Geometric methods for nonlinear optimal control problems

It is the purpose of this paper to develop and present new approaches to optimal control problems for which the state evolution equation is nonlinear. For bilinear systems in which the evolution equation is right invariant, it is possible to use ideas from differential geometry and Lie theory to obtain explicit closed-form solutions.The author wishes to thank Professor A. Krener for many stimulating discussions and in particular for suggesting Theorem 3.3. Also, special thanks are due to the author's thesis advisor Professor R. W. Brockett under whose direction most of the research was done. Finally, the author thanks two anonymous referees for suggestions which have improved the exposition.     

Existence theorems for optimization problems concerning linear,hyperbolic partial differential equations without convexity conditions

Integral representations are obtained for solutions of a Darboux problem in a rectangle and used to prove Neustadt-type existence theorems for optimal control problems with trajectories satisfying linear, hyperbolic partial differential equations with Darboux-type boundary data. The proof bears on the fact that, in this situation, for each generalized solution, there is a usual solution where the functional takes the same value.This work was done in the framework of Research Project AFOSR-69-1662. The author is greatly indebted to Professor L. Cesari for his valuable guidance and constant encouragement during the writing of this paper.     

Duality Theorem for a Three-Phase Partition Problem

In some nonlinear diffusive phenomena, the systems have three or more stable states. Sternberg and Zeimer established the existence of minimal solutions for the problem of partitioning a certain domain Ω⊂ℝ² into three subdomains having least interfacial area. Ikota and Yanagida investigated stability and instability for stationary curves with one triple junction and for stationary binary-tree type interfaces. In this paper, we introduce a new concept of separation of three convex sets by a triangle, define a dual problem to the three-phase partition problem, and present a duality theorem. The author thanks Professor F. Giannessi for valuable comments, especially on Gale and Klee-type separation theorems. This research was partially supported by Kyushu University 21st Century COE Program (Development of Dynamic Mathematics with High Functionality) and by the Grant-in-Aid for General Scientific Research from the Japan Society for the Promotion of Science 14340037.     

On the asymptotic order of accuracy of Tikhonov regularization

In this paper, we obtain a necessary and sufficient condition for a system governed by Dirichlet and Neumann problems for a self-adjoint elliptic operator with an infinite number of variables to have an optimal control of the distributed type which is characterized by a set of inequalities.The authors are indebted to Professor J. L. Lions, President of IRIA/LABORIA, Paris, France, and Professor Jean Pierre Aubin, University of Paris IX, Dauphine, France, for their valuable comments, and also to the referees for their constructive suggestions.     

Existence of solutions to second-order nonlinear discrete elliptic equations

In this paper, we consider a boundary value problem (BVP) for second-order nonlinear partial difference equations on finite lattice domains. Some conditions are established that ensure existence and uniqueness of solutions to the BVP under consideration.     

Single-machine ready times scheduling with group technology and proportional linear deterioration

Scheduling research has increasingly taken the concept of deterioration into consideration. In this paper, we study a single machine group scheduling problem with deterioration effect, where the jobs are already put into groups, before any optimization. We assume that the actual processing times of jobs are increasing functions of their starting times, i.e., the job processing times are described by a function which is proportional to a linear function of time. The setup times of groups are assumed to be fixed and known. For some special cases of minimizing the makespan with ready times of the jobs, we show that the problem can be solved in polynomial time for the proposed model. For the general case, a heuristic algorithm is proposed, and the computational experiments show that the performance of the heuristic is fairly accurately in obtaining near-optimal solutions. The results imply that the average percentage error of the proposed heuristic algorithm from optimal solutions is less than 3%.     

Existence of solutions for a vector variational inequality: An extension of the Hartmann-Stampacchia theorem

A vector variational inequality is studied. The paper deals with existence theorems for solutions under convexity assumptions and without convexity assumptions.This research was partially supported by the Italian Research Council (CNR), Group for Functional Analysis and Applications (GNAFA), and was carried out while the author was Visiting Professor at the Department of Mathematics, University of Pisa, September–November, 1989.     

Implicit Function Theorems for Nondifferentiable Mappings

Sufficient conditions are given for a mapping to be γ-G inverse differentiable. Constrained implicit function theorems for γ-G inverse differentiable mappings are obtained, where the constraint is taken to be either a closed convex cone or a closed subset. A theorem without assuming the γ-G inverse differentiability in a finite-dimensional space is also presented. Communicated by F. A. Potra The author thanks the referees for valuable suggestions concerning the presentation of this paper. He also thanks Professor J. R. L. Webb and Dr. M. French for help.     

Maximal vectors and multi-objective optimization

Maximal vector andweak-maximal vector are the two basic notions underlying the various broader definitions (like efficiency, admissibility, vector maximum, noninferiority, Pareto's optimum, etc.) for optimal solutions of multi-objective optimization problems. Moreover, the understanding and characterization of maximal and weak-maximal vectors on the space of index vectors (vectors of values of the multiple objective functions) is fundamental and useful to the understanding and characterization of Pareto-optimal and weak-optimal solutions on the space of solutions.This paper is concerned with various characterizations of maximal and weak-maximal vectors in a general subset of the EuclideanN-space, and with necessary conditions for Pareto-optimal and weak-optimal solutions to a generalN-objective optimization problem having inequality, equality, and open-set constraints on then-space. A geometric method is described; the validity of scalarization by linear combination is studied, and weak conditioning by directional convexity is considered; local properties and a fundamental necessary condition are given. A necessary and sufficient condition for maximal vectors in a simplex or a polyhedral cone is derived. Necessary conditions for Pareto-optimal and weak-optimal solutions are given in terms of Lagrange multipliers, linearly independent gradients, Jacobian and Gramian matrices, and Jacobian determinants.Several advantages in approaching the multi-objective optimization problem in two steps (investigate optimal index vectors on the space of index vectors first, and study optimal solutions on the specific space of solutions next) are demonstrated in this paper.This work was supported by the National Science Foundation under Grant No. GK-32701.     

