Optimal Control: How to find the cost function from the given dynamics?

Question

Consider the growth equation: $x'(t)=tu(t)$ , with $x(0)=0$ and $x(1)=1$, and with the cost function: $J= \int_0^1 u^2(t) dt$.

I'm wondering how to find the expression $ u^2(t)$ that is inside of the integral?

Answer 1

The cost function $J$ is given with the problem, it is the objective function you are trying to minimize. The constraint on $u$ is that the BVP given has a solution. Integrating, we get that $x(t) = x(0) + \int_0^t su(s)ds = \int_0^t su(s)ds$, so $x(1) = \int_0^1 tu(t)dt = 1$. This restricts the set of functions $u(t)$ that we have to choose from to minimize the objective $J$. In the unconstrained case, it's clear that the minimum of $J$ is $0$, achieved when $u(t)\equiv 0$. However, since this doesn't satisfy the constraint ($\int_0^1 t\cdot 0 dt = 0 \neq 1$), it is not a solution.