# Control as Programming in General Normed Linear Spaces

## Abstract

The theory of necessary conditions for optimal control problems can be presented in various forms. In these lectures we shall present a relatively recent form of that theory in which optimal control problems are considered as mathematical programming problems in infinite dimensional spaces. In Section I we shall define a very general mathematical programming problem and in Theorem I we shall state a necessary condition for that problem, A suitable background and appropriate references for Section I can be found in Halkin and Neustadt.^{[1]} A simple proof of Theorem I is given in Halkin.^{[2]} In Section II we shall study the dynamics of a control problem and in Theorem II we shall prove that for a general class of control problems the assumptions of Theorem I are satisfied.

## Keywords

Optimal Control Problem Integrable Function Differential System Optimal Control Theory Normed Linear Space## Preview

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## References

- [1]H. Halkin and L. W. Neustadt, General necessary conditions for optimization problems, Proc. Nat. Acad. Sciences, 56, 1966, pp. 1066–1071.MathSciNetzbMATHCrossRefGoogle Scholar
- [2]H. Halkin, Nonlinear nonconvex programming in an infinite dimensional space, in Mathematical Theory of Control, A. V. Balakrishan and L. W. Neustadt, eds., Academic Press, 1967, pp. 10-25.Google Scholar
- [3]Halkin, H., “Optimal Control as Programming in Infinite Dimensional Spaces” in C.I.M.E.: Calculus of Variations, Classical and Modern, Edizioni Cremonese, Roma, 1966, 179-192.Google Scholar