Stochastic Control of Tidal Dynamics Equation with Lévy Noise

Agarwal, Pooja; Manna, Utpal; Mukherjee, Debopriya

doi:10.1007/s00245-017-9440-2

Cited by 13 publications

(9 citation statements)

References 36 publications

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“…Suppose the assumption (H 1 ) is satisfied. Let ε > 1 + α, α ∈ (0, 1 2 ). Then there exist constants C > 0 and θ 1 > 0 such that for any t ∈ [0, T ] and x ∈ D(A α ), the following estimate holds:…”

Section: A Priori Estimatesmentioning

confidence: 99%

“…The qualitative properties like, ergodicity ( [14,28]), nonlinear filtering ( [19]) and invariant measure ( [12]) for stochastic Navier-Stokes/Burgers equations with Lévy noise have been studied in the literature. By applying the Lévy type stochastic forces on the state equation, the ergodic control of Navier-Stokes equations is treated in [27] and the optimal control of tidal dynamics model with control on the initial data is discussed in the recent paper [1].…”

Section: Introductionmentioning

confidence: 99%

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Dynamic Programming of Stochastic 2-D Navier-Stokes Equations Forced by Levy Noise

Mohan¹,

Sakthivel²,

Sritharan³

2022

Preprint

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In this article, we study optimal feedback control synthesis of stochastic 2D Navier-Stokes equations perturbed Lévy type noise with distributed stochastic control process acting on the state equation. We use the dynamic programming approach to solve this control problem which involves the study of second order infinite dimensional Hamilton-Jacobi-Bellman (HJB) equation consisting of an integro-differential operator with Lévy measure associated with the stochastic control problem. Using the regularizing properties of the transition semigroup corresponding to the stochastic 2D Navier-Stokes equation, we obtain a smooth solution in weighted function space for the HJB equation and solve the resultant feedback control problem.

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Section: A Priori Estimatesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Dynamic Programming of Stochastic 2-D Navier-Stokes Equations Forced by Levy Noise

Mohan¹,

Sakthivel²,

Sritharan³

2022

Preprint

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show abstract

“…Unlike in the case of optimal control of stochastic Navier-Stokes/Burgers equations with Gaussian noise (see, [23,25,13,6,7,8]), there are only very few works in the case of control of stochastic PDEs with Lévy noise (see, [1,17]). Moreover, in the recent book [4], a phenomenological study of fully developed turbulence and intermittency is carried out and in which it is nicely proposed that the experimental observations of these physical characteristics can be modeled by stochastic Navier-Stokes equations with Lévy noise.…”

Section: Introductionmentioning

confidence: 99%

Dynamic Programming of Stochastic Burgers Equation Driven by Levy Noise

Mohan¹,

Sakthivel²,

Sritharan³

2022

Preprint

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In this work, we study the optimal control of stochastic Burgers equation perturbed by Gaussian and Lévy type noises with distributed control process acting on the state equation. We use the dynamic programming approach for the second order Hamilton-Jacobi-Bellman (HJB) equation consisting of an integro-differential operator with Lévy measure associated with the stochastic control problem. Using the regularizing properties of the transition semigroup corresponding to the stochastic Burgers equation and compactness arguments, we solve the HJB equation and the resultant feedback control problem.

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“…The authors in [3] developed a numerical approach for solving the tidal dynamics problem, based on the splitting methods and the optimal control theory. The global solvability results for stochastic perturbations in bounded and unbounded domains, and asymptotic analysis of solutions is available in [2,16,24,37,40] etc.…”

mentioning

confidence: 99%

“…The dynamic programming method and feedback analysis for an optimal control of the 2D tidal dynamics system is carried out in [26]. The authors in [2] formulated a martingale problem of Stroock and Varadhan associated to an initial value control problem and established the existence of optimal controls. Some optimal control problems in tidal power generation and related problems are considered in [5,32,33,39], etc.…”

mentioning

confidence: 99%

First order necessary conditions of optimality for the two dimensional tidal dynamics system

Mohan¹

2021

MCRF

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<p style='text-indent:20px;'>In this work, we consider the two dimensional tidal dynamics equations in a bounded domain and address some optimal control problems like total energy minimization, minimization of dissipation of energy of the flow, etc. We also examine an another interesting control problem which is similar to that of the data assimilation problems in meteorology of obtaining unknown initial data, when the system under consideration is the tidal dynamics, using optimal control techniques. For these cases, different distributed optimal control problems are formulated as the minimization of suitable cost functionals subject to the controlled two dimensional tidal dynamics system. The existence of an optimal control as well as the first order necessary conditions of optimality for such systems are established and the optimal control is characterized via the adjoint variable. We also establish the uniqueness of optimal control in small time interval.</p>

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Stochastic Control of Tidal Dynamics Equation with Lévy Noise

Cited by 13 publications

References 36 publications

Dynamic Programming of Stochastic 2-D Navier-Stokes Equations Forced by Levy Noise

Dynamic Programming of Stochastic 2-D Navier-Stokes Equations Forced by Levy Noise

Dynamic Programming of Stochastic Burgers Equation Driven by Levy Noise

First order necessary conditions of optimality for the two dimensional tidal dynamics system

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