I have an optimal control problem to maximize the function $J =\int_0^1 x(t)-\alpha u^2(t) dt$ subjects to the system $dx(t)/dt = f(x(t),u(t),t)$ and the initial/final states.
The system $dx(t)/dt$ is an affine control nonlinear system and assume that the optimal control exists. Now I want to investigate the rough scale or relation between the cost $\int_0^1 u^2(t) dt$ and the payoff $\int_0^1 x(t) dt$ with respect to the optimal control u(t).
What methods/literature/keywords can I apply to solve this problem analytically?