## Controllability

In previous chapters we have been concerned with the analysis of linear and nonlinear dynamics. We have developed closed formulas for predicting future states and outputs, and in addition, stability problems have been discussed. On the other hand, in control theory the control problem is to find an input that causes the state or the output to behave in a desired way. As an example, consider again the satellite problem discussed earlier in Example 3.6. Let x
_{1} be a desired future state of the satellite, that is, its desired positions, radial and angular velocities. Find an input function that will drive the state to x
_{1} in a finite time. A more restrictive problem is when an entire trajectory x(t) is given, and we wish to find an input function such that the entire state trajectory coincides with x(t).