Search

A nonlinear optimal control problem in the Lotka–McKendrick population model is studied. It describes rational management of age-structured farmed populations in aquaculture and indoor farms. Employing generalized functions, we prove the impulse nature of optimal harvesting. Exact analytic solutions for sustainable harvesting strategies are obtained and used to analyze the optimal dynamics of harvesting age and rotation under technological innovations. [ABSTRACT FROM AUTHOR]

We consider a nonlinear profit maximization problem in the Lotka–McKendrick model of age-structured harvested population describing farmed populations in agriculture and aquaculture. The control functions are time- and age-dependent harvesting rate and time dependent supply of newborns. We establish the existence of optimal controls with measure-valued harvesting rate by using distributional partial derivatives of functions of bounded variation through the equivalent integrated form to the original problem. [ABSTRACT FROM AUTHOR]

The integral model with finite memory is employed to analyze the timeline of COVID-19 epidemic in the United Kingdom and government actions to mitigate it. The model uses a realistic infection distribution. The time-varying transmission rate is determined from Volterra integral equation of the first kind. The authors construct and justify an efficient regularization algorithm for finding the transmission rate. The model and algorithm are approbated on the UK data with several waves of COVID-19 and demonstrate a remarkable resemblance between real and simulated dynamics. The timing of government preventive measures and their impact on the epidemic dynamics are discussed. [ABSTRACT FROM AUTHOR]

This paper studies the cost-minimizing serial asset replacement problem under changing operating and replacement costs. Technological improvements affect these costs in different ways, which leads to different duration of sequential replacement cycles of the same asset. Therefore, a firm should optimize a chain of consecutive asset replacements using available information about future costs. We offer a novel multi-cycle algorithm for making such replacement decisions and prove that it converges to the infinite-horizon solution when the number of cycles increases. Important qualitative properties of the obtained optimal non-equal-life replacement strategies are derived, and practical recommendations are discussed. [ABSTRACT FROM AUTHOR]

The review surveys and compares state-of-the-art epidemiologic models starting from the celebrated SIR model to recent SEIR Erlang models and integral models with distributed delays. The authors emphasize the flexibility and advantages of integral models in describing government efforts to control the current COVID -19 pandemic. [ABSTRACT FROM AUTHOR]

Solutions from non-smooth functional spaces, including generalized functions and measures, often appear in optimal control theory but are avoided in applications. They are however useful in finding the optimal distribution of investments into new and old capital equipment under improving technology. The corresponding economic problem involves optimal control in a linear Lotka-McKendrik model of age-structured population. Optimal solutions do not exist in normal functional classes and, so, generalized functions are used to construct the solutions. The optimal age-distributions of capital and investment include the Dirac function and are interpreted as instantaneous investment in equipment of certain age. A numerical simulation completes the presentation of the dynamics. [ABSTRACT FROM AUTHOR]

An optimal control problem for nonlinear integral equations of special kind is analyzed. It considers a firm's investment into age-dependent capital under improving technology and limited substitutability among capital of different ages. We prove the existence of solutions and analyze their structure. It is shown that the initially bang-bang optimal investment switches to an interior one and eventually converges to a steady-state trajectory that represents balanced economic growth. The obtained analytic outcomes contribute to better understanding of investment policies under technological change. [ABSTRACT FROM AUTHOR]

The suggested model describes the optimal management of a manufacturing system (a firm, region, or industry) that consumes a scarce natural resource and replaces obsolete productive assets with new resource-efficient ones. The model includes resource-saving embodied technological change, a varying resource price, and environmental quotas on resource consumption. Mathematically, it leads to the infinite-horizon optimal control of nonlinear integral equations with unknown integration limits. A steady-state analysis of the model addresses environmentally sustainable manufacturing and provides important implications for both firm and government decision-making. [ABSTRACT FROM AUTHOR]

We consider the optimal control of nonlinear integral equations with endogenous delay and state constraints, which describe a developing economy subjected to resource constraints. The economy invests in new resource-efficient technologies, invests in new capital, and scraps obsolete capital. We derive the optimality condition and determine long-term asymptotically exponential trajectories that optimally combine scrapping the dirtiest capital and developing new clean technologies. Next, we study the short-term dynamics of the model and show that it leads to a sustainable growth with active resource constraint. [ABSTRACT FROM AUTHOR]

