Nonlinear Programming: A Unified Approach

The high order maximal principle (HMP) which was announced in [11] is a generalization of the familiar Pontryagin maximal principle. By using the higher derivatives of a large class of control vari...

The High Order Maximal Principle and Its Application to Singular Extremals

In dynamic models of energy allocation, assimila ted energy is alloc ated to reproduction, somatic growth , maintenance or storage, and the allocation pattern can change with age . The expected evolutionary outcome is an optimal allocation patte rn, but this depends on the environment experienced during the evolutiona ry pro cess and on the fitness costs and benefits incurred by allocating resources in dif ferent ways. Here we review existing treatments

Dynamic models of energy allocation and investment

The principle of vibrational control is presented. This principle consists in the introduction of such vibrations (with zero mean value) of the dynamic system parameters which modify the properties of the system in a desired manner. Unlike conventional control principles, vibrational control does not require measurements of deviations and disturbances. Classes of dynamic systems are described where the vibrational control is effective for ensuring:. 1) the stability, 2) the desired properties of the transient motions, and 3) the desired properties of the dynamic system response to the external disturbance. Applications of the vibrational control are also considered which include method of alternating-gradient focusing in particle accelerators and vibrational stabilization of glow discharge in powerful continuous gas lasers. Vibrational control principle seems to be useful in cases where feedback and feedforward principles cannot be applied.

Principle of vibrational control: Theory and applications

The main results of this paper are two-fold. The first, Theorem 1, is a generalization of the work of Chow and others concerning the set of locally accessible points of a nonlinear control system. It is shown that under quite general conditions, this set lies on a surface in state space and has a nonemptyinterior in the relative topology of that surface.The second result, Theorem 3, generalizes the bang-bang theorem to nonlinear control systems using higher order control variations as developed by Kelley and others. As a corollary we obtain Halkin’s bang-bang theorem for a linear piecewise analytic control system.

A Generalization of Chow’s Theorem and the Bang-Bang Theorem to Nonlinear Control Problems

This is a survey paper. First, we review various methods of evaluating singular controls. An original proof of Kelley’s second order necessary conditions for optimality is given. High order optimality conditions are obtained by the method of Kopp and Moyer. The latter method is generalized to multidimensional singular controls (Goh’s result). A method of transformations in state space is presented (Kelley, Gurman). A method for investigating singular controls with the aid of a bundle of variations is described. A survey is made of the results which were obtained with the aid of a bundle of variations and matrix impulses in problems with closed control regions. The problem of joining extremals is singled out. The junction of singular and nonsingular extremals is considered. A survey of necessary conditions for optimality of nonsingular controls is presented. The cases of open and closed control regions are considered separately.

High Order Necessary Conditions for Optimality

The qualitative theory of optimal processes

Papers on optimal, control theory, reviewed in Referativnyi Zhurnal “Matematika” during 1971–1975, are surveyed.

Mathematical theory of optimal control

A new approach to the synthesis problem of optimal controls of feedback type is considered. Algorithms of operating optimal controllers which are able to calculate values of optimal feedbacks during each particular control process in real time are described. The algorithm of operating the controller of the first type (continuous controller) is based on solving special (defining) equations in real time using the Newton method. For operating the optimal controller of the second type (discrete controller) the dual method of linear programming is used. In control problems for systems under uncertainty, a nonstochastic model is used, around which guaranteed results are obtained. Under incomplete and inexact information, problems of optimal observation and identification are introduced. Algorithms of operating the optimal estimator and identifier which are able to construct estimates of available information required for operating the optimal output controller in real time are described. Results are illustrated by optimization problems of the fourth order control system.

On the Synthesis Problem for Optimal Control Systems

A model of optimal control for discrete systems and the historical development of the discrete maximum principle are considered. The paper deals with local optimality conditions of the first order, e.g. with a local maximum-principle and a quasi-maximum principle. Furthermore, optimality conditions of higher order, e. g. an optimality condition for singular controls are given.

F. M. Kirillova

Papers

High Order Necessary Conditions for Optimality

The qualitative theory of optimal processes

Mathematical theory of optimal control

On the Synthesis Problem for Optimal Control Systems

On discrete maximum principle