Dual approach for the solution of an advertising control problem

We consider a monopolistic firm that sells seasonal goods. The firm seeks the minimum of the total advertising expenditure during the selling period, given that some previously defined levels of goodwill and sales have to be reached at the end of the period. The only control allowed is on advertising while good-will and sales levels are considered as state variables. More precisely we consider a linear optimal control problem for which the general position condition does not hold so that the application of Pontryagin's maximum principle may not be useful to determine a solution. Therefore, the dual of the problem is studied and solved. Moreover, a necessary and sufficient condition for the feasibility of the primal problem is determined.