One of the important and urgent tasks of microeconomics is the problems of research of the economic system, in which there are restrictions associated with the planned volume of output or the size of the enterprise production capacity. These constraints are set by the requirement that the analyzed trajectories do not leave some given region of the control existence space. Most often, such restrictions for all time points are set in the form of inequalities, and certain requirements are imposed on the function of the phase coordinates of the object, their value at a given time. This problem is classified as an optimal control problem with mixed and phase constraints. In general, this area is of scientific interest and requires consideration. In this case, we study the microeconomic model of the household economy as the most stable object in the conditions of crises. The accumulated savings are subject to a natural phase constraint of non-negativity. This led to the study of the features of the microeconomic formulation of the problem of finding a method for the optimal division of material resources into consumed and accumulated parts, since the imposition of a natural phase restriction on the non-negativity of accumulated savings makes everything much more complicated. Just as in macroeconomics, consumption is optimized, but not in its pure form, but the integral discounted utility of consumption is maximized. The relation equation in this paper differs from a similar macroeconomic equation, since the household exists and survives in crisis conditions in a different way than do social organisms and large enterprises. That is why the article formulates and proves sufficient conditions for solving the problem with a phase constraint.
No Infimum Gap and Normality in Optimal Impulsive Control Under State Constraints