2021
DOI: 10.48550/arxiv.2105.03336
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Neural network architectures using min plus algebra for solving certain high dimensional optimal control problems and Hamilton-Jacobi PDEs

Abstract: Solving high dimensional optimal control problems and corresponding Hamilton-Jacobi PDEs are important but challenging problems in control engineering. In this paper, we propose two abstract neural network architectures which respectively represent the value function and the state feedback characterisation of the optimal control for certain class of high dimensional optimal control problems. We provide the mathematical analysis for the two abstract architectures. We also show several numerical results computed… Show more

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Cited by 4 publications
(21 citation statements)
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“…In this case, Proposition 21 shows that the function (x, t) → V (x, t; p, a, b) defined in (9) and (18) (where p is the slope of Φ) is the unique continuously differentiable solution in the solution set G to the HJ PDE (8) with this linear initial data Φ. Moreover, Proposition 22 shows that the trajectory s → γ(s; x, t, p, a, b) defined by (12), ( 13), ( 14), ( 16), (17), and (19) for different cases is the unique optimal trajectory of the optimal control problem (6), whose optimal value equals V (x, t; p, a, b).…”
Section: One-dimensional Casementioning
confidence: 98%
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“…In this case, Proposition 21 shows that the function (x, t) → V (x, t; p, a, b) defined in (9) and (18) (where p is the slope of Φ) is the unique continuously differentiable solution in the solution set G to the HJ PDE (8) with this linear initial data Φ. Moreover, Proposition 22 shows that the trajectory s → γ(s; x, t, p, a, b) defined by (12), ( 13), ( 14), ( 16), (17), and (19) for different cases is the unique optimal trajectory of the optimal control problem (6), whose optimal value equals V (x, t; p, a, b).…”
Section: One-dimensional Casementioning
confidence: 98%
“…where the right-hand side is well-defined using ( 12), ( 13), ( 14), (16), and (17) for different cases. Now, we consider a general convex initial cost Φ : R → R. The corresponding HJ PDE is solved using the following Hopf-type formula:…”
Section: One-dimensional Casementioning
confidence: 99%
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