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I want to solve a nonlinear optimal control problem which has constraints at specific times. In continuous time setting, the constraint is as follows

$\int_{a}^{b}f(x,t)dt = \eta$ which I approximate using techniques such as Simpson's rule. This becomes:

$\frac{b-a}{6}(f(x)(t=0)+f(x)(t=a+b/2)+f(x)(t=b)) = \eta$

where $\eta$ is some constant, and f is a differentiable function in continuous time and evaluated at $a, a/2$ and $b$.

Upon discretization, I convert this into a discrete problem with a sampling time $k$. Do I need to discretize f(x) ( using matrix exponentials) and evaluate at discrete times as we do for the state equations?

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