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On the complete controllability of linear nonautonomous systems

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Abstract

We suggest a geometric approach to the controllability of nonautonomous linear control systems of the form \(\dot x = A(t)x + B(t)u\), x ∈ ℝn, uU ⊆ ℝm, with conical control constraint set U and continuous matrices A(t) and B(t). We derive two new complete controllability criteria, the first of which is reduced to the analysis of the arrangement of the cones Φ−1(t)B(t)U in the state space of the system [\(\dot \Phi (t) = \left. {A(t)\Phi } \right|(t)\), Φ(0) = E] and the second is based on the existence of appropriate controls bringing zero back to zero. We prove a theorem on the approximation of the control constraint set U by cones with finitely many generators lying inside the cone U with the preservation of the complete controllability property. We present a number of examples illustrating some peculiarities in the evolution of controllability sets of nonautonomous linear systems.

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Original Russian Text © Yu.M. Semenov, 2012, published in Differentsial’nye Uravneniya, 2012, Vol. 48, No. 9, pp. 1265–1277.

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Semenov, Y.M. On the complete controllability of linear nonautonomous systems. Diff Equat 48, 1245–1257 (2012). https://doi.org/10.1134/S0012266112090054

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