Adjoint equation


i(t) =
@H @= f x(t);u(t); i = 1;:::n (4.9)
@xi @x k(i
 Minimum condition
H((x(t);u(t);0;) = min [ H(x(t);u;0;)] = const (4.10)
u2

 Transversality condition: T0 and T1 are tangent spaces to X0 and X1 at x(t0) and
x(t1); respectively then (t) can be selected to satisfy the following transversality
conditions
(t0) ? T0 and (t1) ? T1: (4.11)
Remark 5 In the case of X0 = fx0g and X1 = fx : k(x) = 0; k = 1;2;:::;lg; where k are
di¤erentiable functions (k = 1;2;:::l) the previous condition (4.11) get the following forms:
(t0) is arbitrary (4.12)
lX
(t1) = kr k(x(t1)); where k 2R: (4.13)
k=1
4.3 Control problems with terminal payo¤
In this paper, we will analyze an optimal control problem when the cost takes the following
form t
1Z
C(u) = g(x(t1)) + [f0(x(t);u(t))]dt (4.14)
t0
where g : Rn ! R is a continuously di¤erentiable function that represents some terminal
payo¤at the …nal time t1. In order to convert this problem into a standard optimal control
problem we add an extra variable xn+1(t) such that:
19
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