Generalized Control Systems in the Space of Probability Measures

Cavagnari, Giulia; Marigonda, Antonio; Nguyen, Khai T.; Priuli, Fabio S.

doi:10.1007/s11228-017-0414-y

Cited by 40 publications

(60 citation statements)

References 22 publications

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“…The l.s.c. of the functional J F (·, ·), defined in Definition 3.3(A2), was already proved in Lemma 3 in [16]. By Proposition 3 in [16] we have ρ ∈ A I .…”

Section: Control Sparsity Problemsmentioning

confidence: 68%

“…In [14,16,17] time-optimal control problems in the space of probability measures P(R d ) are addressed, by considering systems without interactions among the agents. The authors were able to extend to this framework some classical results, among which an Hamilton-Jacobi-Bellman (briefly HJB) equation solved by the minimum-time function in a suitable viscosity sense.…”

Section: Introductionmentioning

confidence: 99%

“…In Section 6, we study for each of the considered cases an Hamilton-Jacobi-Bellman equation solved by the value function in some suitable viscosity sense. Finally, in Appendix A we discuss other applications of the framework outlined in Section 4, while in Appendix B we recall some estimates borrowed from [13,16].…”

Section: Introductionmentioning

confidence: 99%

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Generalized dynamic programming principle and sparse mean-field control problems

Cavagnari

Marigonda

Piccoli

2020

Journal of Mathematical Analysis and Applications

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In this paper we study optimal control problems in Wasserstein spaces, which are suitable to describe macroscopic dynamics of multi-particle systems. The dynamics is described by a parametrized continuity equation, in which the Eulerian velocity field is affine w.r.t. some variables. Our aim is to minimize a cost functional which includes a control norm, thus enforcing a control sparsity constraint. More precisely, we consider a nonlocal restriction on the total amount of control that can be used depending on the overall state of the evolving mass. We treat in details two main cases: an instantaneous constraint on the control applied to the evolving mass and a cumulative constraint, which depends also on the amount of control used in previous times. For both constraints, we prove the existence of optimal trajectories for general cost functions and that the value function is viscosity solution of a suitable Hamilton-Jacobi-Bellmann equation. Finally, we discuss an abstract Dynamic Programming Principle, providing further applications in the Appendix.

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“…The l.s.c. of the functional J F (·, ·), defined in Definition 3.3(A2), was already proved in Lemma 3 in [16]. By Proposition 3 in [16] we have ρ ∈ A I .…”

Section: Control Sparsity Problemsmentioning

confidence: 68%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Generalized dynamic programming principle and sparse mean-field control problems

Cavagnari

Marigonda

Piccoli

2020

Journal of Mathematical Analysis and Applications

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“…A significant progress in analysing such problems has been made in the last few years, mainly thanks to modern achievements in geometry and analysis on metric spaces of probability measures [3,39]. The existing works are mainly concentrated in two directions: one collection of studies is devoted to necessary optimality conditions [8-10, 20, 33], another part is focused on the dynamic programming approach [5,6,18,28]. In all cited papers, the driving vector field is assumed to be L ∞ bounded in time variable, which makes the problem relatively regular.…”

Section: Introductionmentioning

confidence: 99%

Impulsive control of nonlocal transport equations

Pogodaev

Staritsyn

2020

Journal of Differential Equations

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The paper extends an impulsive control-theoretical framework towards dynamical systems in the space of measures. We consider a transport equation describing the time-evolution of a conservative "mass" (probability measure), which represents an infinite ensemble of interacting particles. The driving vector field contains nonlocal terms and is affine in control variable. The control is assumed to be common for all the agents, i.e., it is a function of time variable only. The main feature of the addressed model is the admittance of "shock" impacts, i.e. controls, which can be arbitrary close in their influence on each an agent to Dirac-type distributions. We construct an impulsive relaxation of this system and of the corresponding optimal control problem. For the latter we establish a necessary optimality condition in the form of Pontryagin's Maximum Principle.2000 Mathematics Subject Classification: 49K20, 49J45, 93C20

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“…Jn k n k )} converges to (r,ê r # ν). Taking into account (15) and upper semicontinuity of ψ 2 , we get…”

mentioning

confidence: 99%

Krasovskii–Subbotin Approach to Mean Field Type Differential Games

Averboukh

2018

Dyn Games Appl

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The paper is concerned with the feedback approach to the deterministic mean field type differential games. Previously, it was shown that suboptimal strategies in the mean field type differential game can constructed based on functions of time and probability satisfying the stability condition. This property realizes the dynamic programming principle for the constant control of one player. We present the infinitesimal form of this condition involving analogs of the directional derivatives. In particular, we obtain the characterization of the value function of the deterministic mean field type differential game in the terms of directional derivatives and the set of directions feasible by virtue of the dynamics of the game. (2010): 93C30, 49L20, 46G05, 93A15. MSC Classification

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Generalized Control Systems in the Space of Probability Measures

Cited by 40 publications

References 22 publications

Generalized dynamic programming principle and sparse mean-field control problems

Generalized dynamic programming principle and sparse mean-field control problems

Impulsive control of nonlocal transport equations

Krasovskii–Subbotin Approach to Mean Field Type Differential Games

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