Sergey Aseev

Steklov Mathematical Institute, Moscow Russia · Differential Equations

Infinite horizon optimal control problem with general endpoint constraints is reduced to a family of standard problems on finite time intervals containing the value of the conditional cost of the phase vector as a terminal term. New version of the Pontryagin maximum principle containing an explicit characterization of the adjoint variable is obtain...

The paper deals with an infinite-horizon optimal control problem with general asymptotic endpoint constraints. The fulfillment of constraints of this type can be viewed as the minimal necessary condition for the sustainability of solutions. A new version of the Pontryagin maximum principle with an explicitly specified adjoint variable is developed....

Рассматривается задача оптимального управления, в которой интегральный член минимизируемого функционала содержит характеристическую функцию заданного открытого множества нежелательных состояний системы. Постановка данной задачи может рассматриваться как ослабление постановки стандартной задачи оптимального управления с фазовым ограничением. Получен...

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...

We study a free-time optimal control problem for a differential inclusion with mixed-type functional in which the integral term contains the characteristic function of a given open set of “undesirable” states of the system. The statement of this problem can be viewed as a weakening of the statement of the classical optimal control problem with stat...

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 оптима...

Under conditions characterizing the dominance of the discounting factor, a complete version of the Pontryagin maximum principle for an optimal control problem with infinite time horizon and a special asymptotic endpoint constraint is developed. Problems of this type arise in mathematical economics in the studies of growth models.

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...

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⊂Rn which can be interpreted as a “risk” or “dangerous” zone. The statement of the problem can be t...

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 diffe...

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 diffe...

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...

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 ⊂ ℝn in the integral term. The statement of the problem weakens the statement of the classical optimal control problem with state constraints to the case wher...

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 отда...

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 elastic...

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 infinite-horizon optimal control problem with unbo...

We consider a class of infinite-horizon optimal control problems with not necessarily bounded set of control constraints. Using finite-horizon approximations and tools of the Pontryagin maximum principle sufficient conditions for existence and boundedness of an optimal control are developed in the general nonlinear case. Application of the result o...

We consider a class of infinite-horizon optimal control problems with not necessarily bounded set of control constraints. Sufficient conditions for the existence of an optimal control are derived in the general nonlinear case by means of finite-horizon approximations and the tools of the Pontryagin maximum principle. Conditions guaranteeing the uni...

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 certai...

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 ar...

We consider a class of infinite-horizon optimal control problems with not necessarily bounded set of control constraints. Sufficient conditions for the existence of an optimal control are derived in the general nonlinear case by means of finite-horizon approximations and the tools of the Pontryagin maximum principle. Conditions guaranteeing the uni...

A class of infinite-horizon optimal control problems that arise in economic applications is considered. A theorem on the nonemptiness and boundedness of the set of optimal controls is proved by the method of finite-horizon approximations and the apparatus of the Pontryagin maximum principle. As an example, a simple model of optimal economic growth...

The properties of adjoint variables involved in the relations of the Pontryagin maximum principle are investigated for a class of infinite-horizon optimal control problems that arise in the study of economic growth processes. New formulations of the maximum principle in terms of intertemporal prices and the conditional value of the capital are esta...

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 dominati...

The paper deals with the 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 ve...

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...

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 s...

This paper extends optimal control theory to a class of infinite-horizon problems that arise in studying models of optimal dynamic allocation of economic resources. In a typical problem of this sort the initial state is fixed, no constraints are imposed on the behaviour of the admissible trajectories at large times, and the objective functional is...

The 9th of March 2012 was the 90th anniversary of the birth of the prominent Russian mathematician Academician Evgenii Frolovich Mishchenko, one of the creators of modern mathematical control theory and the theory of oscillations.

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...

The paper revisits the issue of necessary optimality conditions for infinite-horizon optimal control problems. It is proved that the normal form maximum principle holds with an explicitly specified adjoint variable if an appropriate relation between the discount rate, the growth rate of the solution and the growth rate of the objective function is...

The paper revisits the issue of necessary optimality conditions for infinite-horizon optimal control problems. It is proved that the normal form maximum principle holds with an explicitly specified adjoint variable if an appropriate relation between the discount rate, the growth rate of the solution and the growth rate of the objective function is...

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 inte...

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 curr...

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...