Work significantly affects human life and health. Overworking may decrease the quality of life and cause direct economic losses. Technological innovations promote the modernization of firms' capital and lead to the improvement of labor productivity in the workplace. This paper investigates the optimal individual choice of work intensity under improving technology embodied in a new equipment. The balanced growth trajectories are obtained and analyzed to find their impact on the work-leisure balance problem and links between the optimal choice of work intensity and the rate of embodied technological change. [ABSTRACT FROM AUTHOR]

This paper performs a systematic study of some nonlinear integral equations with unknown lower limits of integration, which arise in economics, operations research, population biology and environmental sciences. A general investigation technique is developed for systems of such equations and several theorems about the existence and uniqueness of a solution are proven. The links to other functional equations with state-dependent delays are discussed. Applied interpretation of the obtained results is provided. [ABSTRACT FROM AUTHOR]

We perform the steady-state analysis of a nonlinear partial differential equation model that describes the dynamics of a managed size-structured forest. The harvesting policy is to maximize the net benefits from timber production over an infinite planning horizon. The existence and uniqueness of the steady-state trajectories are analysed. Closed-form steady states are obtained in meaningful special cases and are used to estimate how climate change affects the optimal harvesting regime, diameter of cut trees, number of logged trees, and net benefits in the long run. [ABSTRACT FROM AUTHOR]

Climatic changes will affect the dynamics of a forest ecosystem. Consequently, carbon sequestration costs can only be estimated correctly if changes in climatic conditions are considered. This article determines the changes in mitigation costs of an optimal forest management regime in the presence of climatic changes and varying prices, and takes account of substitution processes between timber production and carbon sequestration at the stand level. The study demonstrates that in the presence of climate change the sequestration costs per ton of carbon increase with higher amounts of carbon sequestered per hectare. This finding can be used to identify a threshold for the amount of sequestered carbon per hectare below which the costs of carbon sequestration are hardly influenced by climate change. [Copyright &y& Elsevier]

Abstract: This work investigates the impact of global climate change on the sustainable growth of forest, namely, on its aggregated characteristics such as the number of trees, the basal area, and the amount of carbon sequestrated in the stand. The forest dynamics is described by a nonlinear size-structured population model. The existence of a steady state regime is proven and explicit formulas for the aggregated characteristics are obtained. A numeric simulation on realistic data illustrates and extends the analytic results obtained. [Copyright &y& Elsevier]

The paper analyzes a fleet replacement problem under general assumptions about technological improvement. The corresponding optimization model considers the variable lifetime of assets in deterministic infinite-horizon settings. The optimal dynamics of the asset lifetime and investment is investigated analytically and numerically under continuous technological change and technological shocks that affect the operating and new asset costs. The obtained results contribute to the OR replacement theory and suggest simple rules for the efficient fleet replacement under improving technology. [ABSTRACT FROM AUTHOR]

Optimal control of partial differential equations arises in population ecology, economics, and demography. The consistency of mathematical treatment is demonstrated for the Lotka-McKendrick model and its nonlinear modifications of increasing complexity. The obtained qualitative optimal dynamics show that the models have either the bang-bang structure of optimal controls or follow balanced growth dynamics. [ABSTRACT FROM AUTHOR]

The paper analyzes the optimal maintenance of a "fully manageable" biological population with no natural reproduction. Examples of such populations include dairy farms, horse breeding, gardening, and forestry. Mathematically, the problem leads to the nonlinear optimal control of a first-order partial differential equation with pointwise state constraints. A necessary extremum condition is proven using the abstract optimization theory in Banach spaces. It is shown that the optimal dynamics has a bang–bang structure with three jumps and involves buying new young individuals and selling the old ones when their anticipated future expenses exceed the profit. [ABSTRACT FROM AUTHOR]

Abstract: The paper is devoted to theoretic and numeric investigation of a nonlinear integral equation with an unknown function in the upper limit of integration. This equation appears in optimal replacement problems of engineering and production economics. Its solution is essential for finding the optimal policy of equipment replacement under technological advances. Solvability and qualitative dynamics of the solution are analyzed. Computational algorithm for solving the equation is constructed. A numeric example illustrates the obtained results. [Copyright &y& Elsevier]

The authors analyze the optimal replacement of a single asset under continuous and discontinuous technological change. They consider an infinite-horizon replacement problem with a variable asset lifetime. Due to deterioration, maintenance costs increase when the asset becomes older. Because of technological change, both maintenance and new capital costs decrease in time for a fixed asset age. The dynamics of the optimal asset lifetime are studied analytically and numerically in the cases of one and several technological breakthroughs. The breakthroughs cause irregularities (anticipation echoes) in the asset lifetime before the breakthrough time. It is shown that the optimal lifetime is always smaller when new capital costs decrease faster, when the maintenance costs decrease slower, or when both maintenance and new capital costs decrease faster with the same rate. [ABSTRACT FROM AUTHOR]

