Economic Control Problem Research Articles

Optimization problems for switched systems with impulsive control

By using Impulsive Maximum Principal and three stage optimization method, this paper discusses optimization problems for linear impulsive switched systems with hybrid controls, which includes continuous control and impulsive control. The linear quadratic optimization problems without constraints such as optimal hybrid control, optimal stability and optimal switching instants are addressed in detail. These results are applicable to optimal control problems in economics,mechanics,and management.

Using dynamic programming with adaptive grid scheme for optimal control problems in economics

The study of the solutions of dynamic models with optimizing agents has often been limited by a lack of available analytical techniques to explicitly find the global solution paths. On the other hand, the application of numerical techniques such as dynamic programming to find the solution in interesting regions of the state was restricted by the use of fixed grid size techniques. Following Grüne (Numer. Math. 75 (3) (1997) 319; University of Bayreuth, submitted, 2003), in this paper an adaptive grid scheme is used for finding the global solutions of discrete time Hamilton–Jacobi–Bellman equations. Local error estimates are established and an adapting iteration for the discretization of the state space is developed. The advantage of the use of adaptive grid scheme is demonstrated by computing the solutions of one- and two-dimensional economic models which exhibit steep curvature, complicated dynamics due to multiple equilibria, thresholds (Skiba sets) separating domains of attraction and periodic solutions. We consider deterministic and stochastic model variants. The studied examples are from economic growth, investment theory, environmental and resource economics.

Sufficient conditions and sensitivity analysisfor economic control problems

Second‐order sufficient optimality conditions for nonlinear optimal control problems arereviewed. Applications of these conditions to convex‐concave models of economic controlare presented. On the basis of second‐order conditions, recent stability results are discussedthat ensure differentiability of optimal solutions with respect to perturbation parameters inthe system. Methods for computing the sensitivity differentials of optimal solutions areoutlined and numerical results for two economic control problems are given.

In Optimal Control Problem in Economics with Four Linear Controls

An optimal control problem with four linear controls describing a sophisticated concern model is investigated. The numerical solution of this problem by combination of a direct collocation and an indirect multiple shooting method is presented and discussed. The approximation provided by the direct method is used to estimate the switching structure caused by the four controls occurring linearly. The optimal controls have bang-bang subarcs as well as constrained and singular subarcs. The derivation of necessary conditions from optimal control theory is aimed at the subsequent application of an indirect multiple shooting method but is also interesting from a mathematical point of view. Due to the linear occurrence of the controls, the minimum principle leads to a linear programming problem. Therefore, the Karush–Kuhn–Tucker conditions can be used for an optimality check of the solution obtained by the indirect method.

A new route to cyclical strategies in two-dimensional optimal control models

This paper proposes a new route to rationalize cyclical policies in concave two-state variable, economic control problems. The essential ingredients are: positive growth; a (positive) externality of the stock; and sluggish control, such that actual control is the sum over historical changes. Given these conditions, it is amazingly simple to generate stable limit cycles, even for separable models. A simple renewable resource model is used to verify this claim.

An optimal control problem in economics

The first problem in the economics of natural resources is to find the rate at which to extract the resource in order to optimize its value when there are no extraction costs. It is shown that the existence of an optimal extraction path is not guaranteed by a utility function that is merely (strictly) concave, but that the additional requirement of asymptotic nonlinearity will assure the existence of the desired optimum.

PLEM: A computer program for passive learning, stochastic control experiments

Abstract The benefits of using stochastic control methods to solve real world optimal policy problems in economics needs to be demonstrated. A potentially convincing demonstration for any problem is to compare the actual policy settings over time with those which would have been optimal had stochastic control methods been employed. If the optimal policy is more efficient at meeting targets (or meets them more accurately for the same cost) than the actual policy, then the methods are likely to find their way into the toolboxes of managers, planners, and other decision-makers. While the analytical properties of myriad economic control problems are becoming increasingly well-known, and while numerical methods and computational capacities have grown rapidly in recent years, there is a striking lack of software available to solve actual stochastic control problems. This paper reports on the development of a computer program to perform passive learning control on systems for which new information (about system parameters and forecasts of exogenous variables) is available to decision-makers each period. Upon execution the program computes the optimal path and presents a comparison table of the actual and optimal state and control variables over time, as well as the respective values of the objective functions.

