Optimal control theory of PDEs

Question

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!

Answer 1

Maybe most obviously, there is a whole area of research devoted to numerical analysis of optimal control problems where results like $\|\bar f - f_h\| \in O(h^\alpha)$ as $h \searrow 0$ are of interest. But of course, the functions $f_h$ there are of a particular structure arising from the underlying numerical scheme and this structure is used heavily.

On the other hand, what is used in many papers regarding the numerical approximation, but is also of independent theoretical interest, is the quadratic growth condition.

The basic idea is that a positive definiteness condition of the second derivative $I''(\bar f)$ along the so-called critical cone implies a quadratic growth condition of the type $$\|f - \bar f\|^2 \leq C \bigl(I(f) - I(\bar f)\bigr)$$ for all $f$ close enough to $\bar f$ in the correct norm. The actual conditions are of course more complicated and involved, but this idea sits at the core of it. Such a property is quite useful and there is a lot of research attached to it. I'd recommend to maybe check out the survey paper by Casas and Tröltzsch and start digging from there if that is what you are looking for.