The Experts below are selected from a list of 303 Experts worldwide ranked by ideXlab platform
N. Sakamoto - One of the best experts on this subject based on the ideXlab platform.
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analytical approximation methods for the stabilizing solution of the hamilton Jacobi Equation
IEEE Transactions on Automatic Control, 2008Co-Authors: N. Sakamoto, Arjan J. Van Der SchaftAbstract:In this paper, two methods for approximating the stabilizing solution of the Hamilton-Jacobi Equation are proposed using symplectic geometry and a Hamiltonian perturbation technique as well as stable manifold theory. The first method uses the fact that the Hamiltonian lifted system of an integrable system is also integrable and regards the corresponding Hamiltonian system of the Hamilton-Jacobi Equation as an integrable Hamiltonian system with a perturbation caused by control. The second method directly approximates the stable flow of the Hamiltonian systems using a modification of stable manifold theory. Both methods provide analytical approximations of the stable Lagrangian submanifold from which the stabilizing solution is derived. Two examples illustrate the effectiveness of the methods.
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Analytical Approximation Methods for the Stabilizing Solution of the Hamilton–Jacobi Equation
IEEE Transactions on Automatic Control, 2008Co-Authors: N. Sakamoto, Arjan J. Van Der SchaftAbstract:In this paper, two methods for approximating the stabilizing solution of the Hamilton-Jacobi Equation are proposed using symplectic geometry and a Hamiltonian perturbation technique as well as stable manifold theory. The first method uses the fact that the Hamiltonian lifted system of an integrable system is also integrable and regards the corresponding Hamiltonian system of the Hamilton-Jacobi Equation as an integrable Hamiltonian system with a perturbation caused by control. The second method directly approximates the stable flow of the Hamiltonian systems using a modification of stable manifold theory. Both methods provide analytical approximations of the stable Lagrangian submanifold from which the stabilizing solution is derived. Two examples illustrate the effectiveness of the methods.
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An analytical approximation method for the stabilizing solution of the Hamilton-Jacobi Equation based on stable manifold theory
2007 American Control Conference, 2007Co-Authors: N. Sakamoto, Arjan J. Van Der SchaftAbstract:In this paper, an analytical approximation approach for the stabilizing solution of the Hamilton-Jacobi Equation using stable manifold theory is proposed. The proposed method gives approximated flows on the stable manifold of the associated Hamiltonian system and provides approximations of the stable Lagrangian submanifold. With this method, the closed loop stability is guaranteed and can be enhanced by taking higher order approximations. A numerical example shows the effectiveness of the method.
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An approximation method for the stabilizing solution of the Hamilton-Jacobi Equation for integrable systems using Hamiltonian perturbation theory
Proceedings of the 45th IEEE Conference on Decision and Control, 2006Co-Authors: N. Sakamoto, Arjan J. Van Der SchaftAbstract:In this report, a method for approximating the stabilizing solution of the Hamilton-Jacobi Equation for integrable systems is proposed using symplectic geometry and a Hamiltonian perturbation technique. Using the fact that the Hamiltonian lifted system of an integrable system is also integrable, the Hamiltonian system (canonical Equation) that is derived from the theory of 1st order partial differential Equations is considered as an integrable Hamiltonian system with a perturbation caused by control. Assuming that the approximating Riccati Equation from the Hamilton-Jacobi Equation at the origin has a stabilizing solution, we construct approximating behaviors of the Hamiltonian flows on a stable Lagrangian submanifold, from which an approximation to the stabilizing solution is obtained
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Analysis of the Hamilton-Jacobi Equation in nonlinear control theory by symplectic geometry
Proceedings of the 41st IEEE Conference on Decision and Control 2002., 2002Co-Authors: N. SakamotoAbstract:The geometric property and structure of the Hamilton-Jacobi Equation arising from nonlinear control theory are investigated using symplectic geometry. The generating function of symplectic transforms plays an important role to reveal the structure of the Hamilton-Jacobi Equation. It is seen that many of fundamental properties of the Riccati Equation can be generalized in the Hamilton-Jacobi Equation, and therefore, the theory of the Hamilton-Jacobi Equation naturally contains that of the Riccati Equation.
Arjan J. Van Der Schaft - One of the best experts on this subject based on the ideXlab platform.
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analytical approximation methods for the stabilizing solution of the hamilton Jacobi Equation
IEEE Transactions on Automatic Control, 2008Co-Authors: N. Sakamoto, Arjan J. Van Der SchaftAbstract:In this paper, two methods for approximating the stabilizing solution of the Hamilton-Jacobi Equation are proposed using symplectic geometry and a Hamiltonian perturbation technique as well as stable manifold theory. The first method uses the fact that the Hamiltonian lifted system of an integrable system is also integrable and regards the corresponding Hamiltonian system of the Hamilton-Jacobi Equation as an integrable Hamiltonian system with a perturbation caused by control. The second method directly approximates the stable flow of the Hamiltonian systems using a modification of stable manifold theory. Both methods provide analytical approximations of the stable Lagrangian submanifold from which the stabilizing solution is derived. Two examples illustrate the effectiveness of the methods.
