Pontryagin's maximum principle
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Pontryagin's maximum principle[‚pän·trē′ä·gənz ′mak·sə·məm ‚prin·sə·pəl]
A theorem giving a necessary condition for the solution of optimal control problems: let θ(τ), τ0≤ τ ≤ T be a piecewise continuous vector function satisfying certain constraints; in order that the scalar function S = ∑ ci xi (T) be minimum for a process described by the equation ∂ xi /∂τ = (∂ H /∂ zi )[z (τ), x (τ), θ(τ)] with given initial conditions x (τ0) = x 0 it is necessary that there exist a nonzero continuous vector function z (τ) satisfying dzi / d τ = -(∂ H /∂ xi ). [z (τ), x (τ), θ(τ)], zi (T) = -ci , and that the vector θ(τ) be so chosen that H [z (τ), x (τ), θ(τ)] is maximum for all τ, τ0≤ τ ≤ T.