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The optimal control problem can be formulated in the phase plane. 
If (x, y) are the phase plane coordinates, the equation of motion (3.15) 
takes the form 
(Gree) 
Starting the oscillator at a given displacement with a given velocity is 
equivalent to assigning a given point (x, y) = (a, b) in the phase plane 
as an initial condition for (3.17). The rest state of the oscillator is 
represented in the phase plane by the point (0, 0), the point of zero 
displacement and velocity. Hence, the optimal time control problem is 
one of finding a control u which minimizes the time between states 
(a, b) and (0, 0). In this problem, the cost is given by 
T 
C@) +P | dt (3.18) 
The cost function F(T, x, u) = l. 
Set p = (p, q). Then the Hamiltonian defined bye @rel)) ees 
H =- 1+ py + q(u - x) - A@u- 1) Gu t+ 1) (3.19) 
and, moreover, (1.28) takes the form 
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