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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?

sgyyhzd
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  • What kind of relation do you ave in mind? I suggest that you first consider some simple control system s.t. the optimal control can be computed explicitly and try to figure out how this works. – Dmitry Jun 12 '18 at 08:32
  • I want to know whether the cost linearly(exponentially) increase with the payoff. Assuming the system is $dx/dt=u(t)*x(t)$, what methods can I use to solve it? – sgyyhzd Jun 12 '18 at 10:16

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