A.V. Kryazhimskiy

The Pontryagin Maximum Principle and Transversality Conditions for a Class of Optimal Control Problems with Infinite Time Horizons

This paper suggests some further developments in the theory of first-order necessary optimality conditions for problems of optimal control with infinite time horizons. We describe an approximation technique involving auxiliary finite-horizon optimal control problems and use it to prove new versions of the Pontryagin maximum principle. Special attention is paid to the behavior of the adjoint variables and the Hamiltonian. Typical cases, in which standard transversality conditions hold at infinity, are described. Several significant earlier results are generalized.

On the solvability of problems of guaranteeing control for partially observable linear dynamical systems

This paper is devoted to a specification of the method of open-loop control packages, a universal instrument for verification of the solvability of problems of closed-loop control for partially observable dynamical systems. Under the assumption that the control system and observed signal are linear and the set of the admissible initial states is finite, a structure of the corresponding open-loop control packages is specified and a finite-step backward construction is described, which provides a criterion for the solvability of a problem of guaranteed closed-loop guidance onto a target set at a prescribed time.

On rough inversion of a dynamical system with a disturbance

Abstract A dynamical inversion problem is considered. A regularizing solving algorithm oriented to a quite long time interval of systems functioning is designed. The algorithm is stable with respect to informational noises.

Dealing with femtorisks in international relations

The contemporary global community is increasingly interdependent and confronted with systemic risks posed by the actions and interactions of actors existing beneath the level of formal institutions, often operating outside effective governance structures. Frequently, these actors are human agents, such as rogue traders or aggressive financial innovators, terrorists, groups of dissidents, or unauthorized sources of sensitive or secret information about government or private sector activities. In other instances, influential “actors” take the form of climate change, communications technologies, or socioeconomic globalization. Although these individual forces may be small relative to state governments or international institutions, or may operate on long time scales, the changes they catalyze can pose significant challenges to the analysis and practice of international relations through the operation of complex feedbacks and interactions of individual agents and interconnected systems. We call these challenges “femtorisks,” and emphasize their importance for two reasons. First, in isolation, they may be inconsequential and semiautonomous; but when embedded in complex adaptive systems, characterized by individual agents able to change, learn from experience, and pursue their own agendas, the strategic interaction between actors can propel systems down paths of increasing, even global, instability. Second, because their influence stems from complex interactions at interfaces of multiple systems (e.g., social, financial, political, technological, ecological, etc.), femtorisks challenge standard approaches to risk assessment, as higher-order consequences cascade across the boundaries of socially constructed complex systems. We argue that new approaches to assessing and managing systemic risk in international relations are required, inspired by principles of evolutionary theory and development of resilient ecological systems.

Resource-saving tracking problem with infinite time horizon

We consider the problem on the infinite-duration tracking of a prescribed trajectory of an inaccurately observed control system subjected to an unobservable dynamic disturbance. We construct a solution algorithm that is resource-saving in the sense that the control resources used for solving the problem for small noise values in the state observation channel are little different from the corresponding resources in the “ideal case” where the current values of the dynamic disturbance are available to direct observation.

Resource-saving infinite-horizon tracking under uncertain input

Abstract A feedback controller for approximate tracking a prescribed trajectory of an inaccurately observed dynamical system effected by uncertain non-observable input disturbances over an infinite time interval is constructed. The controller is “resource-saving” in a sense that control resources spent for approximate tracking do not exceed (with some small gaps) those needed for exact tracking in an “ideal” situation where the current values of the input disturbance are fully observable.

Infinite-horizon dynamic programming and application to management of economies effected by random natural hazards

We describe a version of the dynamic programming method, applicable to infinite-horizon discrete-time stochastic processes, and use this technique to solve a stylized problem of management of an economy effected by random natural hazards. Also, we characterize the equilibrium points in a game, in which two economies invest in common prevention measures to mitigate the future impact of natural hazards.

Shadow prices in infinite-horizon optimal control problems with dominating discounts

We consider a nonlinear optimal control problem, in which an integrated discounted utility index is maximized over an infinite time interval. The problem statement is motivated by various optimization problems arising in economics. Assuming that the discount parameter dominates the growth rates in the state variables and in the gradient of the current utility, we develop a version of the Pontryagin maximum principle providing a complete set of necessary optimality conditions and also suggesting an analytic expression for the values of the adjoint variables often viewed as shadow prices in the economic literature. We illustrate the proposed methodology by applying it to the problem of optimal capital accumulation for a stylized model of an enterprise.

On exact stabilization of an uncertain dynamical system

The study is motivated by the problem of stabilizing the concentration of atmospheric carbon, which is widely discussed in the context of global warming nowadays. A key difficulty in the design of stabilization strategies is the uncertainty of the underlying physical model. In the present paper, a general problem setting is suggested and a relevant alanytic framework elaborated. Analysis employs specific qualitative features of an uncertain dynamics, including automatic stabilization of the trajectories in the absence of input disturbances. An asymptotic version of Krasovskiis extremal shift control principle is developed and model-robust strategies stabilizing a state coordinate at a prescribed level are constructed.

A dynamic model of optimal investment in research and development with international knowledge spillovers

We consider a two-country endogenous growth model where an economic follower absorbs part of the knowledge generated in a leading country. To solve a suitably defined infinite horizon dynamic optimization problem an appropriate version of the Pontryagin maximum principle is developed. The properties of optimal controls and the corresponding optimal trajectories are characterized by the qualitative analysis of the solutions of the Hamiltonian system arising through the implementation of the Pontryagin maximum principle.