We explore an optimal control problem in the context of a specified open set representing “undesirable” system states. This problem statement is closely linked to the standard optimal control problem with a state constraint and can be viewed as a relaxation of the latter. The interrelation between these problems is examined. Recently derived necessary first-order optimality conditions for the discussed problem are presented. An illustrative example is given.

An optimal control problem is considered in which the integral term of the functional to be minimized contains the characteristic function of a given open set of undesirable system states. The statement of this problem can be viewed as a weakening of the statement of the standard optimal control problem with a state constraint. Conditions are obtained that guarantee the equivalence of these problems. Two illustrative examples are given.

We present a version of the Pontryagin maximum principle for the general infinitehorizon optimal control problem with an additional specific asymptotic endpoint constraint under weak regularity assumptions. Such problems arise in economics when studying growth models. The proof is based on reducing the original problem to a family of finite-horizon problems for a mixed type functional containing a terminal term in the form of the conditional cost of the phase vector at a finite time. The results are illustrated by an example.

ПРИНЦИП МАКСИМУМА ДЛЯ ЗАДАЧИ ОПТИМАЛЬНОГО УПРАВЛЕНИЯ С АСИМПТОТИЧЕСКИМ КОНЦЕВЫМ ОГРАНИЧЕНИЕМ 1 С.М

Изучается задача оптимального управления для дифференциального включения со свободным временем и функционалом смешанного типа, содержащим в интегральном члене характеристическую функцию заданного открытого множества "нежелательных" состояний системы. Постановка данной задачи может рассматриваться как ослабление постановки классической задачи оптимального управления с фазовым ограничением. При помощи метода аппроксимаций получены необходимые условия оптимальности первого порядка в форме усиленного включения Эйлера-Лагранжа. Приведены достаточные условия их невырожденности и поточечной нетривиальности. Рассмотрен иллюстрирующий пример.

We consider an optimal control problem with a mixed functional and free stopping time. Dynamics of the system is given by means of a differential inclusion. The integral term of the functional contains the characteristic function of a given open set \(M\subset \mathbb {R}^n\) which can be interpreted as a “risk” or “dangerous” zone. The statement of the problem can be treated as a weakening of the statement of the classical optimal control problem with state constraints. We study relationships between these two problems. An illustrative example is presented as well.

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.

The authors present their recently developed complete version of the Pontryagin maximum principle for a class of infinite-horizon optimal control problems arising in economics. The main distinguishing feature of the result is that the adjoint variable is explicitly specified by a formula analogous to 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 an alternative to them. Examples demonstrate the advantages of the proposed version of the maximum principle. In particular, its applications are considered to Halkin’s example, to Ramsey’s optimal economic growth model, and to a basic model for optimal extraction of a non-renewable resource. Also presented is an economic interpretation of the characterization obtained for the adjoint variable. Bibliography: 62 titles.

В статье представлен недавно полученный авторами полный вариант принципа максимума Понтрягина для класса задач оптимального управления с бесконечным горизонтом, возникающих в экономике. Главной отличительной чертой данного результата является определение сопряженной переменной посредством явной формулы, аналогичной формуле Коши для решений линейных дифференциальных систем. В некоторых случаях эта формула влечет выполнение "стандартных" условий трансверсальности на бесконечности. Более того, она может использоваться в качестве их альтернативы. Приведены примеры, иллюстрирующие преимущества предлагаемого варианта принципа максимума. В частности, рассмотрено его применение к примеру Халкина, к модели оптимального экономического роста Рамсея, а также к базовой модели оптимальной эксплуатации невозобновляемого ресурса. Кроме того, дана экономическая интерпретация полученной характеризации сопряженной переменной. Библиография: 62 названия.

Проводится полное строго математически обоснованное исследование оптимальных стратегий инвестирования в производственный капитал и оптимальных режимов эксплуатации невозобновляемого ресурса в известной модели экономического роста Дасгупты-Хила-Солоу-Стиглица при наличии амортизации капитала. При этом рассматриваются различные значения величины отдачи от расширения масштабов производства. Доказан общий результат о существовании оптимального управления. Показано, что в ситуации, когда коэффициент эластичности производства по используемому ресурсу равен единице, оптимальное управление может не существовать. При помощи специального варианта принципа максимума Понтрягина для задач оптимального управления на бесконечном интервале времени охарактеризовано поведение всех возможных оптимальных режимов. Обсуждаются также общие методологические трудности, возникающие в задачах оптимального управления для моделей экономического роста на бесконечном интервале времени, которым не уделяется должное внимание в экономической литературе, в результате чего представленные там решения часто не выглядят строго математически обоснованными. В заключительной части работы дается экономическая интерпретация полученных результатов.

