Toolkit for optimal management of unstable complex system

The article presents a mathematical description of the process of an optimal control over an unstable macroeconomic system, which is a complex model, based on the Leontiefs input-output model. The optimal equation allows setting a balanced growth rate for a macroeconomic system. It is the main problem in the current development of regional and national economies. The methods of an optimal control are generally applicable to stable systems. This article shows that a developing macroeconomic system is unstable and therefore an optimal control over it has its peculiarities. An unstable macrosystem is divided into two subsystems: a stable multidimensional and an unstable one-dimensional. The stable system is optimized via standard methods, where a single growing exponent sets the growth rate of the entire system from the second unstable system. In order to divide the system, the author suggests using a homothetic transformation. To calculate the parameters of an optimal control a Riccati equation is used. The results of solving a matrix of factors determine the cost of restructuring unstable macroeconomic systems with a balanced growth rate. The knowledge of the cost of an optimal control and restructuring creates prerequisites for a more effective process to manage 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.