Pontryagin's maximum principle


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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 ≤ 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 x0) = 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.
References in periodicals archive ?
Lee DH, Milroy IP, Tyler K (1982) Application of Pontryagin's Maximum Principle to the semi-automatic control of rail vehicles.
Shell K (1968) Applications of Pontryagin's Maximum Principle to economics.
Next, applying the Pontryagin's Maximum Principle, we derive necessary conditions for our optimal control and corresponding state variables, including the two control functions.
The adjoint equations are formed by taking the derivative of the Hamiltonian with respect to each of the state variables as follow; By applying the Pontryagin's maximum principle [4] and the existence result of optimal control from [5], we obtain the following theorem:
The adjoint equations and transversality conditions can be obtained by using Pontryagin's Maximum Principle such that
In addition, we compare the performance of the optimal policy with other policies through extensive numerical results, and find that the optimal policy obtained by Pontryagin's Maximum Principle is the best.
Then according to the Pontryagin's Maximum Principle [33, P.
Therefore, the objective of the present work is the optimization of the reactor to produce the elastomeric copolyester copoly(ethylene-polyoxyethylene terephthalate) by using Pontryagin's maximum principle (PMP).
These by-products are the state or dependent variables, which are described through the Pontryagin's maximum principle, according to Westerterp et al.