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Lyapunov functionals and matrices
Annual Reviews in Control, 2010Abstract In this contribution we present some basic results concerning the computation of quadratic functionals with prescribed time derivatives for linear time delay systems. Some lower and upper bounds for the functionals are given. The functionals are defined by special matrix valued functions. These functions are called Lyapunov matrices.
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Robust Minkowski–Lyapunov functions
Automatica, 2020zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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Founded Lyapunov–Bogdanov Functionals
Differential Equations, 2020zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Borukhov, V. T., Kvetko, O. M.
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On the Construction of Lyapunov Functions
SIAM Journal on Applied Mathematics, 1969the same. These results are used to obtain new instability results for the Hill equation which extend the classical results of Lyapunov and Haupt. Finally, we show that a Lyapunov function for w' = 2Bw can be used to algorithmically obtain a Lyapunov function for y' = By along with certain verifiable conditions from which stability properties of y ...
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Lyapunov and Lyapunov-like functions
2007Lyapunov functions are crucial in the present book aims, given the strict relation between Lyapunov functions and invariant sets. In this chapter, basic notions of Lyapunov and Lyapunov-like functions will be presented. Before introducing the main concept, a brief presentation of the class of dynamic models will be considered and some preliminary ...
Franco Blanchini, Stefano Miani
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Weak Converse Lyapunov Theorems and Control-Lyapunov Functions
SIAM Journal on Control and Optimization, 2004Summary: Given a weakly uniformly globally asymptotically stable closed (not necessarily compact) set \({\mathcal A}\) for a differential inclusion that is defined on \(\mathbb{R}^n\), is locally Lipschitz on \(\mathbb{R}^n \backslash{\mathcal A}\), and satisfies other basic conditions, we construct a weak Lyapunov function that is locally Lipschitz on
Christopher M. Kellett, Andrew R. Teel
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Logical composition of Lyapunov functions
International Journal of Control, 2011This article introduces the use of R-functions to compose single Lyapunov functions (LFs) via classic Boolean operators, with the aim to obtain a rich family of non-conventional, generally non-convex functions. The main benefit of the proposed composition is the nice geometric interpretation, since it corresponds to intersection and union operations in
Aldo Balestrino +2 more
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Higher derivatives of Lyapunov functions and cone-valued Lyapunov functions
Nonlinear Analysis: Theory, Methods & Applications, 1996The authors are interested in the stability properties of the trivial solution of the system \(x'=f(t,x)\) and wish to use the higher derivatives of a single Lyapunov function to investigate them. If the resulting comparison system satisfies the required quasimonotone property, one can use a variant of the method of vector Lyapunov functions.
Köksal, S., Lakshmikantham, V.
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Lyapunov Functions and Cone Families
Journal of Statistical Physics, 2012We describe systematically the relation between Lyapunov functions and nonvanishing Lyapunov exponents, both for maps and flows. This includes a brief survey of the existing results in the area. In particular, we consider separately the cases of nonpositive and arbitrary Lyapunov functions, thus yielding optimal criteria for negativity and positivity ...
Barreira, Luis +2 more
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Using Lyapunov Functions to Construct Lyapunov Functionals for Delay Differential Equations
SIAM Journal on Applied Dynamical Systems, 2015Given that a Lyapunov function is known for a particular system, we outline an approach for determining terms in the system that can be replaced by similar terms that include delay, without changing the global stability. The approach is based on adding integral terms to the original Lyapunov function so that the new Lyapunov derivative is still ...
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