I am trying to find the optimal control of the following problem. We have a material point $x(t)\in\mathbb{R}^2$ with mass $m=1$. It can accelerate in any direction with maximum acceleration of $1$, it can also change the acceleration direction instantaneously. The acceleration $\ddot x$ is also the control $u(t)\in\mathbb{R}^2$, so we have $|u|\leq1$. We want to drive the point $x$ into the origin $(0,0)$ in least time. The system also has the additional constraints $x(0) = x_0$, $\dot x(0) = v_0$ and $\dot x(T) = (0,0)$, where $T$ is the moment when $x(t)$ reaches $(0,0)$ for the first time.
I am looking for an analytic solution. I have stumbled upon the same problem, but in 1D. The methods described there does not apply to the 2D case.
I have not proved it, but I am convinced that the solution curve would lie on the unit sphere, so we would have $|u(t)|=1$. Other than that I am stuck. I can not even define a proper minimization function. It also seams that $T=T(x, \dot x, u)$.
Can you give me any pointers or direct me to a reading that solves a similar problem?
Your system is nonlinear, so you can not apply LQR directly. Nevertheless your problem satisfies all the requirements for using LQR excepting for the saturation at the input. I will suggest you to take a look into the next document
An LQR/LQG Theory for Systems with Saturating Actuators