Reduction of delayed optimal control problems to nondelayed problems

Typographical errors are corrected in Ref. 1.     

On optimal control problems with general boundary conditions

It is shown that the necessary optimality conditions for optimal control problems with terminal constraints and with given initial state allow also to obtain in a straightforward way the necessary optimality conditions for problems involving parameters and general (mixed) boundary conditions. In a similar manner, the corresponding numerical algorithms can be adapted to handle this class of optimal control problems.This research was supported in part by the Commission on International Relations, National Academy of Sciences, under Exchange Visitor Program No. P-1-4174.The author is indebted to the anonymous reviewer bringing to his attention Ref. 9 and making him aware of the possible use of generalized inverse notation when formulating the optimality conditions.     

A minimax optimal control problem

A control system x=f(t,x,u) is considered, and a cost functional ess supT ₀tT ₁ G(t, x(t),u(t)) is to be minimized. Necessary conditions for optimality (maximum principle and transversality conditions) are derived. It is also shown that an optimal control is optimal for the corresponding problem on a subinterval of [T ₀,T ₁], if a certain controllability condition is satisfied.     

Structure of minimizers in linear-quadratic Bolza problems of optimal control

In this paper, we study intersections of extremals in a linear-quadratic Bolza problem of optimal control. The structure of the inter-sections is described. We show that this structure implies the semipositive definiteness of the quadratic cost functional. In addition, we derive necessary and sufficient conditions for the existence of minimizers.     

Constrained retarded nonlinear optimal control problems in Banach state space

Necessary conditions are proved for the optimal control of solutions of ordinary and retarded differential equations in a Banach state space, with mixed and pure state restrictions. The treatment includes the possibility of point targets. A generalization of earlier results for finite-dimensional or Hilbert state spaces is obtained.     

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.     

An optimal control problem for systems described by partial differential equations of hyperbolic type with delay

An optimal control problem of the Gourse type with delay is investigated. With a given aim functional, a necessary condition of optimality is formulated and proved in the form of a maximum principle. The proof is based on the reduction of a problem with delay to a problem without delay.The authors thank Prof. G. Leitmann, University of California, Berkeley, for discussions and for his interest in this paper.     

Realistic modelled optimal control problems in aerospace engineering - a challenge to the necessary optimality

A control problem for a hypersonic space vehicle is used to illustrate the need for a generalization of the necessary optimality conditions in the accurate numerical solution of more realistic models for optimal control problems in aerospace engineering.     

Pontryagin's principle for problems with isoperimetric constraints and for problems with inequality terminal constraints

Necessary conditions are derived for optimal control problems subject to isoperimetric constraints and for optimal control problems with inequality constraints at the terminal time. The conditions are derived by transforming the problem into the standard form of optimal control problems and then using Pontryagin's principle.     

Existence theorems in optimal control problems without convexity assumptions

A correction in the statement of Proposition 4.1 of Ref. 1 is given.     

Sufficient conditions for asymptotic optimal control

For a selected family of Lagrange-type control problems involving a nonnegative integral costJ _T (y,u) over the interval [0,T], 0<T<, with system conditions consisting of differential inequalities and/or equalities, the following material is treated: (i) a resumé of relevant necessary conditions and sufficient conditions for a pair (y _T ,u _T ) to minimizeJ _T (y,u); (ii) conditions sufficient for the convergence asT of minimizing pairs (y _T ,u _T ) over [0,T] to a limit pair (y ,u ) over the infinite-time interval [0, ); (iii) conditions sufficient for (y ,u ) to minimize the costJ (y,u) over [0, ); and (iv) conditions sufficient for the optimal cost per unit timeJ _T (y _T ,u _T )/T to have a limit asT.     

Proximal normal analysis approach to optimal control problems in infinite-dimensional spaces

In this paper, we extend the Pontryagin maximum principle and the transversality conditions to a class of optimal control problems for an evolution system of parabolic type through the analysis of proximal normals to the epigraph of suitable value functions. The paper extends previous results of the same authors to nonconvex target situations.This work was supported by MURST of Italy, Fondi 40%, Equazioni di Evoluzione ed Applicazioni Fisico-Matematiche, and Fondi 60%, University of Bari and University of Calabria.     

