A specific type of optimal control problem

Question

I recently encountered the following optimal control problem

\begin{alignat}{3} \min_u &\quad& J = 50(x(2)-0.5)+\int_{0}^{2} u^2(t)dt && & \\ \text{s.t: } &\quad& \frac{dx}{dt} = u, \quad x(0)=a, \quad |u|\leq1 \end{alignat}

I am a novice in optimal control and optimization, in general. I cannot figure to how to solve this, so any help would be appreciated. How does one approach such a control problem with the inequality constraint and the integral? Also $ x(2) $ seems to complicate things. Is there an analytic solution? I thank all helpers.

Answer 1

Integrating $\dot x = u$ with initial condition $x(0) = a$,

$$ x (2) = a + \int_{0}^2 u (t) \, {\rm d} t $$

and, thus, the cost function is

$$ J (u) = (50 a - 25) + \int_{0}^2 \left( u^2 (t) + 50 \, u (t) \right) {\rm d} t $$

Note that

$$\arg\min_{-1 \leq u \leq 1} \left( u^2 + 50 \, u \right) = -1$$

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optimization control-theory optimal-control