The objective has the following form:
t1Z
C(u) = [f0(x(t);u(t))]dt: (4.5)
t0
where f0 is continuously di�erentiable in Rn
: The goal is to minimize C(u) over all
admissible controls u 2
:
4.2 Maximum Principle and transversality conditions
We now de�ne a function H called Hamiltonian which is a function of (x;u;) 2Rn Rm
Rn and;0 2R;
H(x;u;0;) = 0f0(x;u) + 1f1(x;u) + 2f2(x;u) + ::: + nfn(x;u) (4.6)
H(x;u;0;) = 0f0(x;u) +f(x;u): (4.7)
We can then state the Maximum Principle theorem by Pontryagin (see [2]).
Theorem 4 (Pontryagin Maximum Principle) Assuming u is an optimal control for the
above problem and x is the corresponding trajectory, there exists a function
(t) = (1(t);:::;n(t))
and 0  0 satisfying:
 State equation

 @Hx 
i(t) = = f (t)); i = 1;:::;n (4.8)@ i(x
(t);u
i
18
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