In this problem you will extend your knowledge of Hamiltonian systems, general nonlinear systems theory, phase plane analysis, and Euler's method in order to model the following scenario governed by optimal control theory: Suppose you are a lifelong sher who owns a selfsustaining pond of sh. Rather than harvest a xed amount of sh for all time, you decide to variably control how you harvest your sh in order to maximize the utility you obtain from selling the sh at market. One of the principal concepts of economics is the phenomenon of diminishing marginal utility: the more sh you harvest the less meaningful each subsequent one is. We can model this phenomenon with the concave downward function , where 1105) represents the function controlling your sh harvesting. Dene the payoff J = f: dt. In the following steps you will derive the optimal strategy to harvest your sh in order to maximize J, the utility you accumulate over time. write an equation for g. This, along with %, should yield an autonomous system of nonlinear differential equations