Error Analysis for POD Approximations of Infinite Horizon Problems via the Dynamic Programming Approach
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In this paper infinite horizon optimal control problems for nonlinear high-dimensional dynamical systems are studied and a reduced-order model is derived for the dynamical system, using the method of proper orthogonal decomposition (POD).Abstract:
In this paper infinite horizon optimal control problems for nonlinear high-dimensional dynamical systems are studied. Nonlinear feedback laws can be computed via the value function characterized as the unique viscosity solution to the corresponding Hamilton--Jacobi--Bellman (HJB) equation which stems from the dynamic programming approach. However, the bottleneck is mainly due to the curse of dimensionality, and HJB equations are solvable only in a relatively small dimension. Therefore, a reduced-order model is derived for the dynamical system, using the method of proper orthogonal decomposition (POD). The resulting errors in the HJB equations are estimated by an a priori error analysis, which is utilized in the numerical approximation to ensure a desired accuracy for the POD method. Numerical experiments illustrates the theoretical findings.read more
Citations
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Overcoming the curse of dimensionality for some Hamilton–Jacobi partial differential equations via neural network architectures
TL;DR: This article showed that some classes of neural networks correspond to representation formulas of HJ PDE solutions whose Hamiltonians and initial data are obtained from the parameters of the neural networks, which naturally encode the physics contained in some HJ partial differential equations.
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Tensor Decomposition Methods for High-dimensional Hamilton--Jacobi--Bellman Equations
TL;DR: A tensor decomposition approach for the solution of high-dimensional, fully nonlinear Hamilton-Jacobi-Bellman equations arising in optimal feedback control of nonlinear dynamics is presented in this article.
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On some neural network architectures that can represent viscosity solutions of certain high dimensional Hamilton–Jacobi partial differential equations
Jérôme Darbon,Tingwei Meng +1 more
TL;DR: It is proved that under certain assumptions, the two neural network architectures proposed represent viscosity solutions to two sets of HJ PDEs with zero error.
Journal ArticleDOI
An Efficient DP Algorithm on a Tree-Structure for Finite Horizon Optimal Control Problems
TL;DR: A new approach for finite horizon optimal control problems where the value function is computed using a DP algorithm on a tree structure algorithm (TSA) constructed by the time discrete dynamics allowing for the solution of very high-dimensional problems.
Posted Content
Overcoming the curse of dimensionality for some Hamilton--Jacobi partial differential equations via neural network architectures
TL;DR: It is proved that some classes of neural networks correspond to representation formulas of HJ PDE solutions whose Hamiltonians and initial data are obtained from the parameters of the neural networks.
References
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Book
Controlled Markov processes and viscosity solutions
Wendell H. Fleming,H. Mete Soner +1 more
TL;DR: In this paper, an introduction to optimal stochastic control for continuous time Markov processes and to the theory of viscosity solutions is given, as well as a concise introduction to two-controller, zero-sum differential games.
Book
Optimal Control of Systems Governed by Partial Differential Equations
TL;DR: In this paper, the authors consider the problem of minimizing the sum of a differentiable and non-differentiable function in the context of a system governed by a Dirichlet problem.
Book
An introduction to infinite-dimensional linear systems theory
Ruth F. Curtain,Hans Zwart +1 more
TL;DR: This book presents Semigroup Theory, a treatment of systems theory concepts in finite dimensions with a focus on Hankel Operators and the Nehari Problem.
Book
Turbulence, Coherent Structures, Dynamical Systems and Symmetry
TL;DR: In this article, the authors present a review of rigor properties of low-dimensional models and their applications in the field of fluid mechanics. But they do not consider the effects of random perturbation on models.