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ABSTRACT: Received (Day Month Year) Revised (Day Month Year) Communicated by (xxxxxxxxxx) We study approximation strategies for the limit problem arising in the homogeniza-tion of Hamilton-Jacobi equations. They involve first an approximation of the effective Hamiltonian then a discretization of the Hamilton-Jacobi equation with the approximate effective Hamiltonian. We give a global error estimate which takes into account all the parameters involved in the approximation.
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ABSTRACT: Three definitions of viscosity solutions for Hamilton-Jacobi equations on
networks recently appeared in literature ([1,4,6]). Being motivated by various
applications, they appear to be considerably different. Aim of this note is to
establish their equivalence.
01/2013;
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ABSTRACT: Mean field type models describing the limiting behavior, as the number of
players tends to $+\infty$, of stochastic differential game problems, have been
recently introduced by J-M. Lasry and P-L. Lions. Numerical methods for the
approximation of the stationary and evolutive versions of such models have been
proposed by the authors in previous works . Convergence theorems for these
methods are proved under various assumptions
07/2012;
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ABSTRACT: We study approximation schemes for the cell problem arising in homogenization of Hamilton-Jacobi equations. We prove several
error estimates concerning the rate of convergence of the approximation scheme to the effective Hamiltonian, both in the optimal
control setting and as well as in the calculus of variations setting.
Applied Mathematics and Optimization 04/2012; 57(1):30-57. · 0.95 Impact Factor
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ABSTRACT: In the present article, we study the numerical approximation of a system of
Hamilton-Jacobi and transport equations arising in geometrical optics. We
consider a semi-Lagrangian scheme. We prove the well posedness of the discrete
problem and the convergence of the approximated solution toward the
viscosity-measure valued solution of the exact problem.
10/2011;
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ABSTRACT: Aim of this paper is to extend the continuous dependence estimates proved in
\cite{JK1} to quasi-monotone systems of fully nonlinear second-order parabolic
equations. As by-product of these estimates, we get an H\"older estimate for
bounded solutions of systems and a rate of convergence estimate for the
vanishing viscosity approximation. In the second part of the paper we employ
similar techniques to study the periodic homogenization of quasi-monotone
systems of fully nonlinear second-order uniformly parabolic equations. Finally,
some examples are discussed.
09/2011;
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ABSTRACT: We show a large time behavior result for class of weakly coupled systems of
first-order Hamilton-Jacobi equations in the periodic setting. We use a PDE
approach to extend the convergence result proved by Namah and Roquejoffre
(1999) in the scalar case. Our proof is based on new comparison, existence and
regularity results for systems. An interpretation of the solution of the system
in terms of an optimal control problem with switching is given.
04/2011;
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ABSTRACT: This paper concerns periodic multiscale homogenization for fully nonlinear equations of the form u ε + H ε x, x ε , . . . , x ε k , Du ε , D 2 u ε = 0. The operators H ε are a regular perturbations of some uniformly elliptic, convex operator H. As ε → 0, the solutions u ε converge locally uniformly to the solution u of a suitably defined effective problem. The purpose of this paper is to obtain an estimate of the corresponding rate of convergence. Finally, some examples are discussed. MSC 2000: 35B27, 35J60, 49L25.
11/2009;
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ABSTRACT: In this paper we study homogenization for a class of monotone systems of first-order time-dependent periodic Hamilton-Jacobi equations. We characterize the Hamiltonians of the limit problem by appropriate cell problems. Hence we show the uniform convergence of the solution of the oscillating systems to the bounded uniformly continuous solution of the homogenized system.
02/2008;
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SIAM J. Control and Optimization. 01/2008; 47:301-326.
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ABSTRACT: We derive a method for the computation of robust domains of attraction based on a recent generalization of Zubov’s theorem
on representing robust domains of attraction for perturbed systems via the viscosity solution of a suitable partial differential
equation. While a direct discretization of the equation leads to numerical difficulties due to a singularity at the stable
equilibrium, a suitable regularization enables us to apply a standard discretization technique for Hamilton-Jacobi-Bellman
equations. We present the resulting fully discrete scheme and show a numerical example.
