On the existence of sporadically catching-up optimal solutions for infinite-horizon optimal control problems

In this paper, we extend the existence theory of Brock and Haurie concerning the existence of sporadically catching-up optimal solutions for autonomous, infinite-horizon optimal control problems. This notion of optimality is one of a hierarchy of types of optimality that have appeared in the literature to deal with optimal control problems whose cost functionals, described by an improper integral, either diverge or are unbounded below. Our results rely on the now classical convexity and seminormality hypotheses due to Cesari and are weaker than those assumed in the work of Brock and Haurie. An example is presented where our results are applicable, but those of the above-mentioned authors do not.This research forms part of the author's doctoral dissertation, written at the University of Delaware, Newark, Delaware, under the supervision of Professor T. S. Angell.     

A Carathéodory-Hamilton-Jacobi theory for infinite-horizon optimal control problems

In this paper, we extend Carathéodory's concept of equivalent variational problems to infinite-horizon optimal control problems. In such a setting, the usual concept of a minimum need not exist, and we therefore consider a weaker definition of optimality, known as catching up optimality. The extension presented here leads us to a Hamilton-Jacobi theory for infinite-horizon optimal control problems that closely parallels the classical work of Carathéodory as well as providing sufficient conditions for optimality. Finally, we show that the results given here subsume several previously known results as a special case.This research forms part of the author's doctoral dissertation, written at the University of Delaware, Newark, Delaware, under the supervision of Professor T. S. Angell.     

Nonconvex and relaxed infinite-horizon optimal control problems

In this paper, we consider the Lagrange problem of optimal control defined on an unbounded time interval in which the traditional convexity hypotheses are not met. Models of this form have been introduced into the economics literature to investigate the exploitation of a renewable resource and to treat various aspects of continuous-time investment. An additional distinguishing feature in the models considered is that we do not assume a priori that the objective functional (described by an improper integral) is finite, and so we are led to consider the weaker notions of overtaking and weakly overtaking optimality. To treat these models, we introduce a relaxed optimal control problem through the introduction of chattering controls. This leads us naturally to consider the relationship between the original problem and the convexified relaxed problem. In particular, we show that the relaxed problem may be viewed as a limiting case for the original problem. We also present several examples demonstrating the applicability of our results.     

Existence and global asymptotic stability of optimal trajectories for a class of infinite-horizon,nonconvex systems

Sufficient conditions for the existence of optimal trajectories and for the global asymptotic stability of these trajectories are given for a class of nonconvex and nonautonomous systems controlled over an infinite-time horizon. The concept ofG-supported trajectory is introduced. It is shown that, under some assumptions, aG-supported trajectory is overtaking and is globally asymptotically stable. The concept of overtaking trajectory has been previously defined as a notion of optimality on an infinite-time domain. For autonomous systems, under weaker conditions, one guarantees the existence of weakly overtaking trajectories. Finally, it is shown howG-supported trajectories can be obtained, and an application to the study of a pre-predator ecosystem optimally harvested is sketched.This research has been partially supported by the Canada Council, Grant No. S.741122X2, and by the Programme FCAC de la DGES, Ministère de l'Education du Québec, Québec, Canada.     

Existence theorems in optimal control problems without convexity assumptions

A correction in the statement of Proposition 4.1 of Ref. 1 is given.     

Existence theorems in optimal control problems without convexity assumptions

We give existence theorems of solutions for Lagrange and Bolza problems of optimal control. These results are obtained without convexity assumptions on the cost function with respect to the control variable. We extend a Cesari's theorem to cost functions which are nonlinear with respect to the space variable and to problems which are governed by a differential inclusion. Moreover, we consider the case where the control variable belongs to a space of measurable functions and the case where this variable belongs to a Lebesgue space.     

Existence of optimal controls for the diffusion equation

The existence is considered of a boundary control which drives a system governed by the one-dimensional diffusion equation from the zero state to a given final state, and at the same time minimizes a given functional. The problem is first modified to one in which the minimum is sought of a functional defined on a set of Radon measures. The existence of a minimizing measure is demonstrated, and it is shown that this measure may be approximated by a piecewise constant control. Finally, conditions are given under which a minimizing measurable control exists for the unmodified problem.     

On the existence of catching-up optimal solutions for Lagrange problems defined on unbounded intervals

In this paper, we are concerned with the question of the existence of optimal solutions for infinite-horizon optimal control problems of Lagrange type. In such problems, the objective or cost functional is described by an improper integral. As dictated by applications arising in mathematical economics, we do nota priori assume that this improper integral converges. This leads us to consider a weaker type of optimality, known as catching-up optimality. The results presented here utilize the classical convexity and seminormality conditions typically imposed in the existence theory for the case of finite intervals. These conditions are significantly weaker than those imposed by other authors; as a consequence, their existence results are contained as special cases of the results presented here. The method of proof utilizes the Carathéodory-Hamilton-Jacobi theory previously developed by the author for infinite-horizon optimal control problems.This research forms part of the author's doctoral dissertation written at the University of Delaware, Newark, Delaware under the supervision of Professor T. S. Angell.     

