Sufficiency conditions for multiple integral control problems

The construction of multiple integrals which are independent, in the sense that they depend solely on the values of their integrands on the boundary of the domain of integration is described. These integrals are applied to the derivation of a sufficiency condition for multiple integral optimal control problems.This research was supported in part by NSF Grant No. GP-32830.     

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.     

A lower closure theorem for abstract control problems withL p-bounded controls

A lower closure theorem for an abstract control problem is proved. The functional isJ(,u)= _G f ⁰(t, (M)(t),u(t))dt and the state equations areN(t)=f(t, (M)(t),u(t)). It is shown that, if {( _k ,u _k)} is a sequence of admissible controlsu _k and corre-sponding trajectories _k such that lim infJ( _k ,u _k)<+ and such that _k weakly,M _k M strongly,N _k N weakly, and {u _k} is bounded in someL _p norm, then there is a controlu such that (,u) is admissible and lim infJ( _k ,u _k)J(,u).Dedicated to Professor M. R. HestenesThis research was supported by the National Science Foundation, Grant No. GP-33551X.     

General necessary conditions for infinite horizon fractional variational problems

We consider infinite horizon fractional variational problems, where the fractional derivative is defined in the sense of Caputo. Necessary optimality conditions for higher-order variational problems and optimal control problems are obtained. Transversality conditions are obtained in the case state functions are free at the initial time.     

First-order necessary conditions for infinite-horizon variational problems

We establish rigorously several pointwise or asymptotic firstorder necessary conditions for infinite-horizon variational problems in general form, in the framework of continuous time. We obtain several new results, and we extend to general differentiable Lagrangians some results known only in special cases. To realize this aim, we justify two different ways to associate a family of finite-horizon problems to an infinite-horizon problem.The authors thank an anonymous referee for providing important historical references     

Formulation of Euler-Lagrange equations for fractional variational problems

This paper presents extensions to traditional calculus of variations for systems containing fractional derivatives. The fractional derivative is described in the Riemann-Liouville sense. Specifically, we consider two problems, the simplest fractional variational problem and the fractional variational problem of Lagrange. Results of the first problem are extended to problems containing multiple fractional derivatives and unknown functions. For the second problem, we also present a Lagrange type multiplier rule. For both problems, we develop the Euler-Lagrange type necessary conditions which must be satisfied for the given functional to be extremum. Two problems are considered to demonstrate the application of the formulation. The formulation presented and the resulting equations are very similar to those that appear in the field of classical calculus of variations.     

Lower closure and existence theorems for optimal control problems with infinite horizon

Lower closure theorems are proved for optimal control problems governed by ordinary differential equations for which the interval of definition may be unbounded. One theorem assumes that Cesari's property (Q) holds. Two theorems are proved which do not require property (Q), but assume either a generalized Lipschitz condition or a bound on the controls in an appropriateL _p-space. An example shows that these hypotheses can hold without property (Q) holding.     

Existence theorems in optimal control problems without convexity assumptions

We give existence theorems of solutions for Lagrange and Bolza problems of optimal control. These results are obtained without convexity assumptions on the cost function with respect to the control variable. We extend a Cesari's theorem to cost functions which are nonlinear with respect to the space variable and to problems which are governed by a differential inclusion. Moreover, we consider the case where the control variable belongs to a space of measurable functions and the case where this variable belongs to a Lebesgue space.     

Semigroup approach for the approximation of a control problem with unbounded dynamics

In this paper, we approximate a control problem in an infinite-dimensional Hilbert space by means of a sequence of discrete problems. In the differential equation which describes the dynamics, a Lipschitz perturbation of an unbounded linear operator appears. We prove a convergence result of the approximation value functions to the value function of the original problem.This research was supported by Ministero dell'Università e della Ricerca Scientifica e Technologica and by Consiglio Nazionale delle Ricerche, Rome, Italy.     

