A Pontryagin Maximum Principle in Wasserstein spaces for constrained optimal control problems

Abstract

In this paper, we prove a Pontryagin Maximum Principle for constrained optimal control problems in the Wasserstein space of probability measures. The dynamics is described by a transport equation with non-local velocities which are affine in the control, and is subject to end-point and running state constraints. Building on our previous work, we combine the classical method of needle-variations from geometric control theory and the metric differential structure of the Wasserstein spaces to obtain a maximum principle formulated in the so-called Gamkrelidze form.The support of a Borel measure µ ∈ M + (R d ) is defined as the closed set supp(µ) = {x ∈ R d s.t. µ(N ) > 0 for any neighbourhood N of x}. We denote by P c (R d ) ⊂ P(R d ) the set of probability measures with compact support.We say that a sequence (µ n ) ⊂ P(R d ) of Borel probability measures converges narrowly towardsHere C 0 b (R d ) denotes the set of continuous and bounded functions over R d . Remark that the corresponding narrow topology coincides with the restriction to P(R d ) of the weak- * topology in M + (R d ). In the following proposition, we recall a useful convergence property on the measures of Borel sets, usually referred to as the Portmanteau Theorem (see e.g. [7, Proposition 1.62]). Proposition 2.1 (Portmanteau Theorem). Let (µ n ) ⊂ M + (R d ) be a sequence of measures converging in the weak- * topology towards µ ∈ M + (R d ). Then for any Borel set A ⊂ R d such that µ(∂A) = 0, it holds that µ n (A) −→ n→+∞ µ(A).We recall in the following definitions the notion of pushforward of a Borel probability measure through a Borel map, along with that of transport plan.Definition 2.1 (Pushforward of a measure through a Borel map). Given µ ∈ P(R d ) and a Borel mapDefinition 2.2 (Transport plan). Let µ, ν ∈ P(R d ). We say that γ ∈ P(R 2d ) is a transport plan between µ and ν -denoted by γ ∈ Γ(µ, ν) -provided thatfor any pair of Borel sets A, B ⊂ R d . This property can be equivalently formulated in terms of pushforwards by π 1 # γ = µ and π 2 # γ = ν, where the maps π 1 , π 2 : R 2d → R d denote the projection operators on the first and second factor respectively.

Optimal control of the Fokker-Planck equation under state constraints in the Wasserstein space

“…We can argue as in [14] to deduce from this estimate that the map t Ñ mptq is uniformly (in ǫ, δ P p0, 1q) 1 2 -Hölder continuous. Since D m Ψ is bounded we have that Ψ is a Lipschitz function (see for instance [11] Remark 5.27) and therefore there exists γ ą 0 independent of ǫ, δ P p0, 1q such that, for all pt, sq P r0, T s 2 |Ψpmptqq ´Ψpmpsqq| ď γ a |t ´s|.…”

“…In [25,22] the authors use the dynamic programming approach and prove that the value function is the viscosity (in a sense adapted to the infinite dimensional setting) of an HJB equation. Whereas in [1,3,2] the authors prove some adapted forms of the Pontryagin maximum principle. Notice that optimal control problems for the Fokker-Planck equation were previously considered in [15,12] but without constraint.…”

Impulsive control of nonlocal transport equations

“…The necessary optimality condition that we are going to establish (Theorem 4.7) can be formally deduced from a general result by B. Bonnet and F. Rossi [9], who reproduced the standard proof of Pontryagin's Maximum Principle [12] (based on the needle variation) in the context of control systems in the space of probability measures. We propose an alternative (less demanding, in our opinion) approach employing discretization of the initial measure and the application of Ekeland's variational principle.…”

A level-set approach to the control of state-constrained McKean-Vlasov equations: application to renewable energy storage and portfolio selection

“…Next, it has been extended in [24] to a running law constraint for the control of a standard diffusion process with McKean-Vlasov type cost through the control of a Fokker-Planck equation. Several works also consider directly the optimal control of Fokker-Planck equations in the Wasserstein space with terminal or running constraints, such as [8,9] through Pontryagin principle, in the deterministic case without diffusion.…”