A Stochastic Pontryagin Maximum Principle on the Sierpinski Gasket

The Pontryagin Maximum Principle

Theorem (PontryaginMaximum Principle). Suppose a final time T and controlstate pair (û, x̂) on [τ, T ] give the minimum in the problem above; assume that û is piecewise continuous. Then there exist a vector of Lagrange multipliers (λ0, λ) ∈ R × R with λ0 ≥ 0 and a piecewise smooth function p: [τ, T ] → R n such that the function ĥ(t) def =H(t, x̂(t), p(t), û(t)) is piecewise smooth, and one has ̇̂ ...

The Maximum Principle of Pontryagin

Sampling on the Sierpinski Gasket

Mean-Field Pontryagin Maximum Principle

We derive a Maximum Principle for optimal control problems with constraints given by the coupling of a system of ODEs and a PDE of Vlasov-type. Such problems arise naturally as Γ-limits of optimal control problems subject to ODE constraints, modeling, for instance, external interventions on crowd dynamics. We obtain these first-order optimality conditions in the form of Hamiltonian flows in the...

A Short Proof of the Pontryagin Maximum Principle on Manifolds

Applying the Tubular Neighborhood Theorem, we give a short and new proof of the Pontryagin Maximum Principle on a smooth manifold. The idea is as follows. Given a control system on a manifold M , we embed it into an open subset of some Rn, and extend the control system to the open set. Then, we apply the Pontryagin Maximum Principle on Rn to the extended system and project the consequence to M ...