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On a certain class of linear minimum time control problems. (English) Zbl 0659.49013

Summary: Techniques of functional analysis are more powerful than Pontryagin’s maximum principle, because they can be successfully applied to solve many problems in linear control systems where Pontryagin’s approach fails. In the functional analytic approach the control function is essentially required to belong to an appropriate Banach space; so the problems can be formulated as a mapping from this Banach space to another.
In a previous paper [Indian J. Pure Appl. Math. 12, 151–162 (1981; Zbl 0456.49004)] we demonstrated how the minimum time linear control problems can be solved, where admissible control must satisfy a constraint on the norm. The application of this method is straightforward, if the Banach space is reflexive.
In this paper, we demonstrate how to modify the method, when the control space is not reflexive but is the conjugate of some other Banach space. We illustrate the application of the method by means of an example. It should be noted that the example considered can also be tackled by Pontryagin’s maximum principle. We deal with a problem where Pontryagin’s maximum principle is not applicable, in a separate paper.

MSC:

49K27 Optimality conditions for problems in abstract spaces
46B99 Normed linear spaces and Banach spaces; Banach lattices
93B03 Attainable sets, reachability
93C05 Linear systems in control theory
34H05 Control problems involving ordinary differential equations

Citations:

Zbl 0456.49004
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