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Controlled stability

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Summary

A rest solution, p, of a control system is said to possess controlled stability if for each t 1>0 a full nbd. of points of p can be steered to p in time t 1 by solutions of the system. Equivalently, if the system acts in a linear space and p is the origin, this states the domain of « null controllability » is open. We consider systems of the form\(X\left( x \right) + \sum\limits_{i = 2}^m {u_i } \left( t \right)Y^i \left( x \right)\) with either each ui measurable with |ui(t)| ⩽1, or |ui| ≤1 inL 1 or ‖ui‖ ≤1 in the space of regular countably additive measures, and X, Y2, …, Ym analytic vector fields on an analytic n-manifold. If X(p)=0 a necessary condition that the rest solution p possess controlled stability is that the dimension, at p, of the Lie algebra generated by X, Y2, …, Ym be n. First order sufficient conditions are well known and can be stated in terms of subsets of the elements of this Lie algebra. This paper provides two higher order sufficiency tests, together with examples of their applications.

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Entrata in Redazione il 14 giugno 1976.

This research was supported by the National Science Foundation under grant MCS76-04419.

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Hermes, H. Controlled stability. Annali di Matematica 114, 103 (1977). https://doi.org/10.1007/BF02413781

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Keywords

  • Control System
  • Vector Field
  • Linear Space
  • Additive Measure
  • Analytic Vector