We prove the necessary and sufficient conditions for practical stability of nonlinear dynamical system at general phase restrictions. In such a case the Lyapunov function is nondifferentiable. But if the set of initial data is starry compact, then it is possible building Lyapunov function which belongs to differentiable functions class. We also estimate the optimal sets of initial conditions in structural forms for linear system and concrete phase restrictions.
It is well known that, the existence of a Lyapunov function is a sufficient condition for stability, asymptotic stability, or global asymptotic stability of an equilibrium point of an autonomous system $\dot{\mathbf{x}} = f(\mathbf{x})$. In variants of Lyapunov theorems, the condition for a Lyapunov candidate $V$ (continuously differentiable and positive definite function) to be a Lyapunov function is that its time derivative along system trajectories must be negative semi-definite or negative definite. Numerically checking positive definiteness of $V$ is very difficult; checking negative defi…
We exhibit an explicit sufficient condition for the Lyapunov exponents of a linear cocycle over a Markov map to have multiplicity 1. This builds on work of Guivarc'h-Raugi and Gol'dsheid-Margulis, who considered products of random matrices, and of Bonatti-Viana, who dealt with the case when the base dynamics is a subshift of finite type. Here the Markov structure may have infinitely many symbols and the ambient space needs not be compact. As an application, in another paper we prove the Zorich-Kontsevich conjecture on the Lyapunov spectrum of the Teichmüller flow in the space of translation su…
Stability and safety are crucial in safety-critical control of dynamical systems. The reach-avoid-stay objectives for deterministic dynamical systems can be effectively handled by formal methods as well as Lyapunov methods with soundness and approximate completeness guarantees. However, for continuous-time stochastic dynamical systems, probabilistic reach-avoid-stay problems are viewed as challenging tasks. Motivated by the recent surge of applications in characterizing safety-critical properties using Lyapunov-barrier functions, we aim to provide a stochastic version for the probabilistic rea…
An analysis of necessary conditions for the existence of controlled dynamics with an attractor of a specified topological type is given. It uses the Hopf classification by degree for Gauss maps of manifolds to spheres of the same dimension, in this case noting that by a linear homotopy, the degree is the same for the negative of the gradient flow of a Lyapunov function and for the stabilising vector field. The manifolds are Lyapunov level sets. More general contexts are given by considering maps in homology. Some account is taken of cases where the degree is zero and the surjectivity fails. A …
This paper presents necessary and sufficient characterizations of several notions of input to output stability. Similar Lyapunov characterizations have been found to play a key role in the analysis of the input to state stability property, and the results given here extend their validity to the case when the output, but not necessarily the entire internal state, is being regulated. The work is related to partial stability of differential equations (the particular case which arises when there are no external inputs).
In the context of mechanical Lagrangian dynamics, we prove a new Lyapunov instability criterion for a non strict local minimum equilibrium point of a smooth potential where the sufficient condition for instability is the existence of a smooth solution of a certain linear PDE derived from the mechanical Lagrangian governing the dynamics. In the presence of a magnetostatic field, we also give an additional sufficient condition for the motion of a charged particle to be Lyapunov unstable.
The aim of this paper is to present necessary and sufficient conditions for nonuniform power instability property of linear discrete-time systems in Banach spaces. A characterization of the nonuniform power instability in terms of Lyapunov sequences is also given.
In this paper we show that the existence of a Lyapunov-Krasovskii functional is necessary and sufficient condition for the uniform global asymptotic stability and the global exponential stability of time-invariant systems described by neutral functional differential equations in Hale's form. It is assumed that the difference operator is linear and strongly stable, and that the map in the right-hand side of the equation is Lipschitz on bounded sets. A link between global exponential stability and input-to-state stability is also provided. @ The extended version of this paper has been submitted …
We show that the existence of a non-coercive Lyapunov function is sufficient for uniform global asymptotic stability (UGAS) of infinite-dimensional systems with external disturbances provided the speed of decay is measured in terms of the norm of the state and an additional mild assumption is satisfied. For evolution equations in Banach spaces with Lipschitz continuous nonlinearities these additional assumptions become especially simple. The results encompass some recent results on linear switched systems on Banach spaces. Finally, we derive new non-coercive converse Lyapunov theorems and give…
In this paper, a necessary and sufficient condition for the stability of Lyapunov exponents of linear differential system are proved in the sense that the equations satisfy the weaker form of integral separation instead of its classical one. Furthermore, the existence of full nonuniform exponential dichotomy spectrum under the condition of weak integral separateness is also presented.
We give necessary and sufficient conditions for the existence of smooth Lyapunov 1-forms for the flow of a smooth vector field in terms of the behavior of certain locally finite invariant measures. The main statement generalizes a result of Schwartzman, whereas the methods are adapted from work of Sullivan.
We prove the Hölder continuity of the Lyapunov exponent for quasi-periodic Schrödinger cocycles with a $C^2$ cos-type potential and any fixed Liouvillean frequency, provided the coupling constant is sufficiently large. Moreover, the Hölder exponent is independent of the frequency and the coupling constant.
