Advertising a new product in a segmented market

We bring some market segmentation concepts into the statement of the "new product introduction" problem with Nerlove-Arrow's linear goodwill dynamics. In fact, only a few papers on dynamic quantitative advertising models deal with market segmentation, although this is a fundamental topic of marketing theory and practice. In this way we obtain some new deterministic optimal control problems solutions and show how such marketing concepts as "targeting" and "segmenting" may find a mathematical representation. We consider two kinds of situations. In the first one, we assume that the advertising process can reach selectively each target group. In the second one, we assume that one advertising channel is available and that it has an effectiveness segment-spectrum, which is distributed over a non-trivial set of segments. We obtain the explicit optimal solutions of the relevant problems.