Lyapunov function

In the theory of ordinary differential equations (ODEs), Lyapunov functions, named after Aleksandr Lyapunov, are scalar functions that may be used to prove the stability of an equilibrium of an ODE. Lyapunov functions (also called Lyapunov’s second method for stability) are important to stability theory of dynamical systems and control theory. A similar concept appears in the theory of general state space Markov chains, usually under the name Foster–Lyapunov functions. For certain classes of ODEs, the existence of Lyapunov functions is a necessary and sufficient condition for stability. There is no general technique for constructing Lyapunov functions for ODEs, however, depending on formulation type, a systematic method to construct Lyapunov functions for ordinary differential equations using their most general form in autonomous cases was given by Prof. Cem Civelek. Though, in many specific cases the construction of Lyapunov functions is known. For instance, according to a lot of a

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Limit Behaviour of the Voter Model, Exclusion Process and Regenerative Chains

Samuel Schöpfer

Though the following topics seem unlinked, most of the tools used in this thesis are related to random walks and renewal theory. After introducing the voter model, we consider the parabolic Anderson model with the voter model as catalyst. In GÄRTNER, DEN HOLLANDER and MAILLARD [44], the behaviour of the annealed Lyapunov exponents, i.e., the exponential growth rates of the successive moments of the reactant with respect to the catalyst, was investigated. It was shown that these exponents exhibit an interesting dependence on the dimension and on the diffusion constant. In Chapter 3 we address some questions left open in this paper by considering specifically when the Lyapunov exponents are the a priori maximal value. Then, we use exclusion process techniques to show that the evolution of a perturbed threshold voter model is recurrent in the critical case. The key to our approach is to develop the ideas of BRAMSON and MOUNTFORD [9] : we exhibit a Lyapunov-Foster function for the discrete time version of the process. We also make a widespread use of coupling arguments. Finally, using the regenerative scheme of COMETS, FERNÁNDEZ and FERRARI [19], we establish a functional central limit theorem for discrete time stochastic processes with summable memory decay. Furthermore, under stronger assumptions on the memory decay, we identify the limiting variance in terms of the process only. As applications, we define classes of binary autoregressive processes and power-law Ising chains for which the limit theorem is fulfilled.

EPFL2012

Feedback Control Design Maximizing the Region of Attraction of Stochastic Systems Using Polynomial Chaos Expansion

Petr Listov

A feedback control design is proposed for stochastic systems with finite second moment which aims at maximising the region of attraction of the equilibrium point. Polynomial Chaos (PC) expansions are employed to represent the stochastic closed loop system by a higher dimensional set of deterministic equations. By using the PC expanded system representation, the available information on the uncertainty affecting the system explicitly enters the control design problem. Further, this allows Lyapunov methods for deterministic systems to be used to formulate the stability criteria certifying the region of attraction. These criteria are parametrized by the feedback gain and formulated in a polynomial optimization program which is solved using sum-of-squares methods. This approach offers flexibility in the choice of the stochastic feedback law and accounts for input constraints. The application is demonstrated by two numerical examples.

2020

On Decentralized Navigation Schemes for Coordination of Multi-Agent Dynamical Systems

Denis Gillet, Hajir Roozbehani, Sylvain Rudaz

Coordination of autonomous non-point agents in four-way crossings is studied in this work. A control scheme based on artiﬁcial potential functions is proposed in order to coordinate holonomic agents whose aim is to pass through an intersection while avoiding collisions. To do so, the agents are provided with a decentralized navigation function in which the goal and collision functions are decoupled. The local function deﬁned on each agent requires no global knowledge on the desired destination of other agents. Furthermore, a reachability analysis via Lyapunov functions is proposed in order to guarantee marginal stability in our case study. In addition, conﬂict-free navigation and convergence properties are veriﬁed in simulation.

2009

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