We consider an optimal control problem for an autonomous differential inclusion with free terminal time and a mixed functional which contains the characteristic function of a given open set M ⊂ ℝ n in the integral term. The statement of the problem weakens the statement of the classical optimal control problem with state constraints to the case where the presence of admissible trajectories of the system in the set M is physically allowed but undesirable due to safety or instability reasons. Using an approximation approach, necessary conditions for the optimality of an admissible trajectory are obtained in the form of Clarke’s Hamiltonian inclusion. The result involves a nonstandard stationarity condition for the Hamiltonian. As in the case of the problem with a state constraint, the obtained necessary optimality conditions may degenerate. Conditions guaranteeing their nondegeneracy and pointwise nontriviality are presented. The results obtained extend the author’s previous results to the case of a problem with free terminal time and more general functional.

The paper offers a complete mathematically rigorous analysis of the welfare-maximizing capital investment and resource depletion policies in the Dasgupta—Heal—Solow—Stiglitz model with capital depreciation and any returns to scale. We establish a general existence result and show that an optimal admissible policy may not exist if the output elasticity of the resource equals one. We characterize the optimal policies by applying an appropriate version of the Pontryagin maximum principle for infinite-horizon optimal control problems. We also discuss general methodological pitfalls arising in infinite-horizon optimal control problems for economic growth models, which are not paid due attention in the economic literature so that the results presented there often seem not to be rigorously justified. We finish the paper with an economic interpretation and a discussion of the welfare-maximizing policies.

We study an optimal growth model for a single resource based economy. The resource is governed by the standard model of logistic growth, and is related to the output of the economy through a Cobb-Douglas type production function with exogenously driven knowledge stock. The model is formulated as an infinitehorizon optimal control problem with unbounded set of control constraints and non-concave Hamiltonian. We transform the original problem to an equivalent one with simplified dynamics 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 appropriate version of the Pontryagin maximum principle. Our main finding is that only two qualitatively different types of behavior of sustainable optimal paths are possible depending on whether the resource growth rate is higher than the social discount rate or not. An analysis of these behaviors yields general criterions for sustainable and strongly sustainable optimal growth (w. r. t. the corresponding notions of sustainability defined herein).

We consider an optimal control problem for an autonomous differential inclusion with free terminal time in the situation when there is a set M (“risk zone”) in the state space ℝ^n which is unfavorable due to reasons of safety or instability of the system. Necessary optimality conditions in the form of Clarke’s Hamiltonian inclusion are developed when the risk zone M is an open set. The result involves a nonstandard stationarity condition for the Hamiltonian. As in the case of problems with state constraints, this allows one to get conditions guaranteeing nondegeneracy of the developed necessary optimality conditions.

The paper is concerned with the problem of optimization of dynamics of a control system in the situation when there is a set M ("risk zone") in the state space R-n which is unfavorable due to reasons of safety or instability of the system. In the classical setting the presence of such unfavorable set M is modeled usually via introducing an additional state constraint in the problem that means the ban on the presence of the trajectories in the risk zone M. Necessary optimality conditions in the form of Clarke's Hamiltonian inclusion are developed for the corresponding optimal control problem in the case when the system's dynamics is described by an autonomous differential inclusion and the risk zone M is an open set. The main novelty of the result is that it is proved in the most important case when the risk zone M is an open set. There is a natural relation of the problem under consideration to the classical optimal control problem with state constraints in this case. The result obtained involves an additional nonstandard stationarity condition for the Hamiltonian.

The paper offers a complete analysis of the welfare-maximizing capital investment and resource depletion policies in the Dasgupta–Heal–Solow–Stiglitz (DHSS) model with capital depreciation and any returns to scale. We establish a general existence result and show that an optimal admissible policy may not exist if the output elasticity of the resource equals one. We characterize the optimal policies by applying an appropriate version of the Pontryagin maximum principle for infinite-horizon optimal control problems. We finish the paper with an economic interpretation and a discussion of the welfare-maximizing policies. JEL classification: C61; O38; Q01; Q56 2000 Mathematics Subject Classification: 49K15; 49K45; 91B62