Existence theorems for lagrange control problems with unbounded time domain

Existence theorems are proved for usual Lagrange control systems, in which the time domain is unbounded. As usual in Lagrange problems, the cost functional is an improper integral, the state equation is a system of ordinary differential equations, with assigned boundary conditions, and constraints may be imposed on the values of the state and control variables. It is shown that the boundary conditions at infinity require a particular analysis. Problems of this form can be found in econometrics (e.g., infinite-horizon economic models) and operations research (e.g., search problems).The author wishes to thank Professor L. Cesari for his many helpful comments and assistance in the preparation of this paper. This work was sponsored by the United States Air Force under Grants Nos. AF-AFOSR-69-1767-A and AFOSR-69-1662.     

Extremum principles for optimal control in normed linear spaces

Extremum principles intended for use in optimal control are derived in the form of necessary conditions and sufficient conditions, formulated in general normed linear spaces. The method of application is illustrated by several examples involving optimal control problems, mathematical programming problems, lumped-parameter systems, and distributed-parameter systems. The basic theorems provide a unified approach which is applicable to a wide variety of problems in open-loop optimal control.     

A second-order Lagrangian condition for restricted control problems

We derive a second-order condition for optimal control problems defined by ordinary differential equations with time-varyingconvex restrictions on controls and with endpoint constraints. This condition generalizes theaccessory minimum problem of the calculus of variations.This research was partially supported by Grant No. MCS 76-06756 of the National Science Foundation.     

The bang-bang-bang problem revisited

The solution given by Isaacs to the bang-bang-bang problem is shown to be a solution in aweak sense only. A solution in a stronger sense is demonstrated involving the notions of extended closed-loop strategies and of extended value-function. The bang-bang-bang singular surface is interpreted in a new way.The author is grateful to Professor J. V. Breakwell for his criticism and valuable remarks.     

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.     

Periodic solution of a higher dimensional ecological system

The main purpose of this article is to study the periodicity of a Lotka-Volterra''s competition system with feedback controls. Some new and interesting sufficient conditions are obtained for the global existence of positive periodic solutions. Our method is base on combining matrix''s spectral theory and inequality $|x(t)|\leq x(t_{0})+\int_{0}^{\omega }|\dot{x}(t)|{\rm d}t$. Some examples and their simulations show the feasibility of our main result.     

Half Space and Quarter Space Dirichlet Problems for the Partial Differential Equation

A customary, heuristic, method, by which the Poisson integral formula for the Dirichlet problem, for the half space, for Laplace's equation is obtained, involves Green's function, and Kelvin's method of images. Although this heuristic method leads one to guess the correct result, this Poisson formula still has to be verified directly, independently of the method by which it was arrived at, in order to be absolutely certain that a solution of the Dirichlet problem for the half space, for Laplace's equation, has been actually obtained. A similar heuristic method, as seems to be generally known, could be followed in solving the Dirichlet problem, for the half space, for the equation where is a real constant. However, in Part 1, a different, labor-saving, method is used to study Dirichlet problems for the equation. This method is essentially based on what Hadamard called the method of descent. Indeed, it is shown that he who has solved the half space Dirichlet problem for Laplace's equation has already solved the half space Dirichlet problem for the equation In Part 2, the solution formula for the quarter space Dirichlet problem for Laplace's equation is obtained from the Poisson integral formula for the half space Dirichlet problem for Laplace's equation. A representation theorem for harmonic functions in the quarter space is deduced. The method of descent is used, in Part 3, to obtain the solution formula for the quarter space Dirichlet problem for the equation by means of the solution formula for the quarter space Dirichlet problem for Laplace's equation. So that, indeed, it is also shown that he who has solved the quarter space Dirichlet problem for Laplace's equation has already solved the quarter space Dirichlet problem for the " equation" For the sake of completeness and clarity, and for the convenience of the reader, the appendix, at the end of Part 3, contains a detailed proof that the Poisson integral formula solves the half space Dirichlet problem for Laplace's equation. The Bibliography for Parts 1,2, 3 is to be found at the end of Part 1.     

Ergodic Control of Partially Degenerate Diffusions in a Compact Domain

The problem of ergodic control of a reflecting diffusion in a compact domain is analysed under the condition of partial degeneracy, i.e. when its transition kernel after some time is absolutely continuous with respect to the Lebesgue measure on a part of the state space. Existence of a value function and a “martingale dynamic programming principle” are established by mapping the problem to a discrete time control problem. Implications for existence of optimal controls are derived.     

Hierarchical controls in stochastic manufacturing systems with machines in tandem

This paper presents an asymptotic analysis of hierarchical production planning in a manufacturing system with serial machines that are subject to breakdown and repair, and with convex costs. The machines capacities are modeled as Markov chains. Since the number of parts in the internal buffers between any two machines needs to be non-negative, the problem is inherently a state constrained problem. As the rate of change in machines states approaches infinity, the analysis results in a limiting problem in which the stochastic machines capacity is replaced by the equilibrium mean capacity. A method of “lifting” and “modification” is introduced in order to construct near optimal controls for the original problem by using near optimal controls of the limiting problem. The value function of the original problem is shown to converge to the value function of the limiting problem, and the convergence rate is obtained based on some a priori estimates of the asymptotic behavior of the Markov chains. As a result, an error estimate can be obtained on the near optimality of the controls constructed for the original problem.     

Constrained Controllability of Fractional Dynamical Systems

The article deals with the controllability results for fractional dynamical systems with prescribed controls represented by the fractional integrodifferential equation in finite dimensional spaces. Sufficient conditions for the controllability results of nonlinear fractional dynamical systems are obtained using the contraction mapping principle. Examples are included to illustrate the theory.