Optimal control of time-delay systems with distributed parameters

An optimal control problem with a prescribed performance index for parabolic systems with time delays is investigated. A necessary condition for optimality is formulated and proved in the form of a maximum principle. Under additional conditions, the maximum principle gives sufficient conditions for optimality. It is also shown that the optimal control is unique. As an illustration of the theoretical consideration, an analytic solution is obtained for a time-delayed diffusion system.The author wishes to express his deep gratitude to Professors J. M. Sloss and S. Adali for the valuable guidance and constant encouragement during the preparation of this paper.     

An elementary proof of the maximum principle for optimal control problems governed by a Volterra integral equation

An elementary proof of the maximum principle for optimal control problems whose states are governed by Volterra integral equations is given. Our proof is motivated by the work of Michel (Ref. 7) and utilizes only elementary results from analysis and mathematical programming. By appealing to Pontryagin-type perturbations of the controls, the above optimal control problem is effectively reduced to a mathematical programming problem. The results are then obtained by appealing to well-known mathematical programming results.     

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.     

Numerical solution of minimax optimal control problems by multiple shooting technique

This paper presents the application of the multiple shooting technique to minimax optimal control problems (optimal control problems with Chebyshev performance index). A standard transformation is used to convert the minimax problem into an equivalent optimal control problem with state variable inequality constraints. Using this technique, the highly developed theory on the necessary conditions for state-restricted optimal control problems can be applied advantageously. It is shown that, in general, these necessary conditions lead to a boundary-value problem with switching conditions, which can be treated numerically by a special version of the multiple shooting algorithm. The method is tested on the problem of the optimal heating and cooling of a house. This application shows some typical difficulties arising with minimax optimal control problems, i.e., the estimation of the switching structure which is dependent on the parameters of the problem. This difficulty can be overcome by a careful application of a continuity method. Numerical solutions for the example are presented which demonstrate the efficiency of the method proposed.     

On the existence of sporadically catching-up optimal solutions for infinite-horizon optimal control problems

In this paper, we extend the existence theory of Brock and Haurie concerning the existence of sporadically catching-up optimal solutions for autonomous, infinite-horizon optimal control problems. This notion of optimality is one of a hierarchy of types of optimality that have appeared in the literature to deal with optimal control problems whose cost functionals, described by an improper integral, either diverge or are unbounded below. Our results rely on the now classical convexity and seminormality hypotheses due to Cesari and are weaker than those assumed in the work of Brock and Haurie. An example is presented where our results are applicable, but those of the above-mentioned authors do not.This research forms part of the author's doctoral dissertation, written at the University of Delaware, Newark, Delaware, under the supervision of Professor T. S. Angell.     

Application of the method of multipliers to state space constrained optimal control problems with discrete time

The paper deals with convergence conditions of multiplier algorithms for solving optimal control problems with discrete time suggested by J. Bjbvonek in some earlier papers. In this approach the original state space constrained problem is converted into a control-constrained problem by introducing an additional control variable and an equality constraint which is taken into consideration by a multiplier method. Convergence conditions for the multiplier Iteration of global and local nature are given for exact and inexact solution of the subproblems.     

On necessary extremum conditions for finite-dimensional problems with inequality constraints

The finite-dimensional optimization problem with equality and inequality constraints is examined. The case where the classical regularity condition is violated is analyzed. Necessary second-order extremum conditions are obtained that are stronger versions of some available results.     

Augmented penalty functions for delayed control systems

A descent method is given for the numerical solution of delayed optimal control problems with fixed delays by first reducing them to nondelayed problems and then using the technique of augmented penalty functions. The system resulting from the reduction to a nondelayed problem is of higher order than the original system; however, the time is proportionally shorter, and the variational matrices are sparse.