An exact solution is constructed to a nonlinear optimization problem in an integral dynamic model with delay. The problem involves the unknown duration of the delay and has important applications to the optimal replacement of capital equipment under technological change. [ABSTRACT FROM AUTHOR]

Abstract: The paper analyzes nonlinear optimal control of integral–differential equations that describe the optimal management of a forest taking into account intra-species competition and carbon sequestration. The objective function includes the revenues from timber production, operational expenses, and the net benefits from carbon sequestration. A dual system is derived and a necessary extremum condition is established. [Copyright &y& Elsevier]

Optimization problems with nonlinear utility and endogenous capital lifetime are investigated in one- and two-sector modifications of the Solow vintage capital model. Both models have the same balanced growth path in the case of exponential technological change and labour. Turnpike theorems in normal form are proved for the optimal capital lifetime in both models. The differences between the cases of the linear and nonlinear utility are highlighted. [ABSTRACT FROM AUTHOR]

Abstract: The paper analyzes the structure of optimal trajectories in the one-sector vintage capital model that optimizes the endogenous lifetime of age-structured capital under technological change. A new repetition pattern (anticipation echoes) is demonstrated in both optimal investment and optimal capital lifetime in the case of finite-horizon optimization. The anticipation echoes are caused by expectation of the future “no-investment” policy at the end of the planning horizon. It is shown that the anticipation echoes represent a general structural property of dynamic systems with endogenous delay. Mathematically, the problem under study is the optimal control of non-linear integral equations with unknowns in the integration limits. [Copyright &y& Elsevier]

A model of the governmental control of age at retirement uses population age structures, productivity, employment, and money income. The sustainability criterion is a sufficient level of individual money income for all ages. The decision variables are the variable age at retirement and the intensity of technological change. The model is calibrated on real data from the U.S. Census Bureau and Bureau of Labor Statistics between 1950 and 2005 and uses official U.S. population projections for 2006-2050. The main result is that the extrapolation of current U.S. technological trends can compensate negative demographic trends in the United States without increasing the age at retirement. [ABSTRACT FROM AUTHOR]

Abstract: The paper investigates a nonlinear optimal control problem for the Volterra integral equations with an unknown function in the lower limit of integration. The applied interpretation of the problem consists of the optimal renovation of vintage capital under the embodied technological change. An explicit solution is constructed and its properties are analyzed in the case of a nonlinear revenue dependence on product output. [Copyright &y& Elsevier]

The paper analyzes the problem of rational replacement of computing systems. Such systems, especially high-performance computers, are expensive, but quickly become obsolete. A system analysis of the replacement problem is provided and the relevant modeling problems are identified and studied. The obtained results give certain rules for calculating the optimal lifetime of computing systems, which are based on the given rates of technological change in the related computer industry. Two simulation examples estimate the optimal lifetime of PCs and high-performance computers in the current US economy. The results are in good agreement with other recommendations and existing business practice. [ABSTRACT FROM AUTHOR]

Abstract: An optimization problem in an economic vintage capital model with nonlinear utility is investigated. It is described by non-linear Volterra integral equations with an unknown in the limits of integration. The concavity of the problem is proven, the condition for an extremum is established, and their relevance to applications is demonstrated. [Copyright &y& Elsevier]

The nonlinear optimal control problem investigated in this paper is governed by Volterra integral equations with an unknown function in the limit of integration. Problems of such type arise in vintage capital models of mathematical economics and reveal new non-linear phenomena in economic-technological development. The main result of the paper is the explicit solution of the problem in important special cases. To construct these solutions, necessary and sufficient conditions for an extremum are derived. The exact solutions show the behavior of a capital replacement process under technological change. [ABSTRACT FROM AUTHOR]

We formulate and study a nonlinear game of n symmetric countries that produce, pollute, and spend part of their revenue on pollution mitigation and environmental adaptation. The optimal emission, adaptation, and mitigation investments are analyzed in both Nash equilibrium and cooperative cases. Modeling assumptions and outcomes are compared to other publications in this fast-developing area of environmental economics. In particular, our analysis implies that: (a) mitigation is more effective than adaptation in a crowded multi-country world; (b) mitigation increases the effectiveness of adaptation; (c) the optimal ratio between mitigation and adaptation investments in the competitive case is larger for more productive countries and is smaller when more countries are involved in the game. [ABSTRACT FROM AUTHOR]