Uniformly overtaking and weakly overtaking optimal solutions in infinite-horizon optimal control: When optimal solutions are agreeable

In this paper, we investigate the relationship between two classes of optimality which have arisen in the study of dynamic optimization problems defined on an infinite-time domain. We utilize an optimal control framework to discuss our results. In particular, we establish relationships between limiting objective functional type optimality concepts, commonly known as overtaking optimality and weakly overtaking optimality, and the finite-horizon solution concepts of decision-horizon optimality and agreeable plans. Our results show that both classes of optimality are implied by corresponding uniform limiting objective functional type optimality concepts, referred to here as uniformly overtaking optimality and uniformly weakly overtaking optimality. This observation permits us to extract sufficient conditions for optimality from known sufficient conditions for overtaking and weakly overtaking optimality by strengthening their hypotheses. These results take the form of a strengthened maximum principle. Examples are given to show that the hypotheses of these results can be realized.This research was supported by the National Science Foundation, Grant No. DMS-87-00706, and by the Southern Illinois University at Carbondale, Summer Research Fellowship Program.     

Existence of solutions for a class of nonconcave infinite horizon optimal control problems

In this article, we establish the existence of optimal solutions for a large class of nonconvex infinite horizon discrete-time optimal control problems. This class contains optimal control problems arising in economic dynamics which describe a model with nonconcave utility functions representing the preferences of the planner.     

Necessary conditions for infinite-horizon Volterra optimal control problems with time delay

Email: csfvega{at}dm.uba.ar Received on August 17, 2006; Accepted on September 8, 2007 Necessary conditions are proved for optimal control problemsinvolving an infinite horizon and terminal conditions at infinitywhose states are governed by Volterra integral equations withnon-linear time delay.     

An on-line procedure in discounted infinite-horizon stochastic optimal control

We consider the discrete-time, infinite-horizon optimal control problem with discounted cost. We propose a test to detect forecast/planning horizons, and derive an on-line procedure for solving the original problem.     

On some elliptic optimal control problems with state constraints

In this paper optimality conditions will be derived for elliptic optimal control problems with a restriction on the state or on the gradient of the state. Essential tools are the method of transposition and generalized trace theorems and green's formulas from the theory of elliptic differential equations.     

Existence of solutions of a thermoviscoplastic model and associated optimal control problems

A quasistatic, thermoviscoplastic model at small strains with linear kinematic hardening, von Mises yield condition and mixed boundary conditions is considered. The existence of a unique weak solution is proved by means of a fixed-point argument, and by employing maximal parabolic regularity theory. The weak continuity of the solution operator is also shown. As an application, the existence of a global minimizer of a class of optimal control problems is proved.     

Decision and forecast horizons,agreeable plans,and the maximum principle for infinite horizon control problems

This paper establishes a link between the concepts of optimality used in economic theory for infinite horizon planning models, and the concepts of decision and forecast horizons used in several areas of Management Science. It is shown that decision and forecast horizons induce an alternate definition of optimality which is stronger than the concept of ‘agreeable plan’ proposed by Hammond. All concepts of optimality share a common property, namely a Principle of Optimality. In an optimal control framework this implies that the maximum principle will be a necessary condition for optimality according to any of these definitions.     

Existence of an optimal control for stochastic systems governed by ito equations

We give existence theorems for stochastic control problems with a lower semicontinuous cost functional and governed by Ito equations. We prove that two formulations of the fundamental problem are equivalent, one involving nonanticipative controls and the other involving (measurable) feedback controls. We then use the concept ofconvergence in distribution to prove existence for the first problem, and hence for the second as well. While our work has certain similarities with a paper of Kushner, our techniques are different and lead to more general results.     

Necessary conditions of Pareto optimality for multiobjective optimal control problems under constraints

This paper studies multiobjective optimal control problems in presence of constraints in the discrete time framework. Both the finite- and infinite-horizon settings are considered. The paper provides necessary conditions of Pareto optimality under lighter smoothness assumptions compared to the previously obtained results. These conditions are given in the form of weak and strong Pontryagin principles which generalize the existing ones. To obtain some of these results, we provide new multiplier rules for multiobjective static optimization problems and new Pontryagin principles for the finite horizon multiobjective optimal control problems.     

Infinite horizon multiobjective optimal control problems in the discrete time case

This article studies multiobjective optimal control problems in the discrete time framework and in the infinite horizon case. The functions appearing in the problems satisfy smoothness conditions. This article generalizes to the multiobjective case results obtained for single-objective optimal control problems in that framework. The dynamics are governed by difference equations or difference inequations. Necessary conditions of Pareto optimality are presented, namely Pontryagin maximum principles in the weak form and in the strong form. Sufficient conditions are also provided. Other notions of Pareto optimality are defined when the infinite series do not necessarily converge.     

On optimal control problems with state constraints

This paper is concerned with necessary conditions for a general optimal control problem developed by Russak and Tan. It is shown that, in most cases, a further relation between the multipliers holds. This result is of interest in particular for the investigation of perturbations of the state constraint.