Lower Closure Theorems Research Articles

On the Existence of a Classical Optimal Solution and of an Almost Strongly Optimal Solution for an Infinite-Horizon Control Problem

We consider an infinite-horizon optimal control problem with the cost functional described either by an integral over an unbounded interval (a Lebesgue integral) or by a limit of integrals (an improper Lebesgue integral). We prove some theorems on the existence of solutions to such problems. The proofs are based on appropriate lower closure theorems and some extensions of Olech’s theorem on the lower semicontinuity of an integral functional; these extensions cover the cases of functionals described by an integral over an unbounded interval and by a limit of integrals.

Infinite-horizon optimal controls for problems governed by a volterra integral equation with state-and-control-dependent discount factor

In this work, we concern ourselves with the existence of optimal solutions to optimal control problems defined on an unbounded time interval with states governed by a nonlinear Volterra integral equation. These results extend both the work of Baum and others in infinite-horizon control of ordinary differential equations as well as the work of Angell concerning integral equations. In addition, we incorporate into the objective functional (described by an improper integral) a discount factor which reflects a hereditary dependence on both state and control. In this manner, we are able to generalize the recent results of Becker, Boyd, and Sung in which they establish an existence theorem in the calculus of variations with objective functionals of the so-called recursive type. Our results are obtained through the use of appropriate lower-closure theorems and compactness conditions. Examples are presented in which the applicability of our results is demonstrated.

Upper semicontinuity of set-valued functions

Various types of upper semcontinuity properties for set-valued functions have been used in the past to obtain closure and lower closure theorems in optimal control theory as well as selection theorems and fixed-point theorems in topology. This paper unifies these various concepts by using semiclosure operators, extended topologies, and lattice theoretic operations and obtains general closure theorems. In addition, analytic criteria are given for this generalized upper semicontinuity. In particular, set-valued functions which are maximal in terms of certain properties (e.g., maximal monotone multifunctions) are shown to be necessarily upper semicontinuous.

On recent existence theorems in the theory of optimization

A condition recently proposed is shown to imply the weak compactness inH1,1 and actually is equivalent to another condition previously proposed by the authors. Once compactness is proved, then existence theorems follow from lower closure theorems also previously proved by the authors, and extended to Pareto problems. The present analysis adds to the recent work of Goodman concerning the equivalence of seminormality conditions with concepts of convex analysis and lattice theory.

Existence Theorems for Pareto Optimization; Multivalued and Banach Space Valued Functionals

Existence theorems are obtained for optimization problems where the cost functional takes values in an ordered Banach space. The order is defined in terms of a closed convex cone in the Banach space; and in this connection, several relevant properties of cones are studied and they are shown to coincide in the finite dimensional case. The notion of a weak (Pareto) extremum of a subset of an ordered Banach space is then introduced. Existence theorems are proved for extrema for Mayer type as well as Lagrange type problems-in a manner analogous to and including those with scalar valued cost. The side conditions are in the form of general operator equations on a class of measurable functions defined on a finite measure space. Needed closure and lower closure theorems are proved. Also, several analytic criteria for lower closure are provided. Before the appendix, several illustrative examples are given. In the appendix, a criterion (different from the one used in main text) is given and proved, for the Pareto optimality of an element.

Lower closure and existence theorems for optimal control problems with infinite horizon

Lower closure theorems are proved for optimal control problems governed by ordinary differential equations for which the interval of definition may be unbounded. One theorem assumes that Cesari's property (Q) holds. Two theorems are proved which do not require property (Q), but assume either a generalized Lipschitz condition or a bound on the controls in an appropriateLp-space. An example shows that these hypotheses can hold without property (Q) holding.

Existence theorems for Pareto optimization; multivalued and Banach space valued functionals

Existence theorems are obtained for optimization problems where the cost functional takes values in an ordered Banach space. The order is defined in terms of a closed convex cone in the Banach space; and in this connection, several relevant properties of cones are studied and they are shown to coincide in the finite dimensional case. The notion of a weak (Pareto) extremum of a subset of an ordered Banach space is then introduced. Existence theorems are proved for extrema for Mayer type as well as Lagrange type problems-in a manner analogous to and including those with scalar valued cost. The side conditions are in the form of general operator equations on a class of measurable functions defined on a finite measure space. Needed closure and lower closure theorems are proved. Also, several analytic criteria for lower closure are provided. Before the appendix, several illustrative examples are given. In the appendix, a criterion (different from the one used in main text) is given and proved, for the Pareto optimality of an element.

Remarks on lower semicontinuity and lower closure

In the context of a recent geometric condition of Cesari, used in the reduction of seminormality requirements in lower closure theorems, this paper shows that the existence of a strongly convergent selection from the sequence of orientor fields, under Kuratowski property (K), is adequate to guarantee lower closure theorems. This generalization is justified through examples. Several related remarks are made.

Orientor field equations in Banach spaces

This paper focuses on the interplay between seminormality requirements and convergence hypotheses on trajectories in the lower closure theorems for orientor field equations. With the use of a weak form of seminormality, called the intermediate property (Q*), we obtain lower closure theorems (and thereby closure theorems) for orientor field equations in Banach spaces, under the assumption of strong convergence of some coordinates of the trajectories, while only weak convergence is assumed in the others. In the Euclidean case, this requirement of property (Q*) is further reduced to mere Kuratowski property (K), under the usual growth-type conditions. Finally, in the appendix, property (Q*) is investigated in detail.

Closure theorems without seminormality conditions

The authors give a variety of conditions under which there is no need to explicity require seminormality conditions in closure and lower closure theorems, and corresponding lower semicontinuity theorems. Both Lipschitz-type conditions and growth-type conditions are taken into consideration in classesL p andL ∞ .

Lower semicontinuity and lower closure theorems without seminormality conditions

Lower semicontinuity and lower closure theorems without seminormality conditions

Existence and Lower Closure Theorems for Abstract Control Problems

A general control problem that includes, among others, the distributed control problem and the optimal control problem for systems governed by ordinary differential equations is considered. Two lower closure theorems and an existence theorem with hypotheses corresponding to those of the lower closure theorems are proved. The first lower closure theorem was first proved by Cesari. The present proof is different and simpler. The second lower closure theorem dispenses with Cesari’s property (Q), but requires a generalized Lipschitz condition to hold.

Closure, Lower Closure, and Semicontinuity Theorems in Optimal Control

First we discuss in detail the concept of lower closure for Lagrange problems, and we extend in various ways previous closure and lower closure theorems. In particular, we show that the present lower closure theorems and related concepts are extensions to Lagrange problems of optimal control of well-known semicontinuity theorems for free problems of the calculus of variations and the related concept of seminormality. Finally, we prove by a new approach that the convexity condition usually requested in lower closure theorems is, in a suitable sense, both a necessary and sufficient condition for lower closure.

Existence theorems for abstract multidimensional control problems

In the present paper, the author discusses an abstract formulation of control problems involving general operators ℒ :S →V, ℳ :S →Y from a Banach spaceS into spaceV andY of vector functions in a fixed domain with components inL p ,p⩾1. For this general formulation, the author states closure theorems, lower closure theorems, and existence theorems for an optimal solution. It is then shown that the problems of control involving Dieudonné-Rashevski partial differential equations previously considered by the author are particular cases of the present formulation. Finally, it is shown by examples that problems of control involving usual partial differential equations, linear or not, as well as other functional relations, can be framed in the present formulation. The present work concerns problems withdistributed controls. Work concerning problems withdistributed as well as boundary controls is forthcoming.