The differential system (1.28) also holds for this H, that is, 
=, Gel 
x dp 
__ i 
DR Sie tae. (1.28) 
oH 
OS 
One example of a constrained control problem is that of a forced 
harmonic oscillator in which the magnitude of the force is limited. In 
this problem, the force is the control and the process is one of chang- 
ing the velocity and displacement of the harmonic oscillator. It be- 
comes an optimal control problem if one is interested in finding the 
force or control which reduces the oscillator from a given velocity and 
displacement to zero velocity and displacement in minimum time. 
The equation of motion for the forced harmonic oscillator with a 
limited force is simply 
where |F| <M, a given constant. Set x = cz/M, T = wt, and u = F/M 
where w = VYc/M. In terms of these nondimensional variables, the non- 
dimensional form of the equation of motion is 
x Pe Sw (3.15) 
where the control function satisfies the inequality |u| < 1. This 
constraint can also be written in the form 
32 
