S.M. Aseev

The Pontryagin maximum principle and optimal economic growth problems

This monograph is devoted to the theory of the Pontryagin maximum principle as applied to a special class of optimal control problems that arise in economics when studying economic growth processes. The main distinctive feature of such problems is that the control process is considered on an infinite time interval. In this monograph, we develop a new approximation approach to deriving necessary optimality conditions in the form of the Pontryagin maximum principle for problems with infinite time horizon. The attention is focused on the characterization of the behavior of the adjoint variable and the Hamiltonian of a problem at infinity. The approach proposed is applied to the analysis of the problem of optimal economic growth of a technological follower, a country that absorbs, in its technological sector, part of knowledge produced by a technological leader. By optimizing its growth performance, the technological follower dynamically redistributes available labor resources between the manufacturing and research and development (R&D) sectors of the economy. This problem is of independent interest in the endogenous economic growth theory. Moreover, it serves as an illustration of the approximation approach proposed. The main results presented in this monograph are new. They generalize and strengthen many previous studies in this direction.

State constraints in optimal control. The degeneracy phenomenon

In this paper we study the degeneracy phenomenon arising in optimal control problems with state constraints. It is shown that this phenomenon occurs because of the incompleteness of the standard variants of Pontryagins maximum principle for problems with state constraints. The new maximum principle containing some additional information about the behavior of the Hamiltonian at the endtimes is developed. As application we obtain some sufficient and necessary conditions for nondegeneracy and pointwise nontriviality of the maximum principle. The results obtained envelope the optimal control problems with systems described by differential inclusions and ordinary differential equations.

Needle variations in infinite-horizon optimal control

S. M. Aseev and V. M. Veliov - Needle variations in infinite-horizon optimal control A. Berman, F. Goldberg, and R. Shorten - Comments on Lyapunov a -stability with some extensions V. S. Borkar and K. S. Kumar - Small noise large time asymptotics for the normalized Feynman-Kac semigroup Y. Dolgin and E. Zeheb - Linear constraints for convex approximation of the stability domain of a polynomial in coefficients space V. Y. Glizer -- Singular solution of an infinite horizon linear-quadratic optimal control problem with state delays I. Ioslovich and P.-O. Gutman - Time-optimal control of wafer stage positioning using simplified models J. Kogan and Y. Malinovsky -- Robust stability and monitoring threshold functions E. Ocai?½a and P. Cartigny - One dimensional singular calculus of variations in infinite horizon and applications N. P. Osmolovskii - Second order optimality conditions in optimal control problems with mixed inequality type constraints on a variable time interval I. Shafrir and I. Yudovich - An infinite-horizon variational problem on an infinite strip D. Wenzke, V. Lykina, and S. Pickenhain - State and time transformations of infinite horizon optimal control problems A. J. Zaslavski - Turnpike properties of approximate solutions of discrete-time optimal control problems on compact metric spaces A. J. Zaslavski - Turnpike theory for dynamic zero-sum games

On a class of optimal control problems arising in mathematical economics

This paper is devoted to the study of the properties of the adjoint variable in the relations of the Pontryagin maximum principle for a class of optimal control problems that arise in mathematical economics. This class is characterized by an infinite time interval on which a control process is considered and by a special goal functional defined by an improper integral with a discounting factor. Under a dominating discount condition, we discuss a variant of the Pontryagin maximum principle that was obtained recently by the authors and contains a description of the adjoint variable by a formula analogous to the well-known Cauchy formula for the solutions of linear differential equations. In a number of important cases, this description of the adjoint variable leads to standard transversality conditions at infinity that are usually applied when solving optimal control problems in economics. As an illustration, we analyze a conventionalized model of optimal investment policy of an enterprise.

Maximum principle for infinite-horizon optimal control problems under weak regularity assumptions

The paper deals with first order necessary optimality conditions for a class of infinite-horizon optimal control problems that arise in economic applications. Neither convergence of the integral utility functional nor local boundedness of the optimal control is assumed. Using the classical needle variations technique we develop a normal form version of the Pontryagin maximum principle with an explicitly specified adjoint variable under weak regularity assumptions. The result generalizes some previous results in this direction. An illustrative economical example is presented.

Optimal control for sustainable consumption of natural resources

Abstract In this paper we study optimal policies for a central planner interested in maximizing utility in an economy driven by a renewable resource. It is shown that the optimal consumption path is sustainable only when the intrinsic growth rate of the resource is greater than the social discount rate. The model is formulated as an infinite horizon optimal control problem. We deal with the mathematical details of the problem, develop a precise notion for optimality and establish the existence of optimal control at least when the condition for sustainability is met. We apply the appropriate version of the Pontryagin maximum principle and show a numerical simulation of the optimal feedback law. In the end we present the results along with physical interpretations.

The Problem of Optimal Endogenous Growth with Exhaustible Resources Revisited

We study optimal research and extraction policies in an endogenous growth model in which both production and research require an exhaustible resource. It is shown that optimal growth is not sustainable if the accumulation of knowledge depends on the resource as an input, or if the returns to scale in research are decreasing, or the economy is too small. The model is stated as an infinite-horizon optimal control problem with an integral constraint on the control variables. We consider the main mathematical aspects of the problem, establish an existence theorem and derive an appropriate version of the Pontryagin maximum principle. A complete characterization of the optimal transitional dynamics is given.

On some properties of the adjoint variable in the relations of the Pontryagin maximum principle for optimal economic growth problems

For a class of infinite-horizon optimal control problems that appear in studies on economic growth processes, the properties of the adjoint variable in the relations of the Pontryagin maximum principle, defined by a formula similar to the Cauchy formula for the solutions to linear differential systems, are studied. It is shown that under a dominating discount type condition the adjoint variable defined in this way satisfies both the core relations of the maximum principle (the adjoint system and the maximum condition) in the normal form and the complementary stationarity condition for the Hamiltonian. Moreover, a new economic interpretation of the adjoint variable based on this formula is presented.

Optimal economic growth with a random environmental shock

The government in a small open economy uses both an old “dirty,” or “polluting,” technology and a new “clean” technology simultaneously. However, because of climate change, it should take into account that at some stage in the future it will be penalized for production based on the old technology. In this paper, pollution is alleviated through international agreements that restrict polluting activities. The government’s incentives to invest in cleaner technologies are based on productivity of the technology and randomly increasing abatement costs for pollution in future. In contrast to the Schumpeterian model of creative destruction, both technologies can be used simultaneously. The technologies are subject to AK production functions. Assuming that the exogenous environmental shock follows a Poisson process, we use Pontryagin’s maximum principle to find the optimal investment policy. We find conditions under which a rational government should invest all its resources in one technology, while the other is moderately run down, as well as conditions under which it should divide the investments between the technologies in a certain ratio.

Optimal growth in a two-sector economy facing an expected random shock

We develop an optimal growth model of an open economy that uses both an old (“dirty” or “polluting”) technology and a new (“clean”) technology simultaneously. A planner of the economy expects the occurrence of a random shock that increases sharply abatement costs in the dirty sector. Assuming that the probability of an exogenous environmental shock is distributed according to the exponential law, we use Pontryagin’s maximum principle to find the optimal investment and consumption policies for the economy.