Dynamic optimization: The calculus of variations and optimal control in economics and management: Morton I. KAMIEN and Nancy L. SCHWARTZ Volume 4 in: Dynamic Economics: Theory and Applications, North-Holland, New York, 1981, xi + 331 pages, Dfl.90.00

For the last 30 years the optimization of nonsingular control problems has been an Important part of control engineering, and its mathematical theory is well developed and widely known. On the other hand, singular control problems prove more difficult to analyse and—although necessary conditions for optimality of singular controls have been established over the past decade—It is only recently that sufficient, and necessary and sufficient, conditions have been formulated. The purpose of this book Is to collect together all known results in optimal control theory (as well as appropriate computational methods) which can be applied to the singular problems In optimal control and which up to now have been scattered In numerous journals. Complete and self-contained, the volume begins with an historical survey of singular control problems and leads to the presentation of important, recent results in the field. There are specific real-world applications and the authors discuss those avenues of research which require further Investigation. All those involved In the optimization of dynamical systems will welcome the publication of this book. In addition to advanced students, lecturers and research workers in universities, this will include practising mechanical, chemical and electrical engineers, builders, textile technologists, paper scientists and chemists, and many concerned with non-technical fields such as economics and business management Contents An historical survey of singular control problems Introduction. Singular control in space navigation. Method of Mlele via Green's theorem. Linear systems—quadratic cost Necessary conditions for singular optimal control. Sufficient conditions and necessary and sufficient conditions for optimality. References. Fundamental concepts Introduction. The general optimal control problem. The first variation of J. The second variation of J. A singular control problem. References. Necessary conditions for singular optimal control Introduction. The generalized Legendre-Clebsch condition. Jacobson's necessary condition. References. Sufficient conditions and necessary and sufficient conditions tor non-negativity of nonsingular and singular second variations Introduction. Preliminaries. The nonsingular case. Strong positivlty and the totally singular second variation. A general sufficiency theorem for the second variation. Necessary and sufficient conditions for non-negativity of the totally singular second variation. Necessary conditions for optimality. Other necessary and sufficient conditions. Sufficient conditions for a weak local minimum. Existence conditions for the matrix Rlccati differential equation. Conclusion. References. Computational methods for singular control problems Introduction. Computational application of the sufficiency conditions of theorems in the previous chapter. Computation of optimal singular controls. Joining of optimal singular and non-singular sub-arcs. Conclusion. References. Conclusion The Importance of singular optimal control problems. Necessary conditions. Necessary and sufficient conditions. Computational methods. Switching conditions. Outlook for the future Author index. Sublect index.

Singular optimal control problems

This article surveys the usual techniques of nonlinear optimal control such as the Pontryagin Maximum Principle and the conjugate point theory, and how they can be implemented numerically, with a special focus on applications to aerospace problems. In practice the knowledge resulting from the maximum principle is often insufficient for solving the problem, in particular because of the well-known problem of initializing adequately the shooting method. In this survey article it is explained how the usual tools of optimal control can be combined with other mathematical techniques to improve significantly their performances and widen their domain of application. The focus is put onto three important issues. The first is geometric optimal control, which is a theory that has emerged in the 1980s and is combining optimal control with various concepts of differential geometry, the ultimate objective being to derive optimal synthesis results for general classes of control systems. Its applicability and relevance is demonstrated on the problem of atmospheric reentry of a space shuttle. The second is the powerful continuation or homotopy method, consisting of deforming continuously a problem toward a simpler one and then of solving a series of parameterized problems to end up with the solution of the initial problem. After having recalled its mathematical foundations, it is shown how to combine successfully this method with the shooting method on several aerospace problems such as the orbit transfer problem. The third one consists of concepts of dynamical system theory, providing evidence of nice properties of the celestial dynamics that are of great interest for future mission design such as low-cost interplanetary space missions. The article ends with open problems and perspectives.

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Optimal Control and Applications to Aerospace: Some Results and Challenges

We consider minimization of functions that are compositions of convex or prox-regular functions (possibly extended-valued) with smooth vector functions. A wide variety of important optimization problems fall into this framework. We describe an algorithmic framework based on a subproblem constructed from a linearized approximation to the objective and a regularization term. Properties of local solutions of this subproblem underlie both a global convergence result and an identification property of the active manifold containing the solution of the original problem. Preliminary computational results on both convex and nonconvex examples are promising.

/pdf/a-proximal-method-for-composite-minimization-3wni3a85bb.pdf

A proximal method for composite minimization

http://www.optimization-online.org/DB_FILE///2008/11/2153.pdf

Reformulations and algorithms for the optimization of switching decisions in nonlinear optimal control

http://www.optimization-online.org/DB_FILE/2006/11/1511.pdf

The Hybrid Maximum Principle is a consequence of Pontryagin Maximum Principle

We study the notion of α-covering map with respect to certain subsets in metric spaces. Generalizing results from [1] we use this notion to give some coincidence theorems for pairs of single-valued and multivalued maps one of which is relatively α-covering while the other satisfies the Lipschitz condition. These assertions extend some classical contraction map principles. We define the notion of α-covering multimap at a point and give conditions under which the covering property of a multimap at each interior point of a set implies that it is covering on the whole set. As applications we consider the solvability of a system of inclusions and the existence of a positive trajectory for a semilinear feedback control system.

Locally covering maps in metric spaces and coincidence points

This paper deals with optimal control problems for systems affine in the control variable. We consider nonnegativity constraints on the control, and finitely many equality and inequality constraints on the final state. First, we obtain second order necessary optimality conditions. Secondly, we derive a second order sufficient condition for the scalar control case.

http://www.cmap.polytechnique.fr/~aronna/RR-7664.pdf

Quadratic order conditions for bang-singular extremals

This paper deals with optimal control problems for systems affine in the control variable We consider nonnegativity constraints on the control, and finitely many equality and inequality constraints on the final state First, we obtain second order necessary optimality conditions Secondly, we derive a second order sufficient condition for the scalar control case

We consider the class of optimal control problems linear in the control, and study a singular extremal. If the La- grange multipliers are unique, the quadratic order optimality con- ditions have the form of sign definiteness of a quadratic functional (the second variation of Lagrange function) with totally zero Legen- dre coecient. Using the Goh transformation, we convert it to a functional possibly satisfying the strengthened Legendre condition, involving also an additional parameter, and by applying the Hestenes approach, determine its sign definiteness in terms of the conjugate point, i.e. give Jacobi type conditions.

Andrei Dmitruk

Papers

The Hybrid Maximum Principle is a consequence of Pontryagin Maximum Principle

Locally covering maps in metric spaces and coincidence points

Quadratic order conditions for bang-singular extremals

Quadratic order conditions for bang-singular extremals

Jacobi Type Conditions for Singular Extremals