A characterization of dichotomy in terms of boundedness of solutions for some Cauchy problems
We prove that a quadratic matrix of order $n$ having complex entries is dichotomic (i.e. its spectrum does not intersect the imaginary axis) if and only if there exists a projection $P$ on $ mathbb{C}^n$ such that $Pe^{tA}=e^{tA}P$ for all $tge 0$ and for each real number $mu$ and each vector...
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| Format: | Article |
| Language: | English |
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Texas State University
2008-07-01
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| Series: | Electronic Journal of Differential Equations |
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| Online Access: | http://ejde.math.txstate.edu/Volumes/2008/94/abstr.html |
| Summary: | We prove that a quadratic matrix of order $n$ having complex entries is dichotomic (i.e. its spectrum does not intersect the imaginary axis) if and only if there exists a projection $P$ on $ mathbb{C}^n$ such that $Pe^{tA}=e^{tA}P$ for all $tge 0$ and for each real number $mu$ and each vector $b in mathbb{C}^n$ the solutions of the following two Cauchy problems are bounded: $$displaylines{ dot x(t) = A x(t) + e^{i mu t}Pb,quad tgeq 0, cr x(0) = 0 }$$ and $$displaylines{ dot{y}(t)= -Ay(t) + e^{imu t}(I-P)b, quad tgeq 0, cr y(0) = 0,. }$$ |
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| ISSN: | 1072-6691 |