Optimal control for a mixed flow of Hamiltonian and gradient type in space of probability measures

Abstract

Abstract. In this paper we investigate an optimal control problem in the space of measures on R 2 . The problem is motivated by a stochastic interacting particle model which gives the 2-D Navier-Stokes equations in their vorticity formulation as mean-field equation. We prove that the associated Hamilton-Jacobi-Bellman equation, in the space of probability measures, is well-posed in an appropriately defined viscosity solution sense.

Sample-path large deviation principle for a 2-D stochastic interacting vortex dynamics with singular kernel

“…(C.1) is standard and follows from Fubini's lemma and integration by parts against test functions φ ∈ C ∞ (T 2 ) to show the equality in the distribution sense. (C.2) can be proved by a similar argument as the proof of Lemma 7.1 of[21]. Turing to (2), by (1.1),J * η = −∆N * J * η + 1.Noting that T 2 N dx = 0, by Fubini's lemma and integration by partsN * J * J * γ, η = N * J * γ, −∆N * J * η = ∇N * J * γ, ∇N * J * η .…”

“…For singular interaction kernels, Fontbona [23] proved the sample-path LDP for diffusing particles with electrostatic repulsion in one dimension. Applying the method in [20], at least formally, Feng and Świech [21] gave the rate function of sample-path LDP for empirical measures of (1.2), starting from initial data with finite entropy. However, making this approach rigorous is really subtle, requiring an uniqueness theory for infinite-dimensional Hamiltonian-Jocabi equation, especially when only having initial data with finite energy in hand.…”

A Hamilton-Jacobi PDE associated with hydrodynamic fluctuations from a nonlinear diffusion equation

“…This is a relatively new topic and is a step forward compared to earlier studies initiated by Crandall and Lions [4][5][6][7][8][9][10] on Hamilton-Jacobi equations in infinite dimensions, focusing on Hilbert spaces. Our Hamiltonian has a structure which is closer in spirit to the one studied by Crandall and Lions [8] (but with a nonlinear operator and other subtle differences), to Example 9.35 in Chapter 9 and Section 13.3.3 in Chapter 13 of [24] and to Example 3 of [25], or Feng and Swiech [26]. It is different in structure than those studied by Gangbo, Nguyen and Tudorascu [28] and Gangbo and Tudorascu [29] in Wasserstein space; or to those studied with metric analysis techniques by Giga, Hamamuki and Nakayasu [31], Ambrosio and Feng [1], Gangbo and Swiech [30].…”

A Differential Game with a Blind Player

“…Note that other definitions of viscosity solution in the space W have been introducted recently: see for instance [6,7,9,10]. Our definition is closely related to the one of [3], which seems more appropriate for the kind of problem we have to handle.…”

“…The definition of the viscosity solution in this framework is inspired by, but slightly differs from, the one given in [3]. Other definitions of viscosity solution in the Wasserstein space have been used in the literature, in general for more singular dynamics (see for instance [6,7,9,10]). Then the existence of a value (i.e., the fact that the upper value coincides with the lower one) relies on min-max arguments combined with PDE ones: we introduce an auxiliary game in which the uninformed player chooses a strategy by randomizing over a finite set of controls.…”