An Adaptive Max-Plus Eigenvector Method for Continuous Time Optimal Control Problems

Dower, Peter M.

doi:10.1007/978-3-030-01959-4_10

Cited by 6 publications

(5 citation statements)

References 17 publications

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“…is the bivariate idempotent convolution kernel associated with the max-plus primal space fundamental solution semigroup corresponding to the optimal control problem (18), see for example [5,Theorem 2] or [14,Theorem 5]. Given any t ∈ (0,t µ ), this kernel is defined via an optimal TPBVP by…”

Section: Idempotent Convolution Representation For (18)mentioning

confidence: 99%

“…Similarly, ψ is concave if −ψ is convex, and upper closed if −ψ is lower closed, see [16, pp.15-17]. Following [9,14], uniformly semiconvex and semiconcave extended real valued function spaces S −M + and S −M − are defined with respect to operator M of (30),(31) by…”

Section: Idempotent Convolution Representation For (18)mentioning

confidence: 99%

“…The proof of a special case [5,Theorem 11] of Theorem 3.3 employs a homotopy argument to verify the corresponding quadratic representation analogous to (27). Here, motivated by [15,Theorem 2] and [14,Theorem 5], an alternative approach to the proof of Theorem 3.3 as stated is developed by exploiting semiconvex duality. This development commences with the definition of a parameterised terminal cost ϕ :…”

Section: Idempotent Convolution Representation For (18)mentioning

confidence: 99%

“…The correspondence between stationary action and optimal control can break down for longer time horizons due to a loss of concavity of the payoff (14), see Lemma 3.1 and the finite escape property (54) associated with the idempotent representation (25), ( 27), (28), (29). Consequently, for longer horizons, a modified approach is required.…”

Section: Longer Horizonsmentioning

confidence: 99%

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Representation of fundamental solution groups for wave equations via stationary action and optimal control

Dower

McEneaney

2017

2017 American Control Conference (ACC)

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A representation of a fundamental solution group for a class of wave equations is constructed by exploiting connections between stationary action and optimal control. By using a Yosida approximation of the associated generator, an approximation of the group of interest is represented for sufficiently short time horizons via an idempotent convolution kernel that describes all possible solutions of a corresponding short time horizon optimal control problem. It is shown that this representation of the approximate group can be extended to arbitrary longer horizons via a concatenation of such short horizon optimal control problems, provided that the associated initial and terminal conditions employed in concatenating trajectories are determined via a stationarity rather than an optimality based condition. The long horizon approximate group obtained is shown to converge strongly to the exact group of interest, under reasonable conditions. The construction is illustrated by its application to the solution of a two point boundary value problem.Keywords. Optimal control, stationary action, dynamic programming, wave equations, fundamental solution groups. two point boundary value problems. MSC2010: 35L05, 49J20, 49L20.any trajectory of an energy conserving system renders the corresponding action functional stationary in the calculus of variation sense.Consequently, Hamilton's action principle can be interpreted as providing a characterisation of all solutions of an energy conserving or lossless system. This interpretation motivates the development summarised in this work, with Hamilton's action principle applied via an optimal control representation to construct the fundamental solution group corresponding to a lossless wave equation. The specific wave equation of interest is given byẍ

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Section: Idempotent Convolution Representation For (18)mentioning

confidence: 99%

Section: Idempotent Convolution Representation For (18)mentioning

confidence: 99%

Section: Idempotent Convolution Representation For (18)mentioning

confidence: 99%

Section: Longer Horizonsmentioning

confidence: 99%

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Representation of fundamental solution groups for wave equations via stationary action and optimal control

Dower

McEneaney

2017

2017 American Control Conference (ACC)

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“…One may cite the sparse grids approximations of Bokanowski, Garcke, Griebel and Klompmaker [BGGK13], or of Kang and Wilcox [KW17], the tensor decompositions of Dolgov, Kalise and Kunisch [DKK21] or of Oster, Sallandt and Schneider [OSS22], the deep learning techniques applied by Nakamura-Zimmerer, Gong, and Kang [NZGK21] or by Darbon and Meng [DM21]. In the case of structured problems, one may also cite the max-plus or tropical numerical method of McEneaney [McE06,McE07], see also [MDG08,Qu13,Qu14,Dow18] for further developments, and the Hopf formula of Darbon and Osher [DO16], see also [CDOY19,KMH + 18,YD21].…”

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confidence: 99%

A Multi-Level Fast-Marching Method For The Minimum Time Problem

Akian¹,

Gaubert²,

Liu³

2023

Preprint

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We introduce a new numerical method to approximate the solutions of a class of stationary Hamilton-Jacobi (HJ) partial differential equations arising from minimum time optimal control problems. We rely on several grid approximations, and look for the optimal trajectories by using the coarse grid approximations to reduce the search space in fine grids. This may be thought of as an infinitesimal version of the "highway hierarchy" method which has been developed to solve shortest path problems (with discrete time and discrete state). We obtain, for each level, an approximate value function on a sub-domain of the state space. We show that the sequence obtained in this way does converge to the viscosity solution of the HJ equation. Moreover, for our multi-level algorithm, if 0 < γ 1 is the convergence rate of the classical numerical scheme, then the number of arithmetic operations needed to obtain an error in O(ε) is in O(ε − θd γ ), to be compared with O(ε − d γ ) for ordinary grid-based methods. Here d is the dimension of the problem, and θ < 1 depends on d and on the "stiffness" of the value function around optimal trajectories, and the notation O ignores logarithmic factors. When γ = 1 and the stiffness is high, this complexity becomes in O(ε −1 ). We illustrate the approach by solving HJ equations of eikonal type up to dimension 7.

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Verifying Fundamental Solution Groups for Lossless Wave Equations via Stationary Action and Optimal Control

Dower

McEneaney

2020

Appl Math Optim

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An Adaptive Max-Plus Eigenvector Method for Continuous Time Optimal Control Problems

Cited by 6 publications

References 17 publications

Representation of fundamental solution groups for wave equations via stationary action and optimal control

Representation of fundamental solution groups for wave equations via stationary action and optimal control

A Multi-Level Fast-Marching Method For The Minimum Time Problem

Verifying Fundamental Solution Groups for Lossless Wave Equations via Stationary Action and Optimal Control

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