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Analytical Approximation Methods for the Stabilizing Solution of the Hamilton–Jacobi Equation
IEEE Transactions on Automatic Control, 2008Co-Authors: N. Sakamoto, Arjan J. Van Der SchaftAbstract:In this paper, two methods for approximating the stabilizing solution of the Hamilton-Jacobi Equation are proposed using symplectic geometry and a Hamiltonian perturbation technique as well as stable manifold theory. The first method uses the fact that the Hamiltonian lifted system of an integrable system is also integrable and regards the corresponding Hamiltonian system of the Hamilton-Jacobi Equation as an integrable Hamiltonian system with a perturbation caused by control. The second method directly approximates the stable flow of the Hamiltonian systems using a modification of stable manifold theory. Both methods provide analytical approximations of the stable Lagrangian submanifold from which the stabilizing solution is derived. Two examples illustrate the effectiveness of the methods.
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An analytical approximation method for the stabilizing solution of the Hamilton-Jacobi Equation based on stable manifold theory
2007 American Control Conference, 2007Co-Authors: N. Sakamoto, Arjan J. Van Der SchaftAbstract:In this paper, an analytical approximation approach for the stabilizing solution of the Hamilton-Jacobi Equation using stable manifold theory is proposed. The proposed method gives approximated flows on the stable manifold of the associated Hamiltonian system and provides approximations of the stable Lagrangian submanifold. With this method, the closed loop stability is guaranteed and can be enhanced by taking higher order approximations. A numerical example shows the effectiveness of the method.
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An approximation method for the stabilizing solution of the Hamilton-Jacobi Equation for integrable systems using Hamiltonian perturbation theory
Proceedings of the 45th IEEE Conference on Decision and Control, 2006Co-Authors: N. Sakamoto, Arjan J. Van Der SchaftAbstract:In this report, a method for approximating the stabilizing solution of the Hamilton-Jacobi Equation for integrable systems is proposed using symplectic geometry and a Hamiltonian perturbation technique. Using the fact that the Hamiltonian lifted system of an integrable system is also integrable, the Hamiltonian system (canonical Equation) that is derived from the theory of 1st order partial differential Equations is considered as an integrable Hamiltonian system with a perturbation caused by control. Assuming that the approximating Riccati Equation from the Hamilton-Jacobi Equation at the origin has a stabilizing solution, we construct approximating behaviors of the Hamiltonian flows on a stable Lagrangian submanifold, from which an approximation to the stabilizing solution is obtained
Jeff Calder - One of the best experts on this subject based on the ideXlab platform.
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numerical schemes and rates of convergence for the hamilton Jacobi Equation continuum limit of nondominated sorting
Numerische Mathematik, 2017Co-Authors: Jeff CalderAbstract:Non-dominated sorting arranges a set of points in n-dimensional Euclidean space into layers by repeatedly removing the coordinatewise minimal elements. It was recently shown that nondominated sorting of random points has a Hamilton---Jacobi Equation continuum limit. The obvious numerical scheme for this PDE has a slow convergence rate of $$O(h^\frac{1}{n})$$O(h1n). In this paper, we introduce two new numerical schemes that have formal rates of O(h) and we prove the usual $$O(\sqrt{h})$$O(h) theoretical rates. We also present the results of numerical simulations illustrating the difference between the formal and theoretical rates.
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a direct verification argument for the hamilton Jacobi Equation continuum limit of nondominated sorting
Nonlinear Analysis-theory Methods & Applications, 2016Co-Authors: Jeff CalderAbstract:Abstract Nondominated sorting is a combinatorial algorithm that sorts points in Euclidean space into layers according to a partial order. It was recently shown that nondominated sorting of random points has a Hamilton–Jacobi Equation continuum limit. The original proof, given in Calder et al. (2014), relies on a continuum variational problem. In this paper, we give a new proof using a direct verification argument that completely avoids the variational interpretation. We believe this may be generalized to apply to other stochastic homogenization problems for which there is no obvious underlying variational principle.
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a direct verification argument for the hamilton Jacobi Equation continuum limit of nondominated sorting
arXiv: Analysis of PDEs, 2015Co-Authors: Jeff CalderAbstract:Nondominated sorting is a combinatorial algorithm that sorts points in Euclidean space into layers according to a partial order. It was recently shown that nondominated sorting of random points has a Hamilton-Jacobi Equation continuum limit. The original proof relies on a continuum variational problem. In this paper, we give a new proof using a direct verification argument that completely avoids the variational interpretation. We believe this proof is new in the homogenization literature, and may be generalized to apply to other stochastic homogenization problems for which there is no obvious underlying variational principle.
