Optimal Strategies in Game-Control Problems of Timing

The paper addresses the issue of optimal investments in innovations. As an example, investments in the construction of gas pipelines are considered. Rational decisions in choosing the commercialization times (stopping times) can be associated with Nash equilibria in a game between the projects. In this game, the total benefits gained during the pipelines’ life periods act as payoffs and commercialization times as strategies. The goal of this paper is to characterize multiequilibria in the game of timing. The case of two players is studied in detail. A key point in the analysis is the observation that all player’s best response commercialization times concentrate at two instants that are fixed in advance. This reduces decisionmaking to choosing between two fixed investment policies, fast and slow, with the prescribed commercialization times. A description of a computational algorithm that finds all the Nash equilibria composed of fast and slow scenarios concludes the paper

Optimal control for proportional economic growth

The research is focused on the question of proportional development in economic growth modeling. A multilevel dynamic optimization model is developed for the construction of balanced proportions for production factors and investments in a situation of changing prices. At the first level, models with production functions of different types are examined within the classical static optimization approach. It is shown that all these models possess the property of proportionality: in the solution of product maximization and cost minimization problems, production factor levels are directly proportional to each other with coefficients of proportionality depending on prices and elasticities of production functions. At the second level, proportional solutions of the first level are transferred to an economic growth model to solve the problem of dynamic optimization for the investments in production factors. Due to proportionality conditions and the homogeneity condition of degree 1 for the macroeconomic production functions, the original nonlinear dynamics is converted to a linear system of differential equations that describe the dynamics of production factors. In the conversion, all peculiarities of the nonlinear model are hidden in a time-dependent scale factor (total factor productivity) of the linear model, which is determined by proportions between prices and elasticities of the production functions. For a control problem with linear dynamics, analytic formulas are obtained for optimal development trajectories within the Pontryagin maximum principle for statements with finite and infinite horizons. It is shown that solutions of these two problems differ crucially from each other: in finite horizon problems the optimal investment strategy inevitably has the zero regime at the final stage, whereas the infinite horizon problem always has a strictly positive solution. A remarkable result of the proposed model consists in constructive analytical solutions for optimal investments in production factors, which depend on the price dynamics and other economic parameters such as elasticities of production functions, total factor productivity, and depreciation factors. This feature serves as a background for the productive fusion of optimization models for investments in production factors in the framework of a multilevel structure and provides a solid basis for constructing optimal trajectories of economic development

Application of i-Smooth Analysis to Differential Games with Delays

In this paper we present application of i-smooth analysis to approach-evasion linear differential game with delay. The main goal is to show that according to the methodology of i-smooth analysis one can realize extremal shift procedure by the finite dimensional component of the system state. © 2018Russian Foundation for Basic Research, RFBR: 17-01-00636The research was supported by the Russian Foundation for Basic Research (project 17-01-00636)

FEEDBACK DESIGN OF DIFFERENTIAL EQUATIONS OF RECONSTRUCTION FOR SECOND-ORDER DISTRIBUTED PARAMETER SYSTEMS

The paper aims at studying a class of second-order partial differential equations subject to uncertainty involving unknown inputs for which no probabilistic information is available. Developing an approach of feedback control with a model, we derive an efficient reconstruction procedure and thereby design differential equations of reconstruction. A characteristic feature of the obtained equations is that their inputs formed by the feedback control principle constructively approximate unknown inputs of the given second-order distributed parameter system

Collective states in social systems with interacting learning agents

We consider a social system of interacting heterogeneous agents with learning abilities, a model close to Random Field Ising Models, where the random field corresponds to the idiosyncratic willingness to pay. Given a fixed price, agents decide repeatedly whether to buy or not a unit of a good, so as to maximize their expected utilities. We show that the equilibrium reached by the system depends on the nature of the information agents use to estimate their expected utilities.Comment: 18 pages, 26 figure

Some Algorithms for the Dynamic Reconstruction of Inputs

For some classes of systems described by ordinary differential equations, a survey of algorithms for the dynamic reconstruction of inputs is presented. The algorithms described in the paper are stable with respect to information noises and computation errors; they are based on methods from the theory of ill-posed problems as well as on appropriate modifications of N. N. Krasovskii's principle of extremal aiming, which is known in the theory of guaranteed control. © 2011 Pleiades Publishing, Ltd.This work was supported by the Russian Foundation for Basic Research (project no. 09-01-00378), by the Program for Fundamental Research of the Presidium of the Russian Academy of Sciences “Mathematical Theory of Control” (project no. 09-P-1-1014), by the Program for State Support of Leading Scientific Schools of the Russian Federation (project no. NSh-65590.2010.1), and by the Ural–Siberian Integration Project no. 09-S-1-1010