We consider explosions in the generalized recurrent set for homeomorphisms on a compact metric space. We provide multiple examples to show that such explosions can occur, in contrast to the case for the chain recurrent set. We give sufficient conditions to avoid explosions and discuss their necessity. Moreover, we explain the relations between explosions and cycles for the generalized recurrent set. In particular, for a compact topological manifold with dimension greater or equal $2$, we characterize explosion phenomena in terms of existence of cycles. We apply our results to give sufficient c…
We study convergence of nonlinear systems in the presence of an `almost Lyapunov' function which, unlike the classical Lyapunov function, is allowed to be nondecreasing---and even increasing---on a nontrivial subset of the phase space. Under the assumption that the vector field is free of singular points (away from the origin) and that the subset where the Lyapunov function does not decrease is sufficiently small, we prove that solutions approach a small neighborhood of the origin. A nontrivial example where this theorem applies is constructed.
This paper gives further insights about the Lyapunov-Krasovskii characterization of input-tostate stability (ISS) for switching retarded systems on the basis of the results in [I. Haidar and P. Pepe. Lyapunov-krasovskii characterization of the input-to-state stability for switching retarded systems. SIAM Journal on Control and Optimization, 59(4):2997-3016, 2021]. We give new characterizations of the ISS property through the existence of a relaxed common Lyapunov-Krasovskii functional. More precisely, we show that the existence of a continuous Lyapunov-Krasovskii functional whose upper right-h…
The aim of this paper is to study in detail the relations between strong chain recurrence for flows and Lyapunov functions. For a continuous flow on a compact metric space, uniformly Lipschitz continuous on the compact subsets of the time, we first make explicit a Lipschitz continuous Lyapunov function strict -that is strictly decreasing- outside the strong chain recurrent set of the flow. This construction extends to flows some recent advances of Fathi and Pageault in the case of homeomorphisms; moreover, it improves Conley's result about the existence of a continuous Lyapunov function strict…
In this paper we give some necessary and sufficient characterizations for weak exponential instability of evolution operators. Variants for the classical results due to Datko and Lyapunov are obtained.
Stability, reachability, and safety are crucial properties of dynamical systems. While verification and control synthesis of reach-avoid-stay objectives can be effectively handled by abstraction-based formal methods, such approaches can be computationally expensive due to the use of state-space discretization. In contrast, Lyapunov methods qualitatively characterize stability and safety properties without any state-space discretization. Recent work on converse Lyapunov-barrier theorems also demonstrates an approximate completeness or verifying reach-avoid-stay specifications of systems modelle…
Let $Q\rightarrow X$ be a continuous principal bundle whose group $G$ is reductive. A flow $ϕ$ of automorphisms of $Q$ endowed with an ergodic probability measure on the compact base space $X$ induces two decompositions of the flag bundles associated to $Q$. A continuous one given by the finest Morse decomposition and a measurable one furnished by the Multiplicative Ergodic Theorem. The second is contained in the first. In this paper we find necessary and sufficient conditions so that they coincide. The equality between the two decompositions implies continuity of the Lyapunov spectra under pe…
This paper focuses on the fractional difference of Lyapunov functions related to Riemann-Liouville, Caputo and Grunwald-Letnikov definitions. A new way of building Lyapunov functions is introduced and then five inequalities are derived for each definition. With the help of the developed inequalities, the sufficient conditions can be obtained to guarantee the asymptotic stability of the nabla discrete fractional order nonlinear systems. Finally, three illustrative examples are presented to demonstrate the validity and feasibility of the proposed theoretical results.
This contribution presents two exponential stability criteria for linear systems with multiple pointwise and distributed delays. These results (necessary and sufficient conditions) are given in terms of the delay Lyapunov matrix and the fundamental matrix of the system. An important property is that the stability test requires a finite number of mathematical operations.
For a product of i.i.d. random maps or a memoryless stochastic flow on a compact space $X$, we find conditions under which the presence of locally asymptotically stable trajectories (e.g. as given by negative Lyapunov exponents) implies almost-sure mutual convergence of any given pair of trajectories ("synchronisation"). Namely, we find that synchronisation occurs and is stable if and only if the system exhibits the following properties: (i) there is a smallest deterministic invariant set $K \subset X$, (ii) any two points in $K$ are capable of being moved closer together, and (iii) $K$ admits…
We give necessary integral conditions and sufficient ones for the existence of a general concept of $μ$-dichotomy for evolution operators defined on the half-line which includes as particular cases the well-known concepts of nonuniform exponential dichotomy and nonuniform polynomial dichotomy, and also contains new situations. Additionally, we consider an adapted notion of Lyapunov function and use our results to obtain necessary and sufficient conditions for the existence of nonuniform $μ$-dichotomies using these Lyapunov functions.
We propose easily verifiable necessary and sufficient conditions for the linearizability of two-input systems by an endogenous dynamic feedback with a dimension of at most two.
