Impulsive control of nonlocal transport equations

Pogodaev, Nikolay; Staritsyn, Maxim

doi:10.1016/j.jde.2020.03.007

Cited by 21 publications

(6 citation statements)

References 36 publications

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“…A natural direction of future work would be an extension of the discussed framework to the case of nonlocal vector fields involving a convolutional term K * µ, which brings certain analogy with convolutional NNs. The PMP for ensemble control of such nonlinear transport equations does exist [22], but the Hamiltonian system appears to be inseparable into the direct and dual subsystems, which makes the realization of our approach a challenging issue.…”

Section: Discussionmentioning

confidence: 99%

Feedback maximum principle for ensemble control of local continuity equations. An application to supervised machine learning

Staritsyn,

Pogodaev,

Chertovskih

et al. 2021

Preprint

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We consider an optimal control problem for a system of local continuity equations on a space of probability measures. Such systems can be viewed as macroscopic models of ensembles of non-interacting particles or homotypic individuals, representing several different "populations". For the stated problem, we propose a necessary optimality condition, which involves feedback controls inherent to the extremal structure, designed via the standard Pontryagin's Maximum Principle conditions. This optimality condition admits a realization as an iterative algorithm for optimal control. As a motivating case, we discuss an application of the derived optimality condition and the consequent numeric method to a problem of supervised machine learning via dynamic systems.

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Section: Discussionmentioning

confidence: 99%

Feedback maximum principle for ensemble control of local continuity equations. An application to supervised machine learning

Staritsyn,

Pogodaev,

Chertovskih

et al. 2021

Preprint

Self Cite

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“…We also mention the earlier work [45] in which the author independently derived an alternative version of the maximum principle for a very particular instance of problem (P ). This result was subsequently extended in the recent paper [46], in which a full PMP in the spirit of [12] is derived for unconstrained impulsive control problems, by means of discrete approximations combined with Ekeland's principle. We likewise point out that a completely different approach to Pontryagin optimality conditions for mean-field optimal control problems -formulated in terms of McKean-Vlasov dynamics and stochastic differential equations -was independently developed in [17] (see also the recent monograph [18, Vol.I Chapter 6]).…”

Section: Introductionmentioning

confidence: 90%

Necessary Optimality Conditions for Optimal Control Problems in Wasserstein Spaces

Bonnet¹,

Frankowska²

2021

Preprint

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In this article, we derive first-order necessary optimality conditions for a constrained optimal control problem formulated in the Wasserstein space of probability measures. To this end, we introduce a new notion of localised metric subdifferential for compactly supported probability measures, and investigate the intrinsic linearised Cauchy problems associated to non-local continuity equations. In particular, we show that when the velocity perturbations belong to the tangent cone to the set of admissible velocities, the solutions of these linearised problems are tangent to the solution set of the corresponding continuity inclusion. We then make use of these novel concepts to provide a synthetic and geometric proof of the celebrated Pontryagin Maximum Principle for an optimal control problem with inequality final-point constraints. In addition, we propose sufficient conditions ensuring the normality of the maximum principle.

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“…= Jt,T V t • X 0,t by direct computation, and ∂ τ τ =t Xτ,T = − Jt,T Vt by (15). Plugging these expressions to (22), we obtain…”

Section: Appendixmentioning

confidence: 98%

Optimal Route Planning in Steady Planar Convective Flows

Chertovskih¹,

Staritsyn²,

Pereira³

2020

Lecture Notes in Electrical Engineering

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Motivated by the problem of designing robust composite pulses for Bloch equations in the presence of natural perturbations, we study an abstract optimal ensemble control problem in a probabilistic setting with a general nonlinear performance criterion. The model under study addresses meanfield dynamics described by a linear continuity equation in the space of probability measures. For the resulting optimization problem, we derive an exact representation of the increment of the cost functional in terms of the flow of the driving vector field. Relying on the exact increment formula, a descent method is designed that is free of any internal line search. The numerical method is applied to solve new control problems for distributed ensembles of Bloch equations. *The authors acknowledge the financial support of the Foundation for Science and Technology (FCT, Portugal) in the framework of the Associated Laboratory "Advanced Production and Intelligent Systems" (AL ARISE, ref. LA/P/0112/2020), R&D Unit SYSTEC (base UIDB/00147/2020 and programmatic UIDP/00147/2020 funds), and projects SNAP (ref. NORTE-01-0145-FEDER-000085) and MLDLCOV (ref. DSAIPA/CS/0086/2020).

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Impulsive control of nonlocal transport equations

Cited by 21 publications

References 36 publications

Feedback maximum principle for ensemble control of local continuity equations. An application to supervised machine learning

Feedback maximum principle for ensemble control of local continuity equations. An application to supervised machine learning

Necessary Optimality Conditions for Optimal Control Problems in Wasserstein Spaces

Optimal Route Planning in Steady Planar Convective Flows

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