Authors: H. Joshi, G. Herrera, S. Lenhart, and M. Neubert, 2008.
The practical examples in this paper are very similar to those in Neubert (2003), with two exceptions: one, they add an advection term, and two, they consider the case where equilibrium has not been reached (i.e. the system changes as a function of time.) The equation that drives the numerical examples is:
$ u_t = ru \left( 1 - \frac{u}{K}\right) - hu + a_{xx} + bu_x $
The authors solve this using forward-backward iteration. For every set of parameters they show, they get a reserve (sometimes multiple reserves), though the shape and position of the reserves varies with the parameters. There are no chattering controls and no singular controls.
The bulk of the paper is devoted to proving the existence of an optimal control. Apparently, one can't just apply Pontryagin naively in the case of PDEs.
To Do:
1. as usual, try to reproduce the graphics. Are these forward-backward solvers standard issue? Can I write a generic interface? How stable are the numerics?
2. I'd like to understand why Pontryagin can't be applied in this case, and to really understand the proofs. The proofs are not particularly well done, alas, so a better bet might be to revisit the paper from which they are drawn, namely Lions, Optimal Control of Systems Governed by Partial Differential Equations (Springer.)
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