11/2007: pages 277-289;
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ABSTRACT: We consider a stochastic differential equation with an asymptotically stable equilibrium point. We show that the domain of
attraction of the equilibrium, i.e. the set of points which are attracted with positive probability to it, can be characterized
by the solution of a suitable partial differential equation.
Nonlinear Differential Equations and Applications NoDEA 06/2006; 13(2):205-222. · 0.54 Impact Factor
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ABSTRACT: We consider a controlled stochastic system which is exponentially stabilizable in proba-bility near an attractor. Our aim is to characterize the set of points which can be driven by a suitable control to the attractor with either positive probability or with probability one. This will be done by associating to the stochastic system a suitable control problem and the corresponding Zubov equation. We then show that this approach can be used as a basis for numerical computations of these sets.
01/2006; 6(6).
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Discrete and Continuous Dynamical Systems-series B - DISCRETE CONTIN DYN SYS-SER B. 01/2003; 3(3):457-468.
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SIAM J. Control and Optimization. 01/2001; 40:496-515.
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ABSTRACT: We derive a method for the computation of robust domains of attraction based on a recent generalization of Zubov's theorem on representing robust domains of attraction for perturbed systems via the viscosity solution of a suitable partial differential equation. While a direct discretization of the equation leads to numerical difficulties due to a singularity at the stable equilibrium, a suitable regularization enables us to apply a standard discretization technique for Hamilton-Jacobi-Bellman equations. We present the resulting fully discrete scheme and show a numerical example.
09/2000;
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ABSTRACT: : We derive a method for the computation of robust domains of attraction based on a recent generalization of Zubov's theorem on representing robust domains of attraction for perturbed systems via the viscosity solution of a suitable partial differential equation. While a direct discretization of the equation leads to numerical difficulties due to a singularity at the stable equilibrium, a suitable regularization enables us to apply a standard discretization technique for Hamilton-Jacobi-Bellman equations. We present the resulting fully discrete scheme and show a numerical example. 1 Introduction The domain of attraction of an asymptotically stable fixed point has been one of the central objects in the study of continuous dynamical systems. The knowledge of this object is important in many applications modeled by those systems like e.g. the analysis of power systems [1] and turbulence phenomena in fluid dynamics [2, 8, 17]. Several papers and books discuss theoretical [19, 20, 5, 12] a...
06/2000;
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ABSTRACT: We present a generalization of Zubov's method to perturbed differential equations. The goal is to characterize the domain of attraction of a set which is uniformly locally asymptotically stable under all admissible time varying perturbations. We show that in this general setting the straightforward generalization of the classical Zubov's equations has a unique viscosity solution which characterizes the robust domain of attraction as a suitable sublevel set.
05/2000;
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ABSTRACT: We present a generalization of Zubov's method to perturbed differential equations. The goal is to characterize the domain of attraction of a set which is uniformly locally asymptotically stable under all admissible time varying perturbations. We show that in this general setting the straightforward generalization of the classical Zubov's equations has a unique viscosity solution which characterizes the robust domain of attraction as a suitable sublevel set.
05/2000;
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[show abstract]
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ABSTRACT: : We derive a method for the computation of robust domains of attraction based on a recent generalization of Zubov's theorem on representing robust domains of attraction for perturbed systems via the viscosity solution of a suitable partial differential equation. While a direct discretization of the equation leads to numerical difficulties due to a singularity at the stable equilibrium, a suitable regularization enables us to apply a standard discretization technique for Hamilton-Jacobi-Bellman equations. We present the resulting fully discrete scheme and show a numerical example. 1 Introduction The domain of attraction of an asymptotically stable fixed point has been one of the central objects in the study of continuous dynamical systems. The knowledge of this object is important in many applications modeled by those systems like e.g. the analysis of power systems [1] and turbulence phenomena in fluid dynamics [2, 8, 17]. Several papers and books discuss theoretical [19, 20, 5, 12] a...
04/2000;
