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 ̇̂ ...

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 long time ago, since the launch of the first artificial satellite in 1957, controling attitude of satellites has been considered by the designers and engineers of aerospace industry. considering the importance of this issue various methods of control in response to this need have been presented and analyzed until now. in this paper, we propose and analyze a three-axis optimal control on the s...

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 ...

We give a simple proof of the Maximum Principle for smooth hybrid control systems by reducing the hybrid problem to an optimal control problem of Pontryagin type and then by using the classical Pontryagin Maximum Principle.