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a hamilton Jacobi Equation for the continuum limit of nondominated sorting
Siam Journal on Mathematical Analysis, 2014Co-Authors: Jeff Calder, Selim Esedoḡlu, Alfred O HeroAbstract:We show that nondominated sorting of a sequence $X_1,\dots,X_n$ of independent and identically distributed random variables in $\mathbb{R}^d$ has a continuum limit that corresponds to solving a Hamilton--Jacobi Equation involving the probability density function $f$ of $X_i$. Nondominated sorting is a fundamental problem in multiobjective optimization and is equivalent to finding the canonical antichain partition and to problems involving the longest chain among Euclidean points. As an application of this result, we show that nondominated sorting is asymptotically stable under bounded random perturbations in $X_1,\dots,X_n$. We give a numerical scheme for computing the viscosity solution of this Hamilton--Jacobi Equation and present some numerical simulations for various density functions.
Patrick Bernard - One of the best experts on this subject based on the ideXlab platform.
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existence of c 1 1 critical sub solutions of the hamilton Jacobi Equation on compact manifolds
Annales Scientifiques De L Ecole Normale Superieure, 2007Co-Authors: Patrick BernardAbstract:Abstract We offer a simple proof of the existence of a C 1 , 1 solution of the Hamilton–Jacobi Equation in the context of Mather theory. We derive some dynamical consequences of this result. We also prove that the solution can be obtained strict outside of the Aubry set.
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smooth critical sub solutions of the hamilton Jacobi Equation
Mathematical Research Letters, 2007Co-Authors: Patrick BernardAbstract:We establish the existence of smooth critical sub-solutions of the Hamilton-Jacobi Equation on compact manifolds for smooth convex Hamiltonians, that is in the context of weak KAM theory, under the assumption that the Aubry set is the union of finitely many hyperbolic periodic orbits or fixed points.
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smooth critical sub solutions of the hamilton Jacobi Equation
arXiv: Dynamical Systems, 2005Co-Authors: Patrick BernardAbstract:We establish the existence of smooth critical sub-solutions of the Hamilton-Jacobi Equation on compact manifolds for smooth convex Hamiltonians, that is in the context of weak KAM theory, under the assumption that the Aubry set is the union of finitely many hyperbolic periodic orbits.
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existence of c 1 1 critical sub solutions of the hamilton Jacobi Equation on compact manifolds
arXiv: Dynamical Systems, 2005Co-Authors: Patrick BernardAbstract:We prove the existence of a critical subsolution of the Hamilton-Jacobi Equation which is differentiable with Lipschitz differential, improving a previous result of Fathi and Siconolfi. This text is a consolidated version with a minor mistake corrected from the published version.
Chiachun Chou - One of the best experts on this subject based on the ideXlab platform.
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complex quantum hamilton Jacobi Equation with bohmian trajectories application to the photodissociation dynamics of nocl
Journal of Chemical Physics, 2014Co-Authors: Chiachun ChouAbstract:The complex quantum Hamilton-Jacobi Equation-Bohmian trajectories (CQHJE-BT) method is introduced as a synthetic trajectory method for integrating the complex quantum Hamilton-Jacobi Equation for the complex action function by propagating an ensemble of real-valued correlated Bohmian trajectories. Substituting the wave function expressed in exponential form in terms of the complex action into the time-dependent Schrodinger Equation yields the complex quantum Hamilton-Jacobi Equation. We transform this Equation into the arbitrary Lagrangian-Eulerian version with the grid velocity matching the flow velocity of the probability fluid. The resulting Equation describing the rate of change in the complex action transported along Bohmian trajectories is simultaneously integrated with the guidance Equation for Bohmian trajectories, and the time-dependent wave function is readily synthesized. The spatial derivatives of the complex action required for the integration scheme are obtained by solving one moving least s...
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computational method for the quantum hamilton Jacobi Equation bound states in one dimension
Journal of Chemical Physics, 2006Co-Authors: Chiachun Chou, Robert E WyattAbstract:An accurate computational method for the one-dimensional quantum Hamilton-Jacobi Equation is presented. The Mobius propagation scheme, which can accurately pass through singularities, is used to numerically integrate the quantum Hamilton-Jacobi Equation for the quantum momentum function. Bound state wave functions are then synthesized from the phase integral using the antithetic cancellation technique. Through this procedure, not only the quantum momentum functions but also the wave functions are accurately obtained. This computational approach is demonstrated through two solvable examples: the harmonic oscillator and the Morse potential. The excellent agreement between the computational and the exact analytical results shows that the method proposed here may be useful for solving similar quantum mechanical problems.