Optimality Issues for a Class of Controlled Singularly Perturbed Stochastic Systems

Goreac, Dan; Serea, Oana-Silvia

doi:10.1007/s10957-015-0738-4

Cited by 5 publications

(5 citation statements)

References 26 publications

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“…[14]) and are considerably more challenging than our previous ones. The need for such extensions is equally justified by the optimality conditions through methods developed in [16].…”

Section: Linearization and Abstract Dynamic Programmingmentioning

confidence: 99%

Reflected dynamics: Viscosity analysis for L∞ cost, relaxation and abstract dynamic programming

Goreac

Hechaichi

Serea

2021

Journal of Differential Equations

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We study an optimal control problem consisting in minimizing the L ∞ norm of a Borel measurable cost function, in finite time, and over all trajectories associated with a controlled dynamics which is reflected in a compact prox-regular set. The first part of the paper provides the viscosity characterization of the value function for uniformly continuous costs. The second part is concerned with linear programming formulations of the problem and the ensued byproducts as e.g. dynamic programming principle for merely measurable costs.

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“…[14]) and are considerably more challenging than our previous ones. The need for such extensions is equally justified by the optimality conditions through methods developed in [16].…”

Section: Linearization and Abstract Dynamic Programmingmentioning

confidence: 99%

Reflected dynamics: Viscosity analysis for L∞ cost, relaxation and abstract dynamic programming

Goreac

Hechaichi

Serea

2021

Journal of Differential Equations

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“…Motivated by the approach in the forward setting (cf. [29]) as well as the singular perturbations setting in [30], we introduce the following sets…”

Section: Occupation Measuresmentioning

confidence: 99%

“…Remark 13 (i) This kind of relaxation has been recently employed in order to characterize Pontryagin-type optimality criteria in forward Brownian settings. To this purpose, the interested reader is referred to [30]. (ii) In the convex setting, whenever the functions ϕ O have at most quadratic growth, using the gradient estimates in [27,Proposition 4.3], one gets sup y 2 ∈∂ϕ O (y 1 ) |y 2 | ≤ c (1 + |y 1 |) , for some constant c > 0.…”

Section: Occupation Measuresmentioning

confidence: 99%

See 1 more Smart Citation

Infection Time in Multistable Gene Networks. A Backward Stochastic Variational Inequality with Nonconvex Switch-Dependent Reflection Approach

Goreac

Rotenstein

2016

Set-Valued Var. Anal

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International audienceWe investigate a mathematical model associated to the infection time in multistable gene networks. The mathematical processes are of hybrid switch type. The switch is governed by pure jump modes and linked to DNA bindings. The differential component follows backward stochastic dynamics reflected in some mode-dependent nonconvex domains. First, we study the existence of solutions to the resulting stochastic variational inclusions, by reducing the model to a family of ordinary variational inclusions with generalized reflection in semiconvex domains. Second, by considering control-dependent drivers, we hint to some model-selection approach by embedding the controlled backward stochastic variational inclusion in a family of regular measures. Regularity and structural properties of these sets are given

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“…However, the speed of convergence is given by the estimates in Lemma 23 and are less explicit than the Hölder ones exhibited in the cited papers. Second, having stated the equivalent problem on a linear space of measures should prove useful for optimality issues (see [20] or, more recently, [19] in a general Markovian framework or [25] in a Brownian one).…”

Section: Conclusion and Commentsmentioning

confidence: 99%

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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Optimality Issues for a Class of Controlled Singularly Perturbed Stochastic Systems

Cited by 5 publications

References 26 publications

Reflected dynamics: Viscosity analysis for L∞ cost, relaxation and abstract dynamic programming

Reflected dynamics: Viscosity analysis for L∞ cost, relaxation and abstract dynamic programming

Infection Time in Multistable Gene Networks. A Backward Stochastic Variational Inequality with Nonconvex Switch-Dependent Reflection Approach

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

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