A Max-Plus-Based Algorithm for a Hamilton--Jacobi--Bellman Equation of Nonlinear Filtering

Fleming, Wendell H.; McEneaney, William M.

doi:10.1137/s0363012998332433

Cited by 115 publications

(142 citation statements)

References 14 publications

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“…This argument (clearly, a heuristic one) can be extended to equations of a more general form. For a rigorous treatment of (semiring) linearity for these equations see [52,96,122,123] and also [120]. Notice that if h is changed to −h, then we have that the resulting Hamilton-Jacobi equation is linear over R min .…”

Section: The Superposition Principle and Linear Problemsmentioning

confidence: 99%

Maslov dequantization, idempotent and tropical mathematics: A brief introduction

Litvinov

2007

J Math Sci

140

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This paper is a brief introduction to idempotent and tropical mathematics. Tropical mathematics can be treated as a result of the so-called Maslov dequantization of the traditional mathematics over numerical fields as the Planck constant tends to zero taking imaginary values.This paper is a brief introduction, practically without exact theorems and proofs, to the Maslov dequantization and idempotent and tropical mathematics. Our list of references is not complete (not at all). Additional references can be found, e.g., in the electronic archive http://arXiv.org

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Section: The Superposition Principle and Linear Problemsmentioning

confidence: 99%

Maslov dequantization, idempotent and tropical mathematics: A brief introduction

Litvinov

2007

J Math Sci

140

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“…Semiconvex duality is a duality between these spaces of semiconvex and semiconcave functions, that is established via the semiconvex transform [11]. The semiconvex transform is a generalization of the Legendre-Fenchel transform [19], [20], [21], in which convexity is weakened to semiconvexity via a quadratic basis function ϕ : R n ×R n → R. This basis function is defined for all x, z ∈ R n by ϕ(x, z) .…”

Section: A Max-plus Algebra and Semiconvex Dualitymentioning

confidence: 99%

“…One such fundamental solution is the symplectic fundamental solution, which is itself the solution of a (derived) Hamiltonian system of linear ordinary differential equations, see for example [1], [5], [6]. Another fundamental solution is the max-plus dual-space fundamental solution [7], [8], [9], [10], which is constructed by exploiting semiconvex duality [11] and maxplus linearity of the Lax-Oleinik semigroup [12] {pdower,hzhang}@unimelb.edu.au programming evolution operators for an associated optimal control problem, see also [12], [13], [14], [15], [16].…”

Section: Introductionmentioning

confidence: 99%

A max-plus primal space fundamental solution for a class of differential Riccati equations

Dower

Zhang

2017

Math. Control Signals Syst.

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Abstract-A class of differential Riccati equations (DREs) is considered whereby the evolution of any solution can be identified with the propagation of a value function of a corresponding optimal control problem arising in L2-gain analysis. By exploiting the semigroup properties inherited from the attendant dynamic programming principle, a max-plus primal space fundamental solution semigroup of max-plus linear max-plus integral operators is developed that encapsulates all such value function propagations. Using this semigroup, a new one-parameter fundamental solution semigroup of matrices is developed for the aforementioned class of DREs. It is demonstrated that this new semigroup can be used to compute particular solutions of these DREs, and to characterize finite escape times (should they exist) in a relatively simple way compared with that provided by the standard symplectic fundamental solution semigroup.

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“…Here, max-plus methods are appropriate for problems with maximizing controllers and vice-versa. These methods include maxplus basis-expansion approaches [1], [2], [6], [7], [11], [15], [18], as well as the more recently developed curse-of-dimensionality-free methods [11], [16], [17]. However, stochastic control problems have eluded idempotent methods.…”

Section: Introductionmentioning

confidence: 99%

Distributed dynamic programming for discrete-time stochastic control, and idempotent algorithms

McEneaney

2011

Automatica

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Previously, idempotent methods have been found to be extremely fast for solution of dynamic programming equations associated with deterministic control problems. The original methods exploited the idempotent (e.g., max-plus) linearity of the associated semigroup operator. However, it is now known that the curse-of-dimensionality-free idempotent methods do not require this linearity. Instead, it is sufficient that certain solution forms are retained through application of the associated semigroup operator. Here, we see that idempotent methods may be used to solve some classes of stochastic control problems. The key is the use of the idempotent distributive property. This allows one to apply the curse-of-dimensionality-free idempotent approach. We demonstrate this approach for a class of nonlinear, discrete-time stochastic control problems.

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A Max-Plus-Based Algorithm for a Hamilton--Jacobi--Bellman Equation of Nonlinear Filtering

Cited by 115 publications

References 14 publications

Maslov dequantization, idempotent and tropical mathematics: A brief introduction

Maslov dequantization, idempotent and tropical mathematics: A brief introduction

A max-plus primal space fundamental solution for a class of differential Riccati equations

Distributed dynamic programming for discrete-time stochastic control, and idempotent algorithms

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