Nondegenerate first-order necessary conditions for optimality are obtained for the problem (1.1)-(1.4) under different assumptions about controllability at the endpoints. These necessary conditions are obtained in the Hamiltonian form of Clarke [1]. With the help of a smoothing technique [2] the perturbation method in [3] is used to carry the main...

In this paper we develop a constructive method of approximation of a differential inclusion by a sequence of smooth control systems. Combining this with other methods of approximation [7], [17], we reduce the optimal control problem for a differential inclusion with state constraints to the classical optimal control problem without constraints on s...

We consider a nonlinear optimal control problem with an integral functional in which the integrand is the characteristic function of a closed set in the phase space. An approximation method is applied to prove the necessary conditions of optimality in the form of a Pontryagin maximum principle without any prior assumptions on the behavior of the o...

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 n...

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...

We provide steps towards a welfare analysis of a two-country endogenous growth model where a relatively small follower absorbs part of the knowledge generated in the leading country. To solve a suitably defined dynamic optimization problem, an appropriate version of the Pontryagin maximum principle is developed. The properties of optimal controls a...

We provide steps towards analysis of rational bahaviors of innovators acting on a market of a technological product. The situation when a technological leader competes with a large number of identical followers is in the focus. The process of development of new generations of the product is treated as a Poisson-type cyclic stochastic process. The t...

The P problem of optimum controlling is considered that is complicated by the discontinuity of δ integrand of J functional. This circumstance makes it impossible a direct use of the standard methods of optimum controlling theory for obtaining the required conditions of the most favorability. The latter for the P problem in the present work are obta...

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 atten...

With the use of the approximation method, we distinguish two, generally nonconvex, cases where the necessary optimality conditions for problem x ˙(t)=f(x(t),u(t)),u(t)∈U,x(0)=x 0 ,J(x,u)=∫ 0 ∞ e -ρ t gx(t),u(t)dt→max include the basic relations of the maximum principle together with additional conditions on the adjoint variables and the Hamiltonian...

This paper (motivated by recent works on optimization of long-term economic growth)suggests some further developments in the theory of first-order necessary optimality conditionsfor problems of optimal control with infinite time horizons. We describe an approximationtechnique involving auxiliary finite-horizon optimal control problems and useit to...

We provide steps towards a welfare analysis of a two-country endogenous growth modelwhere a relatively small follower absorbs part of the knowledge generated in the leadingcountry. To solve a suitably defined dynamic optimisation problem an appropriate versionof the Pontryagin maximum principle is developed. The properties of optimal controls andth...

In this paper a class of nonlinear infinite horizon optimal control problems arising in mathematical economics is considered. First order necessary optimality conditions in a form of the Pontryagin maximum principle are developed together with some extra conditions on the adjoint function and the behaviour of the Hamiltonian at the infinity. In som...

A generalization of Daverman’s problem on the embedding of S n -like compacta into the Euclidean space is formulated. It is shown that for an arbitrary natural number d a number n=n(d) can be found such that an arbitrary S n -like compactum can be embedded into the space ℝ 2n-d . The proof makes use of a construction of L. S. Pontryagin [C. R. Acad...

Abstract In this paper we investigate a class of nonlinear infinite horizon optimal control problems arising in mathematical economics in consideration of economic growth problems and problems of innovations dynamics. First order necessary optimality conditions in a form of the Pontryagin maximum,principle are developed together with some extra con...

Without Abstract

We present the complete investigation of the degeneracy phenomenon. We obtain the new version of Pontryagin's maximum principle for problems with state constraints. This maximum principle contains some additional information about the behavior of the Hamiltonian (maximum function): the new jump conditions at the endtimes. It turns out that these ju...

In this paper we study the degeneracy phenomenon in optimal control problems with state constraints. It is shown that this phenomenon occurs because of the incompleteness of the standard variants of Pontryagin's maximum principle for problems with state constraints. A new maximum principle containing additional information about the behavior of the...

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 Pontryagin's maximum principle for problems with state constraints. The new maximum principle containing some additional information about the...

Properties of semicontinuous multivalued mappings are studied that are analogous to properties of semicontinuous single-valued functions. Theorems are proved on monotone approximation of semicontinuous multivalued mappings by continuous ones, and a theorem is proved on separating a lower semicontinuous multivalued mapping from an upper semicontinuo...