Another View of the Maximum Principle for Infinite-Horizon Optimal Control Problems in Economics

We present a recently developed complete version of the Pontryagin maximum principle for a class of infinite-horizon optimal control problems arising in economics. The peculiarity of the result is that the adjoint variable is explicitly specified by a formula which resembles the Cauchy formula for solutions of linear differential systems. In certain situations this formula implies the “standard” transversality conditions at infinity. Moreover, it can serve as their alternative. We provide examples demonstrating the ad- vantage of the suggested version of the maximum principle. In particular, we consider its applications to Halkin’s example, to Ramsey’s optimal growth model and to a basic model of optimal extraction of a non-renewable resource. An economic interpretation of the developed characterization of the adjoint variable is also presented

Optimal Endogenous Growth with Exhaustible Resources

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. 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 Optimal Leader’s Investments Strategy in a Cyclic Model of Innovation Race with Random Inventions Times

In this paper, we develop a new dynamic model of optimal investments in R&D and manufacturing for a technological leader competing with a large number of identical followers on the market of a technological product. The model is formulated in the form of the infinite time horizon stochastic optimization problem. The evolution of new generations of the product is treated as a Poisson-type cyclic stochastic process. The technology spillovers effect acts as a driving force of technological change. We show that the original probabilistic problem that the leader is faced with can be reduced to a deterministic one. This result makes it possible to perform analytical studies and numerical calculations. Numerical simulations and economic interpretations are presented as well

Optimal Growth, Renewable Resources and Sustainability

We study a growth model for a single resource-based economy, as an infinite-horizon op-timal control problem. The resource is assumed to be governed by the standard model of logistic growth, and is related to the output of the economy through a Cobb-Douglass type production function with an exogenously driven knowledge stock. The problem involves unbounded controls and the non-concave Hamiltonian. These preclude direct application of the standard existence results and Arrow’s sufficient conditions for optimality. We transform the original optimal control problem to an equivalent one with simplified dy-namics and prove the existence of an optimal admissible control. Then we characterize the optimal paths for all possible parameter values and initial states by applying the ap-propriate version of the Pontryagin maximum principle. Our main finding is that only two qualitatively different types of behavior of sustainable optimal paths are possible de-pending on whether the resource growth rate is higher than the social discount rate or not

In memory of Arkady Viktorovich Kryazhimskiy (1949-2014)

The article is devoted to the description of Academician Arkady Kryazhimskiy's life path. The facts of the scientific biography of Acad. Kryazhimskiy are presented with the emphasis on his outstanding contribution into the theory of dynamic inversion, the theory of differential games, and control theory. His personal talents in different spheres are also marked out

Wave-induced loss of ultra-relativistic electrons in the Van Allen radiation belts.

The dipole configuration of the Earth's magnetic field allows for the trapping of highly energetic particles, which form the radiation belts. Although significant advances have been made in understanding the acceleration mechanisms in the radiation belts, the loss processes remain poorly understood. Unique observations on 17 January 2013 provide detailed information throughout the belts on the energy spectrum and pitch angle (angle between the velocity of a particle and the magnetic field) distribution of electrons up to ultra-relativistic energies. Here we show that although relativistic electrons are enhanced, ultra-relativistic electrons become depleted and distributions of particles show very clear telltale signatures of electromagnetic ion cyclotron wave-induced loss. Comparisons between observations and modelling of the evolution of the electron flux and pitch angle show that electromagnetic ion cyclotron waves provide the dominant loss mechanism at ultra-relativistic energies and produce a profound dropout of the ultra-relativistic radiation belt fluxes

Kinematical Test of Large Extra Dimension in Beta Decay Experiments

The forthcoming experiments on neutrino mass measurement using beta decay, open a new window to explore the Large Extra Dimension model. The Kaluza-Klein tower of neutrinos in Large Extra Dimension contribute to the Kurie function of beta decay that can be tested kinematically. In addition to providing an alternative approach using just the kinematical properties, we show that KATRIN can probe the compactification radius of extra dimensions down to 0.2 \mu m which is better, at least by a factor of two, than the upper limits from neutrino oscillation experiments.Comment: 4 pages, 4 figures; v2: discussion improved, matches the version published at PL