When Advertising Is Infinite Dimensional
Elisa Tacconi (Department of Finance) and Salvatore Federico (Università degli Studi di Milano), in the paper Dynamic Programming for Optimal Control Problems with Delays in the Control Variable (in SIAM, Journal on Control and Optimization, Volume 52, Issue 2, pp. 1203-1236, doi: 10.1137/110840649), study a class of optimal control problems with state constraints, where the state equation is an ordinary differential equation with delays in the control variable.
Optimal control problems with delays in the state and/or control variable have many applications in the field of economics and finance. One of them is to the optimal advertising problem. In their seminal work, Nerlove and Arrow (1962) dealt with the simplest form of this problem (without delay). Good reputation of the products - the so called goodwill - represents the state variable of the problem and the expenditure rate in advertising represents the control variable. By investing into advertisement the firm wants to maximize its profits, and the problem is to find the optimal rate of investment. Pauwels (1977) introduced a delay term in the control variable (the investment rate) and Feichtinger, Hartl and Sethi (1994) underlined the importance of this feature. The main assumption is that consumers have a memory of the past advertising which continues to influence their choices.
From a methodological perspective, this kind of problems presents several difficulties since the dynamical system has basically an infinite dimensional nature. Indeed, a general method to tackle these problems consists in rewriting them in infinite dimensional spaces in order to absorb the delay in the new infinite-dimensional state variable and then to study the infinite dimensional Hamilton-Jacobi-Bellman (HJB) equation associated to the optimal control problem in this space.
The contribution of the paper is the proof of the directional regularity for the solutions of the HJB equation and the construction of the optimal feedback controls of the problem by means of this regularity result.