Groundwater Remediation and Control Theory
The main question I have asked in this research area has been how to
solve optimal control problem for groundwater remediation/clean-up
when there is a random component present. The approach taken is that
of including an explicit stochastic process in the equations
governing the flow of water and contaminants. The equations also take
into account porous media flow, so Darcy's Law applies. Published
work thus far has focussed on analysis of discrete-time
systems, using several variants on dynamic programming.
In the case of organic contaminants, realistic numerical models track
30-40 compounds and take into account additional equations. There are
still some serious limitations in the numerics for the control
problem, even without a stochastic term present. Much of the current
mathematical literature in this area focuses on numerical techniques, as
well as matching of the different scales inherent to the physical
problem. However, analytical techniques can be useful to simplify
the numerics, or for limited cases give a generalized closed form
solution that would be optimal or nearly so.