I'm having trouble with the following problem:
The system
$$\dot{x} = -x+4u$$
where $u=u(t)$ is not subject to any constraint, is to be controlled from $x(0)=2$ to $x(1)=1$ in such a way as to minimize
$$\begin{equation*} J = \int_0^1(x+2xu+u^2)\, \mathrm{d}t \end{equation*}$$
Find the optimal control $u^*$ and the optimal path $x^*$.
I've tried using Pontryagin's maximum principle by writing the Hamiltonian and maximizing it with respect to the control $u$, but it doesn't seem to get me anywhere.
Any help would be appreciated. Thanks.
From the system $u = (\dot{x} + x)/4$, you can then replace it $J$ to get
$$ J(x,\dot{x}) = \int_0^1 dt\; \left[ x + \frac{5}{8}x\dot{x} + \frac{1}{16}\dot{x}^2 + \frac{9}{16}x^2\right] = \int_0^1dt\;\mathcal{L}(x,\dot{x}) $$
which can be minimized by solving
\begin{eqnarray*} \frac{d}{dt}\left(\frac{\partial \mathcal{L}}{\partial \dot{x}} \right) - \frac{\partial \mathcal{L}}{\partial x} &=& 0 \\ \Rightarrow\quad \ddot{x} - 9x &=& 8 \end{eqnarray*}
which is a second order-linear ODE in $x$.
Alternatively you can define
$$ v = \frac{\partial \mathcal{L}}{\partial \dot{x}} $$
and solve the system
\begin{eqnarray*} \dot{x} &=& \frac{\partial \mathcal{H}}{\partial v}\\ -\dot{v} &=& \frac{\partial \mathcal{H}}{\partial x}\\ \end{eqnarray*}
which are two first-order linear ODE, with
$$ \mathcal{H}(x, v) = \dot{x} v - \mathcal{L}(x, \dot{x}) $$