Optimal control of the customer dynamics based on marketing policy

We consider an optimal control problem for a non-autonomous model of ODEs that describes the evolution of the number of customers in some firm. Namely we study the best marketing strategy. Considering a $L^2$ cost functional, we establish the existence and uniqueness of optimal solutions, using an inductive argument to obtain uniqueness on the whole interval from local uniqueness. We also present some simulation results, based on our model, and compare them with results we obtain for an $L^1$ cost functional. For the $L^1$ cost functional the optimal solutions are of bang-bang type and thus easier to implement, because at every moment possible actions are chosen from a finite set of possibilities. For the autonomous case of $L^2$ problem, we show the effectiveness of the optimal control strategy against other formulations of the problem with simpler controls.
Comment: This is a preprint of a paper whose final and definite form is with Appl. Math. Comput. [https://www.journals.elsevier.com/applied-mathematics-and-computation]