This is a somewhat openly phrased question because I am not quite sure what has been done in that direction.

Imagine one has two evolution equations

$$\partial_t u = p(x,\partial_x,f)u$$

$$\partial_t u = \widetilde{p}(x,\partial_x,\widetilde{f})u$$

depending on spatial coordinates $x \in \Omega$.

In optimal control theory one usually studies the problem of finding a control $f$ such that $u$ solves the PDE and $u$ and $f$ minimize a certain functional $I(f,u)$ that has some nice properties.

If this functional satisfies certain properties then basically by simple weak$^*$ compactness one can show the existence of a optimal control solution.

The problem however is that it is usually very hard to access the optimzer explicitly. Usually one only gets some optimality condition that the extremizer $f$ has to satisfy.

I would therefore like to ask:

Given two PDEs, the actual one and a perturbed one as above and the same functional $I$.

Are there any results that estimate the error for the optimizer

$$\left\lVert f-\widetilde{f} \right\rVert$$

in some norm? I am just looking for any results in that direction to get some inspiration of what might be possible.

Any remarks, comments or vague ideas are very welcome!