The article presents a mathematical description of the process of optimal control over an unstable macroeconomic system based on Leontief's input-output model. The optimal equation allows setting a balanced growth rate for a macroeconomic system, which is the main problem in a current development of regional and national economies. Methods of optimal control are generally applicable to the stable systems. This article shows that a developing macroeconomic system is unstable and thus, optimal

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... trol over it has its peculiarities. An unstable macrosystem divides into two subsystems: a stable multidimensional and an unstable dimensional. The stable system is optimized via standard methods, where the growth rate of the entire system is set by a single growing exponent from the second unstable system. In order to divide the system, the author suggests using homothetic transformation; and calculating parameters of optimal control is achieved by solving a Riccati equation. Results of solving a matrix of factors determine the cost of restructuring unstable macroeconomic systems with a balanced growth rate. Knowing the cost of optimal control and restructuring creates prerequisites for a more effective process of managing socio-economic politics in the region and the whole country. These results play a vital role in decision-making processes of management and administrative bodies concerning statistical analyses and managing the economic situation. Results are based on the hypothesis that dynamic models of macroeconomic systems are linear. In practice though actual economic systems are subject to various effects like synergy and self-organization, which cannot be described under the linearity hypothesis. Elaboration upon the problems of optimal control over nonlinear and unstable economic systems is required in future research.