Note that k ak1 oth erwise ak1 k i ck would be

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Unformatted text preview: because they are the basic components that either dictate or govern all other characteristics of A along with the physics of associated phenomena. Copyright c 2000 SIAM Buy online from SIAM Buy from 544 Chapter 7 Eigenvalues and Eigenvectors For example, consider the long-run behavior of a physical system that can be modeled by u = Au. We usually want to know whether the system will eventually blow up or will settle down to some sort of stable state. Might it neither blow up nor settle down but rather oscillate indefinitely? These are questions concerning the nature of the limit lim u(t) = lim eAt c = lim eλ1 t G1 + eλ2 t G2 + · · · + eλk t Gk c, It is illegal to print, duplicate, or distribute this material Please report violations to t→∞ t→∞ t→∞ and the answers depend only on the eigenvalues. To see how, recall that for a complex number λ = x + iy and a real parameter t > 0, D E eλt = e(x+iy)t = ext eiyt = ext (cos yt + i sin yt) . (7.4.8) The term eiyt = (cos yt + i sin yt) is a point on the unit circle that oscillates as a function of t, so |eiyt | = |cos yt + i sin yt| = 1 and eλt = |ext eiyt | = |ext | = ext . This makes it clear that if Re (λi ) < 0 for each i, then, as t → ∞, eAt → 0, and u(t) → 0 for every initial vector c. Thus the system eventually settles down to zero, and we say the system is stable. On the other hand, if Re (λi ) > 0 for some i, then components of u(t) may become unbounded as t → ∞, and we say the system is unstable. Finally, if Re (λi ) ≤ 0 for each i, then the components of u(t) remain finite for all t, but some may oscillate indefinitely, and this is called a semistable situation. Below is a summary of stability. T H IG R Stability Y P Let u = Au, u(0) = c, where A is diagonalizable with eigenvalues λi . • If Re (λi ) < 0 for each i, then lim eAt = 0, and lim u(t) = 0 O C • • t→∞...
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This document was uploaded on 03/06/2014 for the course MA 5623 at City University of Hong Kong.

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