Penalization of Singular Control Problems that Arise in Math Biology

Optimal control problems applied to biological models tend to incorporate an objective function that increases quadratically with respect to the control. However, the principle of parsimony would lead one to assume linear dependence on the control instead. Problems of this form may have a solution that contains a singular subarc, which can be difficult to solve both analytically and numerically. We are interested in solving these types of problems entirely numerically without the computation of the derivatives of the switching function. In this case the numerical solution often resembles a phenomenon called“chattering”. In this talk, we will discuss a method of penalizing such problems by adding a bounded variation term to the objective function. We will then look at the effectiveness of this penalization by applying it to a simple fishery problem where the numerical solution may have instances of chattering occurring for varying habitat sizes.