Okay, so if there is an equilibrium at some point in time, does it really matter from what direction I get there? If I run time backwards, starting from the "ending points" , will the dynamics of my problem still be the same? I mean, will the signs of the partials still hold? For example, in my water pumping problem, costs increase as more water is pumped. If I run time backwards, I am still saying costs increase as water is pumped. The only difference is that in one case pumping decreases over time (to the equilibrium), whereas the other case pumping increases over time to a disequilibrium. As it stands now, when I run the problem forward in time pumping increases to infinity. This is completely unrealistic. However, if I put the negative sign, it goes to a nice equilibrium. I am trying to figure out if this makes any sense!