Interior variations in dynamic optimization problems

In this article, the author examines the properties of interior variations and indicates how to use them in order to formulate the necessary condition of optimality for problems of dynamic optimization, in particular, problems of variational calculus and of optimal control. For optimal control problems, an optimization technique based on interior variations and polynomial approximations is suggested and then illustrated by an explanatory example.     

A note on the use of direct sufficient conditions in optimal control problems

It is always possible to transform a nonautonomous optimal control problem into an autonomous one. However, the direct sufficient conditions may yield no information when applied to this autonomous problem, even though they do allow one to conclude sufficiency when applied to the original nonautonomous problem.This research was supported by the Air Force Office of Scientific Research, under Grant No. AFOSR-76-2923.     

Optimality conditions for fractional variational problems with dependence on a combined Caputo derivative of variable order

We establish necessary optimality conditions for variational problems with a Lagrangian depending on a combined Caputo derivative of variable fractional order. The endpoint of the integral is free, and thus transversality conditions are proved. Several particular cases are considered illustrating the new results.     

A new approach to the Pontryagin maximum principle for nonlinear fractional optimal control problems

In this paper, we discuss a new general formulation of fractional optimal control problems whose performance index is in the fractional integral form and the dynamics are given by a set of fractional differential equations in the Caputo sense. The approach we use to prove necessary conditions of optimality in the form of Pontryagin maximum principle for fractional nonlinear optimal control problems is new in this context. Moreover, a new method based on a generalization of the Mittag–Leffler function is used to solving this class of fractional optimal control problems. A simple example is provided to illustrate the effectiveness of our main result. Copyright © 2016 John Wiley & Sons, Ltd.     

A general finite element formulation for fractional variational problems

This paper presents a general finite element formulation for a class of Fractional Variational Problems (FVPs). The fractional derivative is defined in the Riemann-Liouville sense. For FVPs the Euler-Lagrange and the transversality conditions are developed. In the Fractional Finite Element Formulation (FFEF) presented here, the domain of the equations is divided into several elements, and the functional is approximated in terms of nodal variables. Minimization of this functional leads to a set of algebraic equations which are solved using a numerical scheme. Three examples are considered to show the performance of the algorithm. Results show that as the number of discretization is increased, the numerical solutions approach the analytical solutions, and as the order of the derivative approaches an integer value, the solution for the integer order system is recovered. For unspecified boundary conditions, the numerical solutions satisfy the transversality conditions. This indicates that for the class of problems considered, the numerical solutions can be obtained directly from the functional, and there is no need to solve the fractional Euler-Lagrange equations. Thus, the formulation extends the traditional finite element approach to FVPs.     

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.     

Intersections of extremals in linear-quadratic control problems

In this paper, we study the structure of intersections of extremals in linear control systems with quadratic cost. The structure of the intersections is determined. In particular, it is shown that, in contrast with the classical calculus of variations, these points are not necessarily isolated. The results obtained extend a known criterion from the calculus of variations involving the Jacobi necessary condition.The author would like to thank the referee, whose valuable remarks much improved the exposition of the results.     

On existence problems for the optimal control of certain nonlinear integral equations of Urysohn type

In Refs. 1–3, existence results have been obtained for optimal control problems whose state equations are described by certain nonlinear integral equations of Urysohn type. We generalize and synthesize these results by formulating a general lower closure result from which the results of Refs. 1–3 are shown to follow. In the course of this, we also present a novel and rather abstract treatment of existence problems for variable-time optimal control, quite in the spirit of Ref. 4.     

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.     

Hamilton-Jacobi theory for periodic control problems

The general problem of periodic optimization is considered in this paper. The contribution consists in stating sufficient conditions for the optimality of a control satisfying the periodicity constraint. The result has been achieved by means of the classical Hamilton-Jacobi equation, suitably modified in order to consider the peculiar constraint of the problem. Finally, an interesting application of the theory to the case of linear, time-invariant systems is given.This work was supported by CNR (Consiglio Nazionale delle Ricerche), Rome, Italy.