I am working on some optimal control problems but am having trouble getting started. The problem I am solving is
$$\dot{x}=\left[ {\begin{array}{cc}0 & 1\\ 1 & 0\end{array} } \right]x+ \left[ {\begin{array}{cc}0 \\ 1\end{array} } \right]u$$ with $x(0)=(1,0)$ and I have to find $u$ between $0$ and $1$ that minimizes $\int_0^1{u^2(t)dt} $ under the constraint $x(1)=(0,0)$.
I tried looking at examples online but they all seem different than mine because of the integral function. Correct me if I am wrong but is the goal to simply find a controller, u, such that it takes x from (1,0) to (0,0) while keeping the integral as small as possible? Is there a particular method used to solve these problems or is it all supposed to be intuitive? Thank you.
Here is the general approach:
The system is $\dot{x} = Ax + Bu$, where $(A,B)$ are cc.
Define $Lu = \int_0^1 e^{A(1-s)} B u(s) ds$. $L$ is linear and continuous. Since $(A,B)$ are cc. we see that $L$ is surjective.
The problem reduces to $\min \{ \|u\|^2 | Lu = b \}$, where $b=x(1)-x(0)$.
Since $L^2[0,1] = \ker L \oplus {\cal R} L^*$, we see that the minimum norm solution must be of the form $u = L^* y$ for some $y \in \mathbb{R}^n$. Hence we need to solve $L L^* y = b$, so get $y = (L L^*)^{-1} b$ and so the minimum norm solution is $u = L^*(L L^*)^{-1} b$.
Note that since $L$ is surjective, $(L L^*)^{-1}$ is invertible.
A little work shows that $(L^*y)(t) = B^T e^{A^T(1-t)} y$.
You can simplify & short circuit the computations by diagonalising the system first.