The aim of this section is to give basis to solve analytically or numerically optimal control problems.
In full generality, we consider a system governed by the dynamic:
with $\Y \in R^n$ is the state variable and $\U \in R^m$
The control problems is:
Given $T > 0$, $\Y(0) = \Y_0$ and $\Y(T) = \Y_T$ does it exists $\U : [0,T] \rightarrow R^m$ such that systems steers $\Y_0$ to $\Y_T$ in time $T$.
For optimal control problem, we consider a cost function: