Although this could fall under engineering, I thought the Diff Eq forum was the most relevant. Let me know if I should post elsewhere. I have a fairly basic system for which I'm trying to find a minimum-time optimal control policy. I know there are many ways to do this numerically, but since I'm trying to solve it very rapidly, I'm wondering if there is an analytical solution. My system is a double integrator in 3-dimensions with the control inputs being acceleration. This system is affine because there is a constant forcing term (gravity - however this acceleration does not appear in the control accelerations). The control constraints are simply that the norm of the acceleration components our bounded above (i.e. there is a maximum total acceleration due to a maximum thrust of the system). The cost function is just final time. I want to control my system from one state (A) to another (B) in a minimum amount of time while obeying my max acceleration constraint. It looks very similar to a linear-quadratic regulator, which has an analytical solution. However, it is not a LQR because of the affine term and control constraints. For those who do not study controls, this may be too esoteric. However for control engineers, there may be a commonly known answer of which I am unaware. Thanks!