Control of a Water Tank System with Value Function Approximation

We consider a system of two identical rectangular shaped water tanks. A source of constant water inflow is available, which may only be directed to one tank at a time. The objective is to find a control policy to maximize the final sum of the water levels at some terminal time T, subject to minimum water level constraints on each tank. Water exits each tank corresponding to Toricelli's law (i.e., the velocity depends on the current water level). We derive a closed form dynamic programming solution in discrete time to this problem without the water-level threshold constraints. Subsequently, we implement the value iteration algorithm on a set of support points to find a control policy with the threshold constraints, where a random forest regressor is iteratively used to update the value function. Our results show consistency between the dynamic programming solution and the value iteration solution.