Turnpike and Optimal Trajectories in Integral Dynamic Models with Endogenous Delay

Nonlinear optimal control of dynamic systems with endogenous time delays is analyzed. Such systems have important applications and are described by Volterra integral equations with unknowns in the integration limits. The paper focuses on the structure and asymptotic behavior of solutions to several optimization problems with endogenous delay. It is shown that, in certain cases, a special delay trajectory exists and attracts the optimal solution. In economics, such behavior corresponds to the turnpike properties of the optimal lifetime of capital in vintage capital models. The authors thank Professor F. Chernousko for his kind assistance and Professor W. Trotti for a supporting grant from Prairie View A&M University.     

On the uniform boundedness of solutions of contingent equations

It is proved that, under standard conditions (in particular the convexity of the velocity set), the boundedness of the individual solutions of a contingent equation implies the uniform boundedness of the solutions, and consequently the boundedness of the reachable sets. This result is used to obtain an existence theorem in optimal control theory. By means of an example, it is shown that the convexity condition cannot be omitted.     

A variable-metric method for function minimization derived from invariancy to nonlinear scaling

The effect of nonlinearly scaling the objective function on the variable-metric method is investigated, and Broyden's update is modified so that a property of invariancy to the scaling is satisfied. A new three-parameter class of updates is generated, and criteria for an optimal choice of the parameters are given. Numerical experiments compare the performance of a number of algorithms of the resulting class.The author is indebted to Professor S. S. Oren, Economic Engineering Department, Stanford University, Stanford, California, for stimulating discussions during the development of this paper. He also recognizes the financial support by the National Research Council of Italy (CNR) for his stay at Stanford University.     

Existence of Solutions to Some Equilibrium Problems

The concept of a monotone family of functions, which need not be countable, and the solution of an equilibrium problem associated with the family are introduced. A fixed-point theorem is applied to prove the existence of solutions to the problem.¹The authors are thankful to the referees and Professor S. Schaible for valuable suggestions and comments to make the paper in the present form. The first author thanks the Department of Mathematics, University of Queensland for the kind hospitality shown during his stay ²This research was supported in part by grant No. 83470019 from IPM     

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.This work was done in the framework of Research Project AFOSR-71-2122 at the University of Michigan, Ann Arbor, Michigan. The author wishes to thank Professor L. Cesari for helpful discussions during the writing of this paper.     

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.     

On the existence of optimal controls for quasilinear parabolic partial differential equations

A class of systems governed by quasilinear parabolic partial differential equations with first boundary conditions is considered. Existence of solutions for this class of systems and theira priori estimates are established. Further, a theorem on the existence of optimal controls for the corresponding control problem is obtained. Its proof is based on Filippov's implicit functions lemma. The control restraint setU is taken as a measurable multifunction.The authors wish to thank Professor L. Cesari for his most valuable comments and suggestions. In fact, a condition assumed in the original version of this paper was substantially relaxed by him. For details, see Remark 4.1.     

Quadratic convergence of the Iri-Imai algorithm for degenerate linear programming problems

A local analysis of the Iri-Imai algorithm for linear programming is given to demonstrate quadratic convergence under degeneracy. Specifically, we show that the algorithm with an exact line search either terminates after a finite number of iterations yielding a point on the set of optimal solutions or converges quadratically to one of the relative analytic centers of the faces of the set of optimal solutions including vertices. Mostly, the sequence generated falls into one of the optimal vertices, and it is rare that the sequence converges to the relative analytic center of a face whose dimension is greater than or equal to one.This paper is based on Ref. 1.The author thanks Professor Kunio Tanabe of the Institute of Statistical Mathematics for valuable comments as well as stimulating discussions.     

飞机空中加油问题的研究

讨论了飞机的空中加油问题,通过比较不同方案下主机的最大作战半径,找到了辅机给主机空中加油的最佳方案,分析了存在多个基地时各基地的位置及辅机在各基地的数量,研究了二维坐标下主机的最佳飞行路径,给出了解决问题的办法.整个思路由浅入深,通过考虑辅机数目n分别为1,2,3,4的特殊情况进而探讨一般的规律,包括辅机在基地无加油时间和有加油时间以及主机在空中盘旋等不同假设条件下的复杂情况,并将统计出的数据列表对比,获得了更直观的认识.在最后进行了展望,希望结合实际中的飞机空中加油问题综合考虑风力、风向等随机因素的影响,拓展到3维空间的更复杂模型中,使飞机的空间加油问题变得更有实际意义.     

A dynamic programming approach to the maximum principle of distributed-parameter systems

This paper provides a dynamic programming approach to the maximum principle for the optimal control of systems with distributed parameters. The process of the systems under consideration is governed by a partial differential equation.This paper is based on Chapter 2 of the author's PhD Thesis under the supervision of Professor S. E. Dreyfus to whom the author wishes to express his appreciation.     

A note on “Quality investment and inspection policy in a supplier-manufacturer supply chain”

In a recent paper, Hsieh and Liu (2010) consider quality investment and inspection strategies of one supplier and one manufacturer. They propose solutions of four non-cooperative game models with different degrees of information available about the players’ inspection sampling rates and quality investments. For the two most comprehensive scenarios, we show that these (interior) solutions do not always represent the optimal solutions, as the boundary solutions lead to considerably higher profits. Furthermore, we propose modified algorithms providing the optimal solutions by including the boundary solutions into the consideration.