A.M. Tarasyev

Optimal dynamics of innovation in models of economic growth

A nonlinear model of economic growth which involves production, technology stock, and their rates as the main variables is considered. Two trends (growth and decline) in the interaction between the production and R&D investment are examined in the balanced dynamics. The optimal control problem of R&D investment is studied for the balanced dynamics and the utility function with the discounted consumption. The Pontryagin optimality principle is applied for designing the optimal nonlinear dynamics. An existence and uniqueness result is proved for an equilibrium of the saddle type and the convergence property of the optimal trajectories is shown. Quasioptimal feedbacks of the rational type for balancing the dynamical system are proposed. The growth properties of the production rate, R&D, and technology intensities are examined on the generated trajectories.

Multiequilibrium Game of Timing and Competition of Gas Pipeline Projects

The paper addresses the issue of the optimal investments in innovations with strong long-term aftereffects. As an example, investments in the construction of gas pipelines are considered. The most sensitive part of an investment project is the choice of the commercialization time (stopping time), i.e., the time of finalizing the construction of the pipeline. If several projects compete on the market, the choices of the commercialization times determine the future structure of the market and thus become especially important. Rational decisions in this respect 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 the 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, for all players, the 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 simple algorithm that finds all the Nash equilibria composed of fast and slow scenarios concludes the paper.

Control synthesis in grid schemesfor Hamilton‐Jacobi equations

Grid approximation schemes for constructing value functions and optimal feedbacks inproblems of guaranteed control are proposed. Value functions in optimal control problemsare usually nondifferentiable and corresponding feedbacks have a discontinuous switchingcharacter. Constructions of generalized gradients for local (convex, concave, linear) hullsare adapted to finite difference operators which approximate value functions. Optimal feedbacksare synthesized by extremal shift in the direction of generalized gradients. Bothproblems of constructing the value function and control synthesis are solved simultaneouslyin the single grid scheme. The interpolation problem is analyzed for grid values of optimalfeedbacks. Questions of correlation between spatial and temporal meshes are examined.The significance of quasiconvex properties is clarified for linear dependence of space‐timegrids.The proposed grid schemes for solving optimal guaranteed control problems can beapplied to models arising in mechanics, mathematical economics, differential and evolutionarygames.

On a computational algorithm for solving game control problems

Abstract A positional differential game of approach to a target is considered. The construction of the set of positional absorption (SPA) is studied. Relations are given, on the basis of which an algorithm of approximate computation of the SPA for controlled systems in the plane is developed.

Properties of Hamiltonian Systems in the Pontryagin Maximum Principle for Economic Growth Problems

We consider an optimal control problem with a functional defined by an improper integral. We study the concavity properties of the maximized Hamiltonian and analyze the Hamiltonian systems in the Pontryagin maximum principle. On the basis of this analysis, we propose an algorithm for constructing an optimal trajectory by gluing the dynamics of the Hamiltonian systems. The algorithm is illustrated by calculating an optimal economic growth trajectory for macroeconomic data.

Conjugation of Hamiltonian Systems in Optimal Control Problems

Abstract The optimal control problem with a functional given by an improper integral is considered for models of economic growth. Properties of concavity of the maximized Hamiltonian are examined and analysis of Hamiltonian systems in the Pontryagin maximum principle is implemented including estimation of steady states and conjugation of domains with different Hamiltonian dynamics. On the basis of this analysis an algorithm is proposed for construction of optimal trajectories by sewing dynamics of Hamiltonian systems. The proposed algorithm is illustrated by computer simulations of optimal trajectories in models of economic growth for real macroeconomic data.

Green Growth and Sustainable Development

The book examines problems associated with green growth and sustainable development on the basis of recent contributions in economics, natural sciences and applied mathematics, especially optimal control theory. Its main topics include pollution, biodiversity, exhaustible resources and climate change. The integrating framework of the book is dynamic systems theory which offers a common basis for multidisciplinatory research and mathematical tools for solving complicated models, leading to new insights in environmental issues.

Approximation schemes for constructing minimax solutions of Hamilton-Jacobi equations☆

Abstract A grid algorithm is proposed for constructing the optimal guaranteed result function (which need not be differentiable) in control problems. Wherever it is differentiable, this function satisfies the Isaacs-Bellman equation, which is a first-order partial differential equation of Hamilton-Jacobi type. A convergent finite-difference method is proposed for Hamilton-Jacobi equations. Unlike the classical grid method, in which one approximates the gradients of the unknown function, which need not necessarily exist, this method requires the computation of subdifferentials of locally convex hulls. Underlying the method is the concept of a generalized minimax (viscosity) solution [1–4] of the Hamilton-Jacobi equation, with the corresponding infinitesimal constructions—directional differentials and subdifferentials—replacing the classical derivative.

Construction of a regulator for the Hamiltonian system in a two-sector economic growth model

We consider an optimal control problem of investment in the capital stock of a country and in the labor efficiency. We start from a model constructed within the classical approaches of economic growth theory and based on three production factors: capital stock, human capital, and useful work. It is assumed that the levels of investment in the capital stock and human capital are endogenous control parameters of the model, while the useful work is an exogenous parameter subject to logistic-type dynamics. The gross domestic product (GDP) of a country is described by a Cobb-Douglas production function. As a utility function, we take the integral consumption index discounted on an infinite time interval. To solve the resulting optimal control problem, we apply dynamic programming methods. We study optimal control regimes and examine the existence of an equilibrium state in each regime. On the boundaries between domains of different control regimes, we check the smoothness and strict concavity of the maximized Hamiltonian. Special focus is placed on a regime of variable control actions. The novelty of the solution proposed consists in constructing a nonlinear stabilizer based on the feedback principle. The properties of the stabilizer allow one to find an approximate solution to the original problem in the neighborhood of an equilibrium state. Solving numerically the stabilized Hamiltonian system, we find the trajectories of the capital of a country and labor efficiency. The solutions obtained allow one to assess the growth rates of the GDP of the country and the level of consumption in the neighborhood of an equilibrium position.

A differential game of unlimited duration

Abstract The properties of the value function (VF) in a different game of unlimited duration with depreciating performance functional are studied, and two methods of approximating the VF are compared. The VF does not satisfy a Lipschitz condition, due to the type of functional. It is therefore not possible to prove in the general case differential inequalities for the usual directional derivatives. To overcome this difficulty, a generalization of the directional derivative of a continuous function is proposed. It consists in “smearing” the direction by higher-order increments than the increment of the argument. Necessary and sufficient conditions on the VF are obtained in terms of the directional derivative and of the conjugate derivatives. It is shown that the differential inequalities used to find the viscous solutions and the inequalities of the present paper are equivalent in every position. It is also shown that the method of discrete approximation of the stationary Hamilton-Jacobi equation for control problems is likewise applicable for problems of differential game theory. This method is shown to be equivalent to the familar following procedure. Problems of the present type arise e.g., when modelling processes with depreciating performance factor. Such problems of optimal programmed control were previously studied in /1–3/ in the absence of unmonitored noise, which has to be regarded as an opponent player.