I have managed to find a couple of papers which deal with the optimal control of systems governed by integral equations of the second kind (e.g. here: https://hal.inria.fr/inria-00473952/file/RR-7257.pdf ), however I am interested in the case of a system governed by a Volterra integral equation of the first kind, specifically:
$\frac{1}{\sqrt{t}} e^{-\frac{\bar{x}(t)^2}{4 t}} = \int_0^t \frac{j(t')}{\sqrt{t-t'}} e^{-\frac{ \left[\bar{x}(t)-\bar{x}(t') \right]^2}{4 (t-t')}} dt'$
where here $\bar{x}(t)$ is the control function. The cost functional would most likely be something like:
$$\int_0^T (j(t) + \alpha \dot{\bar{x}}(t)^2) dt$$
Does anyone know if there is any literature out there for solving this kind of problem?
I am obtaining the following approximated solution for small $t$
where $A$ is constant which must be determined using initial conditions.