The Pontryagin maximum principle for solving Fokker–Planck optimal control problems

Breitenbach, Tim; Borzì, Alfio

doi:10.1007/s10589-020-00187-x

Cited by 15 publications

(15 citation statements)

References 36 publications

(115 reference statements)

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“…FPEs are degenerate parabolic PDEs, implying that an optimization problem of transporting population governed by stochastic growth dynamics is formulated as a PDE-constrained optimization problem [23][24]. PDE-constrained optimization based on FPEs has been studied from the standpoint of maximum principle [25] where solving an optimization problem reduces to computing Fokker-Planck and adjoint equations concurrently. PDE-constrained optimization problems based on FPEs have various applications; they are including but are not limited to bilinear optimal control [26], model predictive control [27], costefficient switching [28], pedestrian motion control [29], equilibrium firm dynamics [30], and impulsive mean field games [31].…”

Section: Mathematical Backgroundmentioning

confidence: 99%

“…As in the existing framework of controlling FPEs [25], we solve the optimization problem using a maximum principle combined with an adjoint method. The impulse control assumption induces temporal interface conditions at prescribed discrete times [44][45], but the adjoint of the interface conditions is nontrivial even in the simplified case [34].…”

Section: Objective and Contributionmentioning

confidence: 99%

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Impulsive fishery resource transporting strategies based on an open‐ended stochastic growth model having a latent variable

Yoshioka

Tanaka

Aranishi

et al. 2021

Math Methods in App Sciences

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In inland fisheries, transporting fishery resource individuals from a habitat to spatially apart habitat(s) has recently been considered for fisheries stock management in the natural environment. However, its mathematical optimization, especially finding when and how much of the population should be transported, is still a fundamental unresolved issue. We propose a new impulse control framework to tackle this issue based on a simple but new stochastic growth model of individual fishes. The novel growth model governing individuals' body weights uses a Wright-Fisher model as a latent driver to reproduce plausible growth dynamics. The optimization problem is formulated as an impulse control problem of a cost-benefit functional constrained by a degenerate parabolic Fokker-Planck equation of the stochastic growth dynamics.Because the growth dynamics have an observable variable and an unobservable variable (a variable difficult or impossible to observe), we consider both full-information and partial-information cases. The latter is more involved but more realistic because of not explicitly using the unobservable variable in designing the controls. In both cases, resolving an optimization problem reduces to solving the associated Fokker-Planck and its adjoint equations, the latter being non-trivial. We present a derivation procedure of the adjoint equation and its internal boundary conditions in time to efficiently derive the optimal transporting strategy.We finally provide a demonstrative computational example of a transporting problem of Ayu sweetfish (Plecoglossus altivelis altivelis) based on the latest real data set.

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Section: Mathematical Backgroundmentioning

confidence: 99%

Section: Objective and Contributionmentioning

confidence: 99%

Impulsive fishery resource transporting strategies based on an open‐ended stochastic growth model having a latent variable

Yoshioka

Tanaka

Aranishi

et al. 2021

Math Methods in App Sciences

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“…This fact implies that the NE point u * = (u * 1 , u * 2 ) must fulfil the necessary optimality conditions given by the Pontryagin maximum principle applied to both optimisation problems stated in Eq. 5, alternatively (6).…”

Section: Definition 1 the Functions (U *mentioning

confidence: 99%

“…The sequential quadratic Hamiltonian (SQH) scheme has been recently proposed in [4][5][6][7] for solving nonsmooth optimal control problems governed by differential models. The SQH scheme belongs to the class of iterative methods known as successive approximations (SA) schemes that are based on the characterisation of optimality in control problems by the Pontryagin maximum principle (PMP); see [2,27] and [12] for a recent detailed discussion.…”

Section: Introductionmentioning

confidence: 99%

On the SQH Method for Solving Differential Nash Games

Campana

Borzì

2021

J Dyn Control Syst

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A sequentialquadratic Hamiltonian schemefor solving open-loop differential Nash games is proposed and investigated. This method is formulated in the framework of the Pontryagin maximum principle and represents an efficient and robust extension of the successive approximations strategy for solving optimal control problems. Theoretical results are presented that prove the well-posedness of the proposed scheme, and results of numerical experiments are reported that successfully validate its computational performance.

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“…Recently, Brockett's research programme has received much impetus through novel theoretical and numerical work focusing on deterministic models with random initial conditions and the corresponding Liouville equation [5,6], and in the case of a linear Boltzmann equation [7]. The modelling and simulation of FP ensemble optimal control problems has been investigated in view of their large applicability [12,[33][34][35]. However, in comparison to the amount of work on FP control problems with quadratic objectives [2,11,23], much less effort has been put in the analysis of FP ensemble optimal control problems.…”

Section: Introductionmentioning

confidence: 99%

Second-order analysis of Fokker–Planck ensemble optimal control problems

Körner¹,

Borzì²

2022

ESAIM: COCV

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Ensemble optimal control problems governed by a Fokker–Planck equation with space–time dependent controls are investigated. These problems require the minimisation of objective functionals of probability type and aim at determining robust control mechanisms for the ensemble of trajectories of the stochastic system defining the Fokker–Planck model. In this work, existence of optimal controls is proved and a detailed analysis of their characterization by first– and second–order optimality conditions is presented. For this purpose, the well–posedness of the Fokker–Planck equation, and new estimates concerning an inhomogeneous Fokker–Planck model are discussed, which are essential to prove the necessary regularity and compactness of the control–to–state map appearing in the first–and second–order analysis.

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The Pontryagin maximum principle for solving Fokker–Planck optimal control problems

Cited by 15 publications

References 36 publications

Impulsive fishery resource transporting strategies based on an open‐ended stochastic growth model having a latent variable

Impulsive fishery resource transporting strategies based on an open‐ended stochastic growth model having a latent variable

On the SQH Method for Solving Differential Nash Games

Second-order analysis of Fokker–Planck ensemble optimal control problems

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