PLEM: A Computer Program for Passive Learning Stochastic Control Experiments

Abstract The benefits of using stochastic control methods to solve real world optimal policy problems in economics needs to be demonstrated. A potentially convincing demonstration for any problem is to compare the actual policy settings over time with those which would have been optimal had stochastic control methods been employed. If the optimal policy is more efficient at meeting targets (or meets them more accurately for the same cost) than the actual policy, then the methods are likely to find their way into the toolboxes of managers, planners, and other decision-makers. While the analytical properties of myriad economic control problems are becoming increasingly well-known, and while numerical methods and computational capacities have grown rapidly in recent years, there is a striking lack of software available to solve actual stochastic control problems. This paper reports on the development of a computer program to perform passive learning control on systems for which new information (about system parameters and forecasts of exogenous variables) is available to decision-makers each period. Upon execution the program computes the optimal path and presents a comparison table of the actual and optimal state and control variables over time, as well as the respective values of the objective functions.

Quadratic open-loop optimal control of economic systems

This paper exploits the formal analogy between least-squares estimation and quadratic open-loop control to impose a variety, of restrictions, to gain insight into problems of numeric instability, and to propose indexes of policy effectiveness and the importance of instrument costs to the solution. In addition, it provides an open-loop stochastic control solution with computational advantages in certain situations. Although the emphasis is on control problems in economics, the results are believed to be of general interest.

Optimal maintenance and production rates for a machine: A nonlinear economic control problem

This paper derives qualitative properties (such as monotonicity and limiting behavior) of the optimal policy for a class of nonlinear optimal control problems. The problem structure is characteristic of certain economic investment problems where two control instruments influence the rate of deterioration of a capital good. An example for this is the problem of determining the optimal production and maintenance rates for a machine when the effectiveness of the maintenance expenditures is subject to decreasing returns to scale, and where the marginal deterioration is an increasing function of the production rate. A stability analysis in both of the phase planes state-costate and state-control is carried out and it is shown that the Jacobian determinants in both cases are identical. Also provided is a sensitivity analysis of the equilibrium. Furthermore other applications of the model in the areas of advertising and pollution control models are proposed.

Theory of Value and Some Problems of Economic Control

Basing on the Marxian theory of value and utility, the paper discusses the problem of whether it is possible to evaluate individual enterprises' operation. The evaluations unavoidably reflect not only the results of the staff's activity of an enterprise but also the properties of an economy as a whole. Such evaluations are not needed for the centralized control of long-term intersectoral processes.

Qualitative asymptotic synthesis in simple optimal control problems

Many infinite horizon optimal control problems in economics have scalar state and control variables. Under frequently invoked assumptions, it is sometimes very simple to tell whether the control in such a problem is increasing or decreasing in the state near equilibrium.

The Principle of a Guaranteed Result in Problems of Economic Control

The paper is concerned with applying the principle of the best guaranteed result to control with uncertainties. In these problems, the disturbances affect the set of feasible controls, which is characteristic of economic problems, as well as the performance criterion. The effectiveness of three control techniques is compared: (i) programmed control, (ii) programmed control with additional information, (iii) control with complete information. The comparison is made in terms of a maximal guaranteed estimate of the performance criterion, of actual values of the criterion for all possible disturbances, and of the possibility of satisfying conditions of control feasibility. An antagonistic generalized game is formulated which is related to the control problem; a saddle point is defined. It is found that a saddle point exists in dynamic input-output models with independent disturbances.

Comments on "A nonlinear economic control problem with a linear feedback solution"

In a recent paper by Luenberger[1] it was shown that a special nonlinear economic control problem had a unique global solution expressible in linear feedback form and that the optimal return function was affine. It is shown here that these results may be extended to a much wider class of economic problems.

A nonlinear economic control problem with a linear feedback solution

This paper discusses a special optimal control problem having a solution expressible in linear feedback form. The problem structure is characteristic of certain economic investment problems where the control influences the rate of deterioration of a capital good. An example of an economic problem of this type is the determination of the optimal maintenance policy for a housing unit when the effectiveness of maintenance is subject to decreasing returns to scale. Mathematically, the problem is characterized by a dynamic system that is nonlinear with respect to both the state and the control and by a linear objective function. Both the control and the state are restricted to be nonnegative and to lie below given upper bounds. The problem possesses a unique global optimal solution expressible in linear feedback form. The associated control law is derived from the solution of an auxiliary ordinary differential equation that can be solved backward in time. In addition to the linear control law, the problem possesses other important linearity properties. The closed-loop system is linear; that is, although the system itself is nonlinear, when the optimal control law is substituted for the control, the resulting system is linear with respect to the state. And, the optimal return function is affine; that is, the optimal value that can be attained from a given point is an affine function of the value of the state at that time.

Some characteristic features of optimal control problems in economic theory

This paper formulates the system equations, state and control space constraints, and a criterion functional for an elementary example of a problem in economic growth, and discusses some further interpretation of the underlying economic structure. Several examples are presented to illustrate particular features of control problems in economics; references to futher examples, and to more general work in mathematical economics, are cited.