Connectedness of the set of nondominated outcomes in multicriteria optimization

Sufficient conditions for the set of nondominated outcomes of a multicriteria optimization problem to be connected are given.The author would like to thank Professors G. Leitmann and P. L. Yu for many critical remarks.The contents of this article are taken from the author's doctoral dissertation written at the University of California at Berkeley under the supervision of Professor G. Leitmann.     

Optimal threat strategies in collective bargaining

A model of collective bargaining in differential games, similar to one considered by Leitmann (Ref. 1), is proposed. Optimal threat strategies are then found and compared with Leitmann's solution. Nash-optimal threat strategies are also found and compared with the other two solutions.The author would like to thank Professor G. Leitmann of the University of California for bringing the author's attention to this subject.     

Characterization of nondominated controls in terms of solutions of weighting problems

The problem of continuously dynamic multiobjective optimization, or multiobjective control, is discussed. The concepts of nondominated controls and viable controls are introduced. For a class of nonlinear dynamic systems, the convexity of their controlled Carathéodory trajectories is proved. Based on this convexity, sufficient conditions are given for the solution of a multiobjective control problem to be obtained in terms of solutions of weighting problems.This work was partly done at the Huazhong University of Science and Technology, Wuhan, China. The author is indebted to Professor Chen Ting for his advice. The author wishes to thank Professor G. Leitmann for his favorable comments. The author is also grateful to Ms. Mary S. Linn at the University of Kansas, who helped him improve the English presentation of this paper.     

Leitmann’s direct method for fractional optimization problems

Based on a method introduced by Leitmann [G. Leitmann, A note on absolute extrema of certain integrals, Int. J. Non-Linear Mech. 2 (1967) 55-59], we exhibit exact solutions for some fractional optimization problems of the calculus of variations and optimal control.     

Note on the nonexceptionality of Brunovský's counterexample

When dealing with the time-optimal problem for linear control systems, there may be a difference between optimal open-loop control and corresponding synthesized feedback control, since in the latter case one is led to allow for generalized (Filippov) solutions. In this note, it is shown that the set of two-dimensional linear control systems with a convex polyhedron as control domain, which exhibit such paradoxical behavior (completely characterized by Brunovský), has a nonempty interior, in a natural and appropriately defined topology on the space of all such linear control systems.This work was done while the author was visiting the Department of Mechanical Engineering, University of California, Berkeley, California, supported by a fellowship of the Italian CNR (Consiglio Nazionale delle Ricerche). While keeping full responsibility for the present work, the author wishes to thank Professor G. Leitmann for his invaluable suggestions, comments, and encouragement.     

On bounding the effective conductivity of isotropic composite materials

In this paper inequalities for the effective conductivity of isotropic composite materials are derived. These inequalities depend on several coefficients characterizing the microstructure of composites. The obtained coefficients can be exactly calculated for models of a two-component aggregate of multisized, coated ellipsoidal inclusions, packed to fill all space. As a result, new bounds for effective conductivity, considerably narrower than those of Hashin-Shtrikman, are established for such models of composite materials.     

Linear pursuit-evasion games with bounded state spaces

The problem of linear pursuit-evasion games with bounded state spaces is considered. Some sufficient conditions for optimality are established, and an example is given.This research was carried out while the author was a Visiting Associate Research Engineer at the University of California at Berkeley. The research was supported by the Office of Naval Research, Grant No. N00014-69-A-0200-1012. The author would like to express his gratitude to Professor G. Leitmann for discussions and for making possible his visit at Berkeley.     

Control synthesis for discrete time systems with control and state bounds in the presence of disturbances

The problem of control in the presence of unknown but limited disturbance for a discrete-time linear system with polyhedral input and state bounds is investigated. Two problems are considered: that of reaching an assigned target set in the state space; and that of keeping the state in a given region using the available controls. In both cases, a solution is given via linear programming. A computational procedure for the control synthesis is proposed which can be implemented to obtain a feedback control.The author thanks Professor G. Leitmann for his helpful suggestions.     

