The Pontryagin Maximum Principle
نویسندگان
چکیده
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 ̇̂ h(t) = Ht(t, x̂(t), p(t), û(t)) a.e., (a) Adjoint equations: −ṗ(t) = Hx(t, x̂(t), p(t), û(t)) a.e., ̇̂ x(t) = Hp(t, x̂(t), p(t), û(t)) a.e., (b) State equations: û(t) ∈ Û(t, x̂(t), p(t)) a.e., (c) Maximum condition: ( ĥ(T ), −p(T ) ) = λ0∇l0(T, x̂(T )) + M ∑
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