2nd Order Optimal Control Problem

Question

I'm working on a homework problem in optimal controls and my plant model is described as: $$\ddot{x}(t) = u(t)$$ The performance index (cost function) is described by: $$J = 1/2\int_0^5u^{2}(t)dt\,$$ And the boundary conditions are $$x(0) = 0,$$ $$x(5)=0,$$ $$\dot{x}(0)=2$$ $$\dot{x}(5)=0$$ I'm not sure how to start this problem when I need $\dot{x}(t)$ to start with. Do I just integrate the plant model?

Answer 1

This was a stupid question where I had a brain-fart. What confused me was that description of the system, $\ddot{x}(t)=u(t)$, looked like the state equation.

The correct states for this system would be: $$x_1(t) = x(t)$$ $$x_2(t) = \dot{x}(t) = \dot{x}_1(t)$$

After rewriting in this form, I would be able to proceed with solving for the optimal control solution using the state equation... $$\dot{x}_1(t)=x_2(t)$$ $$\dot{x}_2(t)=u(t)$$