Optimisation theory is a wide subject that was introduced in this chapter as a way toward optimal control theory. Use the given introduction to establish the minimum of

\[f(\mathbf{x})=x_{1}\]

subject to the conditions \[
\begin{aligned}
& \left(x_{1}-1ight)^{2}+x_{2}^{2}=1 \\
& \left(x_{1}+1ight)^{2}+x_{2}^{2}=1 \end{aligned}
\]
where \[
\mathbf{x}=\left(\begin{array}{ll}
x_{1} & x_{2}
\end{array}ight)^{T}
\]