Lyapunov Measure for Almost Everywhere Stability

Vaidya, Umesh; Mehta, Prashant G.

doi:10.1109/tac.2007.914955

Cited by 114 publications

(113 citation statements)

References 26 publications

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“…The use of occupation measures and related concepts has a long history in the fields of Markov decision processes and stochastic control; see, e.g., [23,12]. Applications to deterministic control problems were, to the best of our knowledge, first systematically treated 1 in [38] and enjoyed a resurgence of interest in the last decade; see, e.g., [37,27,42,15] and references therein. However, to the best of our knowledge, this the first time these methods are applied to region of attraction computation.…”

Section: Introductionmentioning

confidence: 99%

Convex Computation of the Region of Attraction of Polynomial Control Systems

Henrion

Korda

2014

IEEE Trans. Automat. Contr.

280

355

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We address the long-standing problem of computing the region of attraction (ROA) of a target set (typically a neighborhood of an equilibrium point) of a controlled nonlinear system with polynomial dynamics and semialgebraic state and input constraints. We show that the ROA can be computed by solving an infinite-dimensional convex linear programming (LP) problem over the space of measures. In turn, this problem can be solved approximately via a classical converging hierarchy of convex finite-dimensional linear matrix inequalities (LMIs). Our approach is genuinely primal in the sense that convexity of the problem of computing the ROA is an outcome of optimizing directly over system trajectories. The dual infinite-dimensional LP on nonnegative continuous functions (approximated by polynomial sum-of-squares) allows us to generate a hierarchy of semialgebraic outer approximations of the ROA at the price of solving a sequence of LMI problems with asymptotically vanishing conservatism. This sharply contrasts with the existing literature which follows an exclusively dual Lyapunov approach yielding either nonconvex bilinear matrix inequalities or conservative LMI conditions. The approach is simple and readily applicable as the outer approximations are the outcome of a single semidefinite program with no additional data required besides the problem description.

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Section: Introductionmentioning

confidence: 99%

Convex Computation of the Region of Attraction of Polynomial Control Systems

Henrion

Korda

2014

IEEE Trans. Automat. Contr.

280

355

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“…The only caveat is that an a priori knowledge of the attractor is required in that case. Otherwise, the eigenfunctions of the Koopman operator might also be obtained-without integration of the trajectories-through other methods similar to those developed in [15] (e.g. discretization method, convex linear programming methods).…”

Section: A the Methodsmentioning

confidence: 99%

“…Also, they naturally yield two dual methods for stability analysis. In the case of a fixed point x * , while a Lyapunov function decreases under the action of U t , a Lyapunov density (or Lyapunov measure) decreases (almost everywhere) under the action of P t [15]. The Lyapunov density was initially introduced in [14] as a function C 1 (X \ {x * }) that satisfies ∇ · (F ρ) > 0, a property which precisely corresponds to the action of the Perron-Frobenius infinitesimal generator L P ρ < 0, according to (4) (see also [13]).…”

Section: B Remark On Dualitymentioning

confidence: 99%

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A spectral operator-theoretic framework for global stability

Mauroy

Mezić

2013

52nd IEEE Conference on Decision and Control

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Abstract-The global description of a nonlinear system through the linear Koopman operator leads to an efficient approach to global stability analysis. In the context of stability analysis, not much attention has been paid to the use of spectral properties of the operator. This paper provides new results on the relationship between the global stability properties of the system and the spectral properties of the Koopman operator. In particular, the results show that specific eigenfunctions capture the system stability and can be used to recover known notions of classical stability theory (e.g. Lyapunov functions, contracting metrics). Finally, a numerical method is proposed for the global stability analysis of a fixed point and is illustrated with several examples.

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“…In [8], almost everywhere stability problem for discrete time dynamical systems is studied using linear transfer operators, in particular Koopman and Perron-Frobenius (P-F) operators. The Lyapunov measure is introduced as a new tool to verify the weaker notion of almost everywhere stability of an attractor set in nonlinear systems.…”

Section: Introductionmentioning

confidence: 99%

Stability in the almost everywhere sense: A linear transfer operator approach

Rajaram

Vaidya

Fardad

et al. 2010

Journal of Mathematical Analysis and Applications

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The problem of almost everywhere stability of a nonlinear autonomous ordinary differential equation is studied using a linear transfer operator framework. The infinitesimal generator of a linear transfer operator (Perron-Frobenius) is used to provide stability conditions of an autonomous ordinary differential equation. It is shown that almost everywhere uniform stability of a nonlinear differential equation, is equivalent to the existence of a non-negative solution for a steady state advection type linear partial differential equation. We refer to this non-negative solution, verifying almost everywhere global stability, as Lyapunov density. A numerical method using finite element techniques is used for the computation of Lyapunov density.

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Lyapunov Measure for Almost Everywhere Stability

Cited by 114 publications

References 26 publications

Convex Computation of the Region of Attraction of Polynomial Control Systems

Convex Computation of the Region of Attraction of Polynomial Control Systems

A spectral operator-theoretic framework for global stability

Stability in the almost everywhere sense: A linear transfer operator approach

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