)_1. In some earlier papers (see [1], [2], [3]) concerning second-order systems, particular attention was given to considering estimates of regions of asymptotic stability obtained from Vw by varying the weight function w. For certain weight functions and certain second-order systems, one obtains better estimates than with the single estimate provided by Vw with w = 1. In certain cases (see [1], [2]) optimal estimates over certain subclasses of weight functions may be found.…”

mentioning

confidence: 99%

“…In 1963, Walter Leighton published a paper [7] in which he provided Liapunov functions for general classes of second-and third-order differential equations. In a subsequent paper [3], the current author and Walter Leighton considered Liapunov functions for second-order systems that were more general than those given in Leighton's 1963 paper. Further, this latter paper gave a class of weighted Liapunov functions for certain second-order systems.…”

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confidence: 99%

Weighted Liapunov functions for a class of third-order autonomous differential equations

Anderson¹

1987

Quart. Appl. Math.

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1. Introduction. In 1963, Walter Leighton published a paper [7] in which he provided Liapunov functions for general classes of second-and third-order differential equations. In a subsequent paper [3], the current author and Walter Leighton considered Liapunov functions for second-order systems that were more general than those given in Leighton's 1963 paper. Further, this latter paper gave a class of weighted Liapunov functions for certain second-order systems. Further work utilizing weighted Liapunov functions for second-order equations was published by the current author in [1] and by S. Duchich and the current author in [2], A separate paper by A. Skidmore [8] extended Leighton's 1963 work on third-order equations to fourth-order equations, but no "weighting" was considered in either paper.The purpose of this paper is to extend Leighton's work (in [7]) on third-order systems by providing a class of weighted Liapunov functions for an equation

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Liapunov functions for autonomous systems of second order

Cited by 12 publications

References 8 publications

Weight functions for a class of Liapunov functions in the plane

Weight functions for a class of Liapunov functions in the plane

The p-Laplace Heat Equation With a Source Term: Self-similar Solutions Revisited

Weighted Liapunov functions for a class of third-order autonomous differential equations

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