On the control of a dynamical system defined by a decreasing one-dimensional set-valued function

Agricultural production can be described by discrete time as there is harvest in every year only once. The agricultural production is uncertain because of the weather and the ever changing technology. At the same time, the sector prefers stability which is reflected in the small changes in the prices. The uncertainty of the price may be modeled by a set-valued function in a single product market. The independent variable is the price expectation of the producer which is the future value of the price estimated by the producer. It can be assumed that the set-valued function is decreasing because in the case of higher price expectation, greater quantity appears on the market and thus the real market price becomes the lower. The stability of the market may require some control. In this paper the existence of an appropriate control to reach a target interval and to keep the trajectory in the interval is investigated from mathematical point of view. Necessary and sufficient conditions are given for the existence of the viable solution. The “striped structure” of the dynamical system is explored as well.