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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?

lord voldemort
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  • This is a less generally state problem for PMP applied to an LTI system. A slightly more generally state question is stated here and my answer to that question will likely also answer your question. – Kwin van der Veen Mar 30 '24 at 17:54
  • @KwinvanderVeen I have a LTI system with a dimension of four. I know the general solution for the systems will be $e^{At}z(0)$ but the problem is that it is not easy to find the matrix exponential by hand. – lord voldemort Mar 30 '24 at 18:00
  • The resulting 4 by 4 matrix contains quite a few zeros. Using this it should be quite doable by hand to find the eigen decomposition and using that the matrix exponential. – Kwin van der Veen Mar 30 '24 at 21:22
  • Also note that it is a Hamiltonian matrix, so if $\lambda$ is an eigenvalue of that matrix then so will be $-\lambda$, $\bar{\lambda}$ and $-\bar{\lambda}$ (though note that $\lambda=\bar{\lambda}$ when $\lambda\in\mathbb{R}$). – Kwin van der Veen Mar 30 '24 at 21:33

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