Pontryagin's maximum principle

Pontryagin's maximum principle [‚pän·trē′ä·gənz ′mak·sə·məm ‚prin·sə·pəl]

(mathematics)

A theorem giving a necessary condition for the solution of optimal control problems: let θ(τ), τ₀≤ τ ≤Tbe a piecewise continuous vector function satisfying certain constraints; in order that the scalar functionS= ∑c_ix_i(T) be minimum for a process described by the equation ∂x_i/∂τ = (∂H/∂z_i)[z(τ),x(τ), θ(τ)] with given initial conditionsx(τ₀) =x⁰it is necessary that there exist a nonzero continuous vector functionz(τ) satisfyingdz_i/dτ = -(∂H/∂x_i). [z(τ),x(τ), θ(τ)],z_i(T) = -c_i, and that the vector θ(τ) be so chosen thatH[z(τ),x(τ), θ(τ)] is maximum for all τ, τ₀≤ τ ≤T.