Abstract
Controlled dynamic systems with an entropy operator (DSEO) are considered. Mathematical models of such systems were used to study the dynamic properties in demo-economic systems, the spatiotemporal evolution of traffic flows, recurrent procedures for restoring images from projections, etc. Three problems of the study of DSEO are considered: the existence and uniqueness of singular points and the influence of control on them; stability in “large” of the singular points; and optimization of program control with linear feedback. The theorems of existence, uniqueness, and localization of singular points are proved using the properties of equations with monotone operators and the method of linear majorants of the entropy operator. The theorem on asymptotic stability of the DSEO in “large” is proven using differential inequalities. Methods for the synthesis of quasi-optimal program control and linear feedback control with integral quadratic quality functional, and ensuring the existence of a nonzero equilibrium, were developed. A recursive method for solving the integral equations of the DSEO using the multidimensional functional power series and the multidimensional Laplace transform was developed. The problem of managing regional foreign direct investment is considered, the distribution of flows is modeled by the corresponding DSEO. It is shown that linear feedback control is a more effective tool than program control.
Highlights
Dynamic systems with an entropy operator (DSEOs) form a class of nonlinear systems in which the nonlinearity is described by a perturbed mathematical programming problem with an entropy objective function
The mathematical model of the entropy operator reduces to a perturbed mathematical programming problem, which significantly complicates the study of the operator itself and the corresponding dynamic system
This paper considers the dynamic properties of controlled positive dynamic systems with an entropy operator (DSEO) with the constrained argmax entropy function
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Constraints can be specified as a system of equalities or inequalities In the latter case, the mathematical model of the entropy operator reduces to a perturbed mathematical programming problem, which significantly complicates the study of the operator itself and the corresponding dynamic system. This paper considers the dynamic properties of controlled positive DSEOs with the constrained argmax entropy function. Control in such systems affects the entropy operator. Optimal control design problems in the two classes mentioned are considered using an integral quadratic criterion and constraints associated with the existence, uniqueness, and stability of the equilibrium of positive DSEOs. The developed methods are applied to study the qualitative properties and optimize a stochastic regional exchange system for foreign direct investment
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