Delay-independent stabilization of uncertain linear systems of neutral type

In this paper, we show that a sufficient condition for the delay-independent stabilizability of linear delay systems, which had been obtained by Amemiya et al., is also valid for linear neutral systems with measurable state variables by a new differential-difference inequality.The authors express their appreciation to Professor G. Leitmann for his useful comments.     

Minimax approach to structural optimization problems

The problem of determining the shape of an elastic body which is optimal for a whole class of loads is formulated. Its general solution scheme, based on a minimax approach, is indicated. The optimization problem of an elastic beam is studied for both the simply supported case and the cantilevered case, and some specific properties of the optimal shape are determined.The author is grateful to Professor A. I. Lurie, Professor F. L. Chernousko, and Professor G. S. Shapiro for useful discussions of the results.     

Finite-dimensional perturbations of Volterra operators

This paper is a review by the author of his dissertation, presented in competition for the academic degree of Doctor of Physicomathematical Sciences. This dissertation was defended on December 20, 1973 at the Scientific Council of the Institute of Mathematics, Siberian Branch, Academy of Sciences of the USSR. The examining committee consisted of Professor A. G. Kostyuchenko, Doctor of Physicomathematical Sciences; Professor A. F. Leont'ev, Corresponding Member of the Academy of Sciences of the USSR; and Professor M. K. Fage, Doctor of Physicomathematical Sciences.Translated from Matematicheskie Zametki, Vol. 16, No. 4, pp. 669–680, October, 1974.     

A discrete method of optimal control based upon the cell state space concept

A discrete method of optimal control is proposed in this paper. The continuum state space of a system is discretized into a cell state space, and the cost function is discretized in a similar manner. Assuming intervalwise constant controls and using a finite set of admissible control levels (u) and a finite set of admissible time intervals (), the motion of the system under all possible interval controls (u, ) can then be expressed in terms of a family of cell-to-cell mappings. The proposed method extracts the optimal control results from these mappings by a systematic search, culminating in the construction of a discrete optimal control table.The possibility of expressing the optimal control results in the form of a control table seems to give this method a means to make systems real-time controllable.Dedicated to G. LeitmannThe material is based upon work supported by the National Science Foundation under Grant No. MEA-82-17471. The author is also indebted to Professor G. Leitmann for his many helpful comments.     

On existence of optimal controls

An optimal control is shown to exist for a system when the Hamiltonian is a strictly convex function of the control. It is proven that a system satisfying this condition must have state equations that are linear in the control and a cost functional whose integrand is strictly convex in the control.The authors wish to thank Professor G. Leitmann for fruitful discussion of this problem.     

Exact relations for effective tensors of composites: Necessary conditions and sufficient conditions

Typically the elastic and electrical properties of composite materials are strongly microstructure dependent. So it comes as a nice surprise to come across exact formulae for effective moduli that are universally valid no matter what the microstructure. Such exact formulae provide useful benchmarks for testing numerical and actual experimental data and for evaluating the merit of various approximation schemes. They can also be regarded as fundamental invariances existing in a given physical context. Classic examples include Hill's formulae for the effective bulk modulus of a two‐phase mixture when the phases have equal shear moduli, Levin's formulae linking the effective thermal expansion coefficient and effective bulk modulus of two‐phase mixtures, and Dykhne's result for the effective conductivity of an isotropic two‐dimensional polycrystalline material. Here we present a systematic theory of exact relations embracing the known exact relations and establishing new ones. The search for exact relations is reduced to a search for matrix subspaces having a structure of special Jordan algebras. One of many new exact relations is for the effective shear modulus of a class of three‐dimensional polycrystalline materials. We present complete lists of exact relations for three‐dimensional thermoelectricity and for three‐dimensional thermopiezoelectric composites that include all exact relations for elasticity, thermoelasticity, and piezoelectricity as particular cases. © 2000 John Wiley & Sons, Inc.     

