? Pontryagin's minimum principle

From: Cheng Cosine (acosine_at_ms13.url.com.tw)
Date: 03/30/05

Date: Tue, 29 Mar 2005 19:47:43 -0700

Hi:

The Pontryagin's min principle says:

Let the plant dx/dt = f(x,u,t) have an associated cost index of

J(t0) = phi(x(T),T)+Integral( L(x,u,t), t= t0 to T), where

the final state must satisfy psi(x(T),T) = 0 and x(t0) is given.

Now suppose the control u(t) is constrained to lie in an

admissible region, which might be defined by a requirement

that its magnitude be less than a given value. It is shown by

Pontryagin that in this case, the optimal conditions for unconstraint

case sitll hold but the stationary condition is replaced by:

H(x_opt, u_opt, landa_opt, t) <= H(x_opt, u_opt+du, landa_opt, t) for

all admissible du, where _opt denotes optimal quantities, and H is

the Hamiltonian.

But why this is true (i.e. how to prove it), and what does it differ

with simply saying that the min is reached when one uses the largest

u satisfies the constraint? That is, suppose magnitue of u <= u_cons

then simply use u_cons?

Thanks,
by Cheng Cosine
Mar/29/2k5 Ut