How do I approach this problem? I am unsure of the conditions that need to be met in order to apply the Principle of Optimality.
\begin{equation*} \underset{u}{\text{min}}\frac{1}{2}\int_{t_0}^{t_1} 3x(t)^2 + u(t)^2 \, dt \end{equation*} \begin{align*} \text{subject to}\quad &x'(t) = x(t) + u(t) \text{ and } x(t_0) = x_0 \end{align*} For what values of $t_0, t_1$ and $x_0$ can we apply the Principle of Optimality to solve the problem?
Thanks ahead.
--- Additional Information ---
This is how the theorem was defined in the textbook:
theorem
This is the example the question refers to: example
