Abstract
For homogeneous bilinear control systems, the control sets are characterized using a Lie algebra rank condition for the induced systems on projective space. This is based on a classical Diophantine approximation result. For affine control systems, the control sets around the equilibria for constant controls are characterized with particular attention to the question when the control sets are unbounded.
Highlights
We will study controllability properties of affine control systems of the form m x(t) = Ax(t) + ui (t)(Bi x(t) + ci ) + d, i =1 (1.1)where A, B1, . . . , Bm ∈ Rn×n and c1, . . . , cm, d are vectors in Rn
The set of admissible controls is U = {u ∈ L∞(R, Rm) |u(t) ∈ for almost all t } or the set Upc of all piecewise constant functions defined on R with values in
The monograph Elliott [13] emphasizes the use of matrix Lie groups and Lie semigroups and contains a wealth of results on the control of bilinear control systems
Summary
We use a classical result on Diophantine approximations which allows us to require only the accessibility rank condition on Sn−1 in the interior of S D. This result is illustrated by two-dimensional examples. For systems satisfying the accessibility rank condition on projective space, the control sets on the unit sphere and on Rn \ {0} are characterized in Theorem 3.12 and Theorem. Corollary 3.21 characterizes controllability on Rn \ {0} for systems satisfying only the accessibility rank condition on Pn−1 using a recent result by Cannarsa and Sigalotti [7, Theorem 1] which shows that here approximate controllability implies controllability.
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