The existence of the solution to the control problem of advertising expenses with distributed lag

The dynamic optimal control problem of promotion expenses is analyzed in the paper. The model takes into account response lags of consumers, the response originates from advertising influence and previous purchases. The considered model describes accumulated advertising effect in a time period of advertising impact and accumulated effect of previous purchases in time period of impact of previous purchases. It makes difference to classical optimization dynamic models (Nerlove–Arrow, Vidale–Wolfe etc.) which analyze advertising impact. The optimization problem is formulated as a system of nonlinear integral equations of Volterra type and integral criterion functional. These equations are a special case which is produced by boundaries of the initial problem. The theorem of existence of solution of these equations is proved. Also the theorem of existence of solution of the problem, which maximizes a total company profit for the planning period under restrictions, is examined. These restrictions include boundaries of advertising budget and the functional dependence which demonstrates a reaction of a target audience. The properties of a demand function, which depends on the accumulated advertising and the accumulated goodwill, are considered there. The problem of practical applying of this model is discussed.