Concerning Liapunov functions for linear systems— I†

Sreedhar, N.

doi:10.1080/00207177008905892

Cited by 5 publications

(2 citation statements)

References 5 publications

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“…For example, LaSalle theorem guarantees (under some conditions) the convergence of ẏ = f (y) to an invariant set, but not to a single point, provided f (x) T ∇V (x) ≤ 0 everywhere [10]. Convergence to 0 can then be guaranteed under the additional assumption that d dt V (y(t)) = f (y(t)) T ∇V (y(t)) is not uniformly zero along any trajectory other than that staying at 0 [13]. For more detail on various cases of unforced systems, we refer the reader to [11] as a starting point.…”

Section: Introductionmentioning

confidence: 99%

Trajectory convergence from coordinate-wise decrease of general energy functions

Hendrickx¹,

Gerencsér²

2022

Preprint

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We consider arbitrary trajectories subject to a coordinate-wise energy decrease: the sign of the derivative of each entry is never the same as that of the corresponding entry of the gradient of some convex energy function. We show that this simple condition guarantees convergence to a point, to the minimum of the energy functions, or to a set where its Hessian has very specific properties. This extends and strengthens recent results that were restricted to quadratic energy functions.

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Section: Introductionmentioning

confidence: 99%

Trajectory convergence from coordinate-wise decrease of general energy functions

Hendrickx¹,

Gerencsér²

2022

Preprint

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“…Convergence to a single point is only guaranteed under additional conditions, such as d dt V (x(t)) being sufficiently negative, by one of the original Lyapunov theorems. Another result for time-varying systems guarantees convergence to 0 if d dt V (x(t)) ≤ 0 and is not identically 0 on any trajectory other than that staying at 0 [21]. For a survey on various cases of unforced systems we direct the reader to [19] as a starting point.…”

Section: Introductionmentioning

confidence: 99%

Trajectory Convergence From Coordinate-Wise Decrease of Quadratic Energy Functions, and Applications to Platoons

Hendrickx

Gerencsér

Fi̇dan

2020

IEEE Control Syst. Lett.

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We consider trajectories where the sign of the derivative of each entry is opposite to that of the corresponding entry in the gradient of an energy function. We show that this condition guarantees convergence when the energy function is quadratic and positive definite and partly extend that result to some classes of positive semi-definite quadratic functions including those defined using a graph Laplacian. We show how this condition allows establishing the convergence of a platoon application in which it naturally appears, due to deadzones in the control laws designed to avoid instabilities caused by inconsistent measurements of the same distance by different agents.

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Linear systems of ordinary differential equations

Izobov¹

1976

J Math Sci

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The present survey deals chiefly with the Lyapunov theory of characteristic exponents of the general form of linear differential systems,with bounded piecewise continuous coefficients and particularly with its most recent advances. A sufficiently complete bibliography on such other independent branches of the theory of linear systems as "Linear Systems with Periodic Coefficients" (already surveyed in part [469] in the series General Mechanics) and the "Analytical Theory of Linear Systems" can be found in Sec. 15.

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Concerning Liapunov functions for linear systems— I†

Cited by 5 publications

References 5 publications

Trajectory convergence from coordinate-wise decrease of general energy functions

Trajectory convergence from coordinate-wise decrease of general energy functions

Trajectory Convergence From Coordinate-Wise Decrease of Quadratic Energy Functions, and Applications to Platoons

Linear systems of ordinary differential equations

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