The cacao swollen shoot virus disease (CSSVD) is among the most economically damaging diseases of cacao trees and accounts for almost 15–50% of harvest losses in Ghana. This virus is transmitted by several species of mealybugs (Pseudococcidae, Homoptera) when they feed on cacao plants. One of the mitigation strategies for CSSVD investigated at the Cocoa Research Institute of Ghana (CRIG) is the use of mild-strain cross-protection of cacao trees against the effects of severe strains. In this study, simple deterministic, delay, and stochastic ordinary differential equation-based models to describe the dynamic of the disease and spread of the virus are suggested. Model parameters are estimated using detailed empirical data from CRIG. The modeling outcomes demonstrate a remarkable resemblance between real and simulated dynamics. We have found that models with delay approximate the data better and this agrees with the knowledge that CSSVD epidemics develop slowly. Also, since there are large variations in the data, stochastic models lead to better results. We show that these models can be used to gain useful informative insights about the nature of disease spread. [ABSTRACT FROM AUTHOR]

Abstract The owner of an asset often transfers the right to use or exploit that asset to an agent in exchange for a rent. A limited time of the license and the failure of the owner's commitment to compensate the agent for any asset improvement are likely to lead to underinvestment (holdup). In this study, we analyze the optimal length a contract would need to have to maximize the owner's income in the short- and long-run. We determine the design of a sequence of renegotiation-proof, overlapping, fixed time contracts that allows eliminating the hold-up problem. The obtained outcomes are tested and illustrated on a specific problem (land lease and soil quality). Numeric simulation demonstrates that the most severe version of the hold-up problem arises when the lease contract is not long enough for farmers to make any investment in the soil quality (less than 3 years on calibrated data). Highlights • We study the optimal duration and renewal of fixed-time contracts if a hold-up problem is present. • The agent's investment and cooperative benefits are intertwined. • We determine the minimum contract length for possible solutions of the hold-up problem. • Overlapping fixed time contracts with advance notice provide a solution of the hold-up problem. • We define the optimal length of the contract and advance notice time. [ABSTRACT FROM AUTHOR]

The article discusses various reports published within the issue, including an article by Laura-Iulia Anita and Sebastian Anita on the asymptotic dynamics of a generic nonlinear age structured population model and another by Natali Hritonenko from Prairie on sustainable economic-demographic development under the observed technological and demographic trends.

Abstract: Age-structured and size-structured population models with a distributed harvesting control are considered and their qualitative behaviors are compared. The maximum principle and bang–bang structure of the optimal control are established for a size-structured forestry model with carbon sequestration benefits. [Copyright &y& Elsevier]

Abstract: Solvability and stability of the nonlinear Volterra integral equations with the variable lower limit of integration are investigated. Obtained results are applied to two integral models that describe the dynamics of one- and two-species age-structured populations with intra- and inter-species competitions. [Copyright &y& Elsevier]

Abstract: Vintage capital models such as Solow model, Ramsey model, two-factor model, and models with endogenous technological change are discussed to demonstrate a variety of applications of nonlinear integral models to mathematical economics and finance. For each model an optimal control problem is formulated and basic results about the existence and asymptotic behavior of a solution are provided. [Copyright &y& Elsevier]

Habitat loss and harvesting of non-timber forest products (NTFPs) significantly affect the population dynamics. In this paper, we propose a general mathematical modelling approach incorporating the impact of habitat size reduction and non-lethal harvesting of NTFP on population dynamics. The model framework integrates experimental data of Pentadesma butyracea in Benin. This framework allows us to determine the rational non-lethal harvesting level and habitat size to ensure the stability of the plant ecosystem, and to study the impacts of distinct levels of humidity. We suggest non-lethal harvesting policies that maximize the economic benefit for local populations. [ABSTRACT FROM AUTHOR]

Abstract: Several R&D-based models of endogenous economic growth are investigated under the Solow-like assumption of fixed allocation of resources across activities. We identify model parameters that lead to explosive dynamics and analyze various economic techniques to avoid it. The techniques include adding stricter constraints on model trajectories and limiting factors in technology equation. In particular, we demonstrate that our vintage version of the well-known R&D-based model of economic growth [C.I. Jones, R & D-based models of economic growth, J. Polit. Econom. 103 (1995) 759–784] exhibits the same balanced dynamics as the original model. [Copyright &y& Elsevier]