Linearization Techniques for Controlled Piecewise Deterministic Markov Processes; Application to Zubov’s Method

Goreac, Dan; Serea, Oana-Silvia

doi:10.1007/s00245-012-9169-x

Cited by 17 publications

(29 citation statements)

References 23 publications

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“…Now, let us consider ( ) to be a sequence of standard molli…ers Then, using the Remark 2.2 (i) and (25), the convoluted function W ";h := W ";h; satisfy :…”

Section: Remark 32mentioning

confidence: 99%

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Optimality Issues for a Class of Controlled Singularly Perturbed Stochastic Systems

Goreac

Serea

2015

J Optim Theory Appl

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The present paper aims at studying stochastic singularly perturbed control systems. We begin by recalling the linear (primal and dual) formulations for classical control problems. In this framework, we give necessary and su¢cient support criteria for optimality of the measures intervening in these formulations. Motivated by these remarks, in a …rst step, we provide linearized formulations associated to the value function in the averaged dynamics setting. Second, these formulations are used to infer criteria allowing to identify the optimal trajectory of the averaged stochastic system.

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“…Now, let us consider ( ) to be a sequence of standard molli…ers Then, using the Remark 2.2 (i) and (25), the convoluted function W ";h := W ";h; satisfy :…”

Section: Remark 32mentioning

confidence: 99%

“…In particular, this can be applied to larger classes of systems (e.g. Piecewise Deterministic Markov Processes presenting a mild pathdependance; see [25,26] for the linearization techniques). For this particular class, little is available in the literature.…”

Section: Perspectivesmentioning

confidence: 99%

Optimality Issues for a Class of Controlled Singularly Perturbed Stochastic Systems

Goreac

Serea

2015

J Optim Theory Appl

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“…They appear under this form in [20] and are needed to infer the uniform continuity of the value function. The assumption (A4) is needed in the Appendix of [16] to provide stability properties of viscosity solutions. Roughly speaking, (A4b) states that the probability of exiting the ball centered at the initial point is zero as the radius increases to ∞.…”

Section: Controlled Pdmpsmentioning

confidence: 99%

“…For further details on these assumptions as well as for connections with stochastic gene networks, the reader is referred to [15] and [16].…”

Section: Controlled Pdmpsmentioning

confidence: 99%

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Uniform Assymptotics in the Average Continuous Control of Piecewise Deterministic Markov Processes : Vanishing Approach

Goreac¹,

Serea

2014

ESAIM: ProcS

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A Piecewise Deterministic Markov Toy Model for Traffic/Maintenance and Associated Hamilton–Jacobi Integrodifferential Systems on Networks

2015

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We study optimal control problems in infinite horizon when the dynamics belong to a specific class of piecewise deterministic Markov processes constrained to star-shaped networks (corresponding to a toy traffic model). We adapt the results in [35] to prove the regularity of the value function and the dynamic programming principle. Extending the networks and Krylov's "shaking the coefficients" method, we prove that the value function can be seen as the solution to a linearized optimization problem set on a convenient set of probability measures. The approach relies entirely on viscosity arguments. As a by-product, the dual formulation guarantees that the value function is the pointwise supremum over regular subsolutions of the associated Hamilton-Jacobi integrodifferential system. This ensures that the value function satisfies Perron's preconization for the (unique) candidate to viscosity solution.Mathematics Subject Classification. 49L25, 93E20, 60J25, 49L20 Acknowledgement. The authors would like to thank the anonymous referees for constructive remarks allowing to improve the manuscript. v δ (x, γ) := inf α,X x,γ,α · ∈network E ∞ 0 e −δt l Γ x,γ,α t (X x,γ,α t , α t ) dt . Tel. : +33 (0)1 60 95 75 27, Fax : +33 (0)1 60 95 75 45 ‡ Acknowledgement. )) ds ,where α 2 ∈ L 0 (R + × R m × E; A). We set (X x 0 ,γ 0 ,α t , Γ x 0 ,γ 0 ,α t ) = (y Υ 1 (t; τ 1 , Y 1 , α 2 ) , Υ 1 ) , if t ∈ [τ 1 , τ 2 ) .The post-jump location (Y 2 , Υ 2 ) satisfies

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Linearization Techniques for Controlled Piecewise Deterministic Markov Processes; Application to Zubov’s Method

Cited by 17 publications

References 23 publications

Optimality Issues for a Class of Controlled Singularly Perturbed Stochastic Systems

Optimality Issues for a Class of Controlled Singularly Perturbed Stochastic Systems

Uniform Assymptotics in the Average Continuous Control of Piecewise Deterministic Markov Processes : Vanishing Approach

A Piecewise Deterministic Markov Toy Model for Traffic/Maintenance and Associated Hamilton–Jacobi Integrodifferential Systems on Networks

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