Optimal conditions of impulsive processes in application to the problems of economic dynamics

The article presents an applied model of the optimal distribution in time of advertising costs of two goods, which is mathematically formalized as a problem of optimal impulsive control. Bounding on top of investment’s temps in the advertising does not exclude the possibility of aggressive advertising and formally leads to the need to consider the problem in an extended, impulse statement. A mathematical singularity of this problem is the well-posedness of noncompliance condition of the impulsive control transition (the Frobenius-type well-posedness condition). This fact considerably complicates the investigation of the problem and means the following: each impulsive control corresponds not to one trajectory, but to a set of generalized solutions of the problem’s dynamical system from the class of bounded variation functions. The control in this problem of pulse optimization is not only measures, but also a set of limit controls for each moment of the measure jump. They make it possible to single out an individual generalized trajectory and to construct, if it’s necessary, a conventional suboptimal solution. In this article we applied the corresponding maximum principle and the quadratic necessary optimality conditions of particular controls for qualitative analysis of the presented model.