I would like to ask the general method / principles that is followed for solving problems like the following:
Let $\dot{x} = u$, $x \in \mathbb{R}$, $0 \le u \le 1$, $x(0)=x_0$. Minimise $$J(u) = \frac{1}{10} x^2(T) + \frac{1}{2} \int_0^T (-x^2(t) + u^2(t)) \textrm{d} t$$
Most of the textbooks on Optimal Control focus mostly on the proofs of the underlying theorems. Any suggestions on books that deal with practical problems like the above, preferably with solved examples, are more than welcome. Thank you very much.
Since $J$ is a functional (i.e. a map from a function space into the reals), the standard method of finding a minimizer is called calculus of variations. The idea is that, like in calculus, a minimum can be found by looking at where the derivative vanishes; anytime we change the function $x$ by a small amount, $J$ should increase compared to its value on $x$.
More concretely, let $\eta(t)$ be any continuously differentiable function with $\eta(0) = \eta(T) = 0$. Using $\eta$, we can deviate $x$ by considering the function $x + \epsilon\eta$, where $|\epsilon|$ is assumed to be small. Then $L(x + \epsilon\eta)$ has a minimum at at $\epsilon = 0$, so $\left.\frac{d}{d\epsilon}\right|_{\epsilon=0}L(x + \epsilon\eta) = 0$ where $$L(x + \epsilon\eta) = \frac{1}{10}(x + \epsilon\eta)^2(T) + \frac{1}{2}\int_0^T(-(x + \epsilon\eta)^2 + (\dot{x} + \epsilon\dot{\eta})^2)dt. $$ The first term reduces to $\frac{1}{10}x^2$ since $\eta(T) = 0$, and so $$ \frac{d}{d\epsilon}L(x + \epsilon\eta) = \frac{1}{2}\int_0^T(-2\eta(x + \epsilon\eta) + 2\dot{\eta}(\dot{x} + \epsilon\dot{\eta}))dt $$ $$\implies 0 = \left.\frac{d}{d\epsilon}\right|_{\epsilon=0}L(x + \epsilon\eta) = -\int_0^T\eta x + \int_0^T\dot{\eta}\dot{x}.$$ Using integration by parts and the fact that $\eta(0) = \eta(T) = 0$, we have $$\int_0^T\dot{\eta}\dot{x} = \left.\dot{x}\eta\right|_0^T - \int_0^T\eta\ddot{x} dt = -\int_0^T\eta\ddot{x} dt$$ $$ \implies \int_0^T(\ddot{x} + x)\eta dt$$ for $any$ function $\eta$; the only way this can be true is if $x'' + x = 0$ on the interval $[0,T]$. This is a differential equation that can be solved by $x = ae^t + be^{-t}$ for any constants $a, b$. We can then use the initial conditions to determine the proper values of $a$ and $b$.