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 Pontrjagin maximum principle solves the problem of optimal control of a continuous deterministic system. The discrete maximum principle solves the problem of optimal control of a discrete-time deterministic system. The maximum principle changes the problem of optimal control to a two point boundary value problem which can be completely solved only in special tasks. It was probably the reaso...

Many science and engineering problems can be formulated as optimization problems that are governed by m-flow type PDEs (multitime evolution systems) and by cost functionals expressed as multiple integrals or curvilinear integrals. Our paper discuss the m-flow type PDEconstrained optimization problems, focussing on a simplified multitime maximum principle. This extends the simplified single-time...