Null controllable sets and reachable sets for nonautonomous linear control systems

Roberta  Fabbri; Sylvia Novo; Carmen Núñez; Rafael  Obaya

doi:10.3934/dcdss.2016042

Article Contents

2016, Volume 9, Issue 4: 1069-1094. Doi: 10.3934/dcdss.2016042

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Null controllable sets and reachable sets for nonautonomous linear control systems

1.
Dipartimento di Matematica e Informatica "Ulisse Dine", Università di Firenze, Via di Santa Marta 3, 50139 Firenze
2.
Departamento de Matemática Aplicada, E. Ingenierías Industriales, Universidad de Valladolid, Paseo del Cauce 59, 47011 Valladolid
3.
Departamento de Matemática Aplicada, Universidad de Valladolid, Paseo del Cauce 59, 47011 Valladolid

Received: June 30, 2015

Revised: September 30, 2015

Published: July 2016

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Abstract

Under the assumption of lack of uniform controllability for a family of time-dependent linear control systems, we study the dimension, topological structure and other dynamical properties of the sets of null controllable points and of the sets of reachable points. In particular, when the space of null controllable vectors has constant dimension for all the systems of the family, we find a closed invariant subbundle where the uniform null controllability holds. Finally, we associate a family of linear Hamiltonian systems to the control family and assume that it has an exponential dichotomy in order to relate the space of null controllable vectors to one of the Lagrange planes of the continuous hyperbolic splitting.

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Mathematics Subject Classification: 37B55, 34H05, 37N35, 34D09, 49J15.

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