Abstract
For the bilinear control system\(\dot x = \left( {A + uB} \right)x\),x ∈ ℝn,u ∈ ℝ, whereA is ann ×n essentially nonnegative matrix, andB is a diagonal matrix, the following controllability problem is investigated: can any two points with positive coordinates be joined by a trajectory of the system? Forn>2, the answer is negative in the generic case: hypersurfaces in ℝn are constructed that are intersected by all the trajectories of the system in one direction.
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References
W. M. Boothby, “Some comments on positive orthant controllability of bilinear systems,”SIAM J. Control Optim.,20, No. 5, 634–644 (1982).
Yu. L. Sachkov, “Positive orthant controllability of two- and three-dimensional bilinear systems,”Differentsial'nye Uravneniya [Differential Equations],29, 361–363 (1993).
A. Bacciotti, “On the positive orthant controllability of two-dimensional bilinear systems,”Systems and Control Letters,3, 53–55 (1983).
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Translated fromMatematicheskie Zametki, Vol. 58, No. 3, pp. 419–424, September, 1995.
The author wishes to express gratitude to Professor A. F. Filippov for his attention to this work.
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Sachkov, Y.L. Positive orthant scalar controllability of bilinear systems. Math Notes 58, 966–969 (1995). https://doi.org/10.1007/BF02304774
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DOI: https://doi.org/10.1007/BF02304774