Leontief-type systems and applied problems

The article presents a set of main results obtained in recent years in analytical and numerical studies of various problems for systems of the Leontief type that is a finite-dimensional analogue of Sobolev type equations. The key factor in achieving certain success was the presence of applied problems, the study of each of which was of independent interest. The article presents three mathematical models based on the Leontief type system: a degenerate balance dynamic model of a manufacturing enterprise, a degenerate balance model of the cell cycle, and a mathematical model of a complex measuring device. As a class of problems, we consider the Schowalter–Sidorov initial problem for a Leontief-type system and a number of optimal control problems for it. Numerical methods for solving such problems are briefly outlined, and results on the convergence of approximate solutions to the exact one are shown. Particular attention is paid to the problem of restoring a dynamically distorted input signal from an observed output signal in the presence of noise. The mathematical model of a complex measuring device is constructed as a Leontief-type system, the initial state of which reflects the Showalter–Sidorov condition. The main position of the theory of optimal dynamic measurements is modelling the desired input signal as a solution to the problem of optimal control with minimization of the penalty functional, in which the discrepancy between the simulated and observed output (or observed) signal is estimated. The presence of noise at the output of the measuring device leads to the need to use digital filters in numerical algorithms. The article is of a review nature and provides an analysis of the development of research into Leontief-type systems.