Mathematical Sciences: Second-Order Optimality Conditions for Problems with Constraints

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9404591 Zeidan A general optimal control problem is considered in which the control and/or state constraints are of the form of an inclusion in a convex set. The accessory problem when the data functions are differentiable with Lipschitz derivatives will be derived. A new term is expected to appear and hence, this is a generalization of the envelope- like effect (introduced by Kawasaki). The results will be expressed only in terms of the original data. New second- order necessary conditions for strong local minimality of the above problem form the second part of this project. This is a long-standing problem. The key idea of this research is based on suitably modifying the time transformation, introduced by A. Ya. Dubovitskii and A. A. Milyutin, in order to obtain a new problem to which the second-order necessary conditions for weak local optimality (developed in the first part of the proposal) could be applied. The third part of this project is to develop the sufficiency theory for the case when pure state inequality constraints and control set constraints are present. This research involves a modified Riccati equation and the study of connections between the three concepts: Riccati equation, coupled points, and positive definiteness of the second variations. This project lies at the heart of optimal control theory, a subject which was developed in the 1950's mainly to deal with problems emanating from engineering, natural resources, and economics. This field, which is a branch of optimization, could be described as follows: Find a candidate that produces the smallest cost (or the largest return) while meeting given requirements. The latter are referred to as constraints. "First order optimality conditions'' serve to produce a list of possible candidates for the smallest cost. In practice, many of them do not furnish the least cost. Hence, there is a need for another test to screen the candidates and to obtain a short list. This test evok es what are known as "second order conditions''. In this project, second-order optimality conditions for optimal control problem with very general constraints will be derived. In fact, several recent problems in aerospace science, industrial process control, robotics and mathematical economics can be formulated as optimal control problems with such general constraints (state and/or control set). However, in this complicated setting no satisfactory second order conditions are presently available. Thus, when accomplished, this research will provide new tools to assist engineers and economists in the selection of optimal strategies. ***