I want to find the optimal control $u^*(t)$ for the following problem: \begin{align*} \dot{x}_1(t) &= x_2(t), & x_1(0) &= 3 \\ \dot{x}_2(t) &= -2x_1(t) + 5u(t), & x_2(0) &= 5 \end{align*} which minimizes: \begin{equation} J = \frac{1}{2} \int_{0}^{2} [x_1^2(t) + u^2(t)] \, dt \end{equation}
Step 1: Define the Hamiltonian The Hamiltonian $H$ for this problem is given by:
$H = \frac{1}{2}(x_1^2 + u^2) + p_1 x_2 + p_2(-2x_1 + 5u)$
Step 2: Apply the Pontryagin's Minimum Principle The necessary conditions for optimality are derived from the Hamiltonian:
$\dot{x}_1 = \frac{\partial H}{\partial p_1} = x_2$
$\dot{x}_2 = \frac{\partial H}{\partial p_2} = -2x_1 + 5u$
$\dot{p}_1 = -\frac{\partial H}{\partial x_1} = -x_1 + 2p_2$
$\dot{p}_2 = -\frac{\partial H}{\partial x_2} = -p_1$
$\frac{\partial H}{\partial u} = 0 \Rightarrow u + 5p_2 = 0 \Rightarrow u = -5p_2$
$p_1(2) = 0$
$p_2(2) = 0$
I'm supposed to solve this systems of equations by hand. Can you suggest any method to solve this sytem by hand? Or is there any other approach which is simple for hand calculation?