Hierarchical multi-objective decision systems for general resource allocation problems

We consider optimization methods for hierarchical power-decentralized systems composed of a coordinating central system and plural semi-autonomous local systems in the lower level, each of which possesses a decision making unit. Such a decentralized system where both central and local systems possess their own objective function and decision variables is a multi-objective system. The central system allocates resources so as to optimize its own objective, while the local systems optimize their own objectives using the given resources. The lower level composes a multi-objective programming problem, where local decision makers minimize a vector objective function in cooperation. Thus, the lower level generates a set of noninferior solutions, parametric with respect to the given resources. The central decision maker, then, parametric with respect to the given resources. The central decision maker, then, chooses an optimal resource allocation and the best corresponding noninferior solution from among a set of resource-parametric noninferior solutions. A computational method is obtained based on parametric nonlinear mathematical programming using directional derivatives. This paper is concerned with a combined theory for the multi-objective decision problem and the general resource allocation problem.The authors are indebted to Professor G. Leitmann for his valuable comments and suggestions.     

A class of trilinear differential games

This paper characterizes a class ofN-person, general sum differential games for which the optimal strategies only depend upon remaining playing time. Such strategies can be easily characterized and determined, and the optimal play can be easily analyzed.We acknowledge the helpful comments of G. Leitmann and an anonymous referee.     

Differential games without pure strategy saddle-point solutions

In some two-player, zero-sum differential games, pure strategy saddle-point solutions do not exist. For such games, the concept of a minmax strategy is examined, and sufficient conditions for a control to be a minmax control are presented. Both the open-loop and the closed-loop cases are considered.The research was partially supported by ONR under Contract No. N00014-69-A-0200-12. An earlier version of this paper was presented at the Eleventh Annual Allerton Conference on Circuit and System Theory, Monticello, Illinois, 1973.The author wishes to acknowledge his many valuable discussions of this problem with Professor G. Leitmann and also to thank one of the reviewers for his suggestions for simplifying the proof of Theorem 2.1.     

Impulsive optimal control with finite or infinite time horizon

We consider a dynamical system subjected to feedback optimal control in such a way that the evolution of the state exhibits both sudden jumps and continuous changes. Previously obtained necessary conditions (Ref. 1) for such impulsive optimal feedback controls are generalized to admit the case of infinite time horizon; this generalization permits application to a wider class of problems. The results are illustrated by application to a version of the innkeeper's problem.Dedicated to G. Leitmann     

Some fundamental qualifications of optimal strategies and transition surfaces in differential games

At each point of a regular region of a differential game, there are two tangent cones which are complementary to each other. The two vectograms defined by the pair of pure optimal strategies must be contained separately in the two tangent cones. The velocity vector resulting from the selection of the optimal strategies consequently must represent a semipermeable direction. These conditions, which reveal a fundamental separating property of the optimal velocity vector, are weaker than that of Isaac's main equation. Moreover they hold even on singular surfaces in the regular regions. The conditions also reveal a necessary condition for a differential game to have a regular region. Each isovalued surface in the regular region is essentially a semipermeable surface. A transition surface arises only when the resulting directed isovalued surface of the previous optimal strategies cannot have a smooth semipermeable extension at the singular surface. This observation yields a necessary condition for a transition surface to occur. The localization of transition surfaces is then possible. Finally, a jump and smooth condition of the isovalued surfaces is given.This research was supported in part by the Systems Analysis Program, The University of Rochester, under the Bureau of Naval Personnel Contract No. N00022-70-C-0076. It was also supported in part by the Center for Naval Analyses of the University of Rochester. The author is grateful to Professors R. Isaacs, M. Freimer, and H. Gerber for their helpful discussion during the research. He is especially grateful to Professor G. Leitmann, University of California at Berkeley, for his very helpful comments and suggestions.     

A dual extremum principle for a population equation

A dual extremum principle for the Verhulst-Pearl population equation is constructed using a complementary variational technique. The dual formulation utilizes a minimum principle recently developed by Leitmann to convert the functional optimization problem into a parameter optimization problem.This research was supported in part by NASA Grant No. NGR-36-010-024. The first author would like to thank Dr. W. Stadler, Department of Mechanical Engineering, University of California, Berkeley, for his valuable suggestions.