Using an endogenous growth model in which learning by doing, although bounded in each good, exhibits spillovers across goods, this paper investigates the dynamic effects of international trade. Examining an LDC and a DC, the latter distinguished by a higher initial level of knowledge, under autarky and free trade, I find that under free trade the LDC (DC) experiences rates of technical progress and GOP growth less than or equal (greater than or equal) to those enjoyed under autarky. Unless the LDC's population is several orders of magnitude greater than that of the DC and the initial technical gap between the two economies is not large, the LDC will be unable to catch up with its trading partner. Hence, in terms of technical progress and growth, the LDC experiences dynamic losses from trade, whilst the DC experiences dynamic gains. However, since technical progress abroad can improve welfare at home, LDC consumers may enjoy - higher intertemporal utility along the free trade path. In the case of DC consumers, as long as their economy is not overtaken by the LDC they will enjoy both more rapid technical progress and the traditional static gains from trade, and hence experience an unambiguous improvement in intertemporal welfare.

Learning by Doing and the Dynamic Effects of International Trade

Nonlinear Programming: A Unified Approach

This paper gives a survey of the various forms of Pontryagin’s maximum principle for optimal control problems with state variable inequality constraints. The relations between the different sets of optimality conditions arising in these forms are shown. Furthermore, the application of these maximum principle conditions is demonstrated by solving some illustrative examples.

A survey of the maximum principles for optimal control problems with state constraints

Singular perturbations and order reduction in control theory - An overview

This paper discusses typical applications of singular perturbation techniques to control problems in the last fifteen years. The first three sections are devoted to the standard model and its time-scale, stability and controllability properties. The next two sections deal with linear-quadratic optimal control and one with cheap (near-singular) control. Then the composite control and trajectory optimization are considered in two sections, and stochastic control in one section. The last section returns to the problem of modeling, this time in the context of large scale systems. The bibliography contains more than 250 titles.

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Applications of Singular Perturbation Techniques to Control Problems

https://ntrs.nasa.gov/api/citations/19700002504/downloads/19700002504.pdf

New necessary conditions of optimality for control problems with state-variable inequality constraints

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

A class of singular control problems is made nonsingular by the addition of an integral quadratic functional of the control to the cost functional; a parameter \epsilon > 0 multiplies this added functional. The resulting nonsingular problem is solved for a monotonically decreasing sequence \{\epsilon; \epsilon_{1} > \epsilon_{2} > ... > \epsilon_{k} > 0\} . As k \rightarrow \infty and \epsilon_{k} \rightarrow 0 the solution of the modified problem tends to the solution of the original singular problem. A variant of the method which does not require that \epsilon \rightarrow 0 is also presented. Four illustrative numerical examples are described.

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Computation of optimal singular controls

The performance of a process for example, how an aircraft consumes fuel can be enhanced when the most effective controls and operating points for the process are determined. This holds true for many physical, economic, biomedical, manufacturing, and engineering processes whose behavior can often be influenced by altering certain parameters or controls to optimize some desired property or output. Primer on Optimal Control Theory provides a rigorous introduction to analyzing these processes and finding the best modes of control and operation for them. It makes optimal control theory accessible to a large class of engineers and scientists who are not mathematicians but have a basic mathematical background and need to understand the sophisticated material associated with optimal control theory. The book presents the important concepts of weak and strong control variations leading to local necessary conditions as well as global sufficiency of Hamilton Jacobi Bellman theory. It also gives the second variation for local optimality where the associated Riccati equation is derived from the transition matrix of the Hamiltonian system. These ideas lead naturally to the development of H2 and H-infinity synthesis algorithms. Audience: This book will enable applied mathematicians, engineers, scientists, biomedical researchers, and economists to understand, appreciate, and implement optimal control theory at a level of sufficient generality and applicability for most practical purposes and will provide them with a sound basis from which to proceed to higher mathematical concepts and advanced systems formulations and analyses. Contents: List of Figures; Preface; Chapter 1: Introduction; Chapter 2: Finite-Dimensional Optimization; Chapter 3: Systems with General Performance Criteria; Chapter 4: Terminal Equality Constraints; Chapter 5: Linear-Quadratic Control Problem; Chapter 6: Linear-Quadratic Differential Games; Appendix: Background; Bibliography; Index.

David H. Jacobson

Papers

New necessary conditions of optimality for control problems with state-variable inequality constraints

Singular Optimal Control Problems

Computation of optimal singular controls

Primer on optimal control theory

Necessary and sufficient conditions for optimality for singular control problems - A limit approach