Stability Analysis of Monotone Systems via Max-separable Lyapunov Functions
ACCESS Linnaeus Centre（ACCESS林奈乌斯中心）；Department of Automatic Control, School of Electrical Engineering, KTH Royal Institute of Technology（瑞典皇家理工学院电气工程学院自动控制系）
It has been shown that the Kullback-Leibler divergence is a Lyapunov function for the replicator equations at evolutionary stable states, or ESS. In this paper we extend the result to a more general class of game dynamics. As a result, sufficient conditions can be given for the asymptotic stability of rest points for the entire class of incentive dynamics. The previous known results will be can be shown as corollaries to the main theorem.
Linear Lyapunov Functions for Nonlinear Compartmental Systems
Chalmers University of Technology（查尔姆斯理工大学）；Mines Paris – PSL University（巴黎矿业高等学院-巴黎文理研究大学）；University of Gothenburg（哥德堡大学）；Pázmány Péter Catholic University（帕兹马尼·彼得天主教大学）；Université Paris-Saclay（巴黎萨克雷大学）
We present a stability analysis framework for the general class of discrete-time linear switching systems for which the switching sequences belong to a regular language. They admit arbitrary switching systems as special cases. Using recent results of X. Dai on the asymptotic growth rate of such systems, we introduce the concept of multinorm as an algebraic tool for stability analysis. We conjugate this tool with two families of multiple quadratic Lyapunov functions, parameterized by an integer T >= 1, and obtain converse Lyapunov Theorems for each. Lyapunov functions of the first family associ…
In this paper, we study random Blaschke products, acting on the unit circle, and consider the cocycle of Perron-Frobenius operators acting on Banach spaces of analytic functions on an annulus. We completely describe the Lyapunov spectrum of these cocycles. As a corollary, we obtain a simple random Blaschke product system where the Perron-Frobenius cocycle has infinitely many distinct Lyapunov exponents, but where arbitrarily small natural perturbations cause a complete collapse of the Lyapunov spectrum, except for the exponent 0 associated with the absolutely continuous invariant measure. That…
We find necessary and sufficient conditions for a dynamical system to be topologically conjugate to any given substitution minimal system, thus extending the results in [CKL] for the Morse and Toeplitz substitutions.
In this work we first provide sufficient conditions to assure the persistence of some zeros of functions having the form $$g(z,\varepsilon)=g_0(z)+\sum_{i=1}^k \varepsilon^i g_i(z)+\mathcal{O}(\varepsilon^{k+1}),$$ for $|\varepsilon|\neq0$ sufficiently small. Here $g_i:\mathcal{D}\rightarrow\mathbb{R}^n$, for $i=0,1,\ldots,k$, are smooth functions being $\mathcal{D}\subset \mathbb{R}^n$ an open bounded set. Then we use this result to compute the bifurcation functions which controls the periodic solutions of the following $T$-periodic smooth differential system $$ x'=F_0(t,x)+\sum_{i=1}^k \vare…
We give a sufficient condition for existence of an exponential dichotomy for a general linear dynamical system (not necessarily invertible) in a Banach space, in discrete or continuous time. We provide applications to the backward heat equation with a potential varying in time and to the heat equation with a finite number of slowly moving potentials. We also consider the Klein-Gordon equation with a finite number of potentials whose centres move at sub-light speed with small accelerations.
On the attractor in a high-dimensional neural network dynamics of reservoir computing: Lyapunov analysis viewpoint
Faculty of Economics, Rissho University（立教大学经济学部）；Graduate School of Environment, Life, Natural Science and Technology, Okayama University（冈山大学环境·生命·自然科学和技术研究院）；Graduate School of Business Administration, Hitotsubashi University（一桥大学商学研究科）
We give necessary and sufficient conditions to have measurable and continuous eigenfunctions for linearly recurrent Cantor dynamical systems. We also construct explicitly an example of linearly recurrent system with nontrivial Kronecker factor and a trivial maximal equicontinuous factor.
This paper is concerned with the stabilization problem of singular fractional order systems with order $α\in(0,2)$. In addition to the sufficient and necessary condition for observer based control, a sufficient and necessary condition for output feedback control is proposed by adopting matrix variable decoupling technique. The developed results are more general and efficient than the existing works, especially for the output feedback case. Finally, two illustrative examples are given to verify the effectiveness and potential of the proposed approaches.
Lyapunov's indirect method is an attractive method for analyzing stability of non-linear systems since only the stability of the corresponding linearized system needs to be determined. Unfortunately, the proof for finite-dimensional systems does not generalize to infinite-dimensions. In this paper a unified approach to Lyapunov's indirect method for infinite-dimensional system is described. It is shown how existing sufficient conditions fit this framework and a new sufficient condition is presented.
In this paper, We study the stability of solutions of fuzzy differential equations by Lyapunov's second method. By using scale equations and comparison principle for Lyapunov - like functions, we give some sufficient criterias for the stability and asymptotic stability of solutions of fuzzy differential equations.
