7. Conclusions
In this paper we have considered an optimal control problem for a nonlinear system of ordinary differential equations that describes the evolution of the number of regular customers and referral customers in some firm. The aim is to study, considering several types of behaviour for the population, the best marketing strategy in the decision to invest in referrals programs. The existence and uniqueness of optimal solutions was established for an L2 cost functional model. Some simulation results of such model were presented and compared with the ones obtained for the model with an L1 cost functional. The optimal solutions for the problem with linear lagrangian are of bang-bang type. While performing the numerical simulations, we have noticed that, for some values of the cost weights κ1, κ2 and κ3, the solutions for the quadratic objective model are slightly better than the ones for the linear objective model. Nevertheless, the strategy obtained for the linear objective model is easier to implement, since at each time interval the possible actions are taken from a finite set of possibilities, and thus may be more appealing to the marketing managers. For the autonomous case of quadratic cost functional model, we have shown the effectiveness of the optimal control strategy over the constant control strategy, a heuristic control strategy and the no control.
