109 if k0 a converges it follows k that k0 j must

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Unformatted text preview: s illegal to print, duplicate, or distribute this material Please report violations to [email protected] eAt = P .. .J e P−1 with eJ t .. t . ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ =⎜ ⎜ ⎜ ⎜ ⎜ ⎝ teλt te 2! eλt eλt teλt .. ··· t e (k − 1)! .. . . . .. . . . eλt t2 eλt 2! teλt D E eλt while setting f (z ) = e zt in (7.9.9) produces s e At ki −1 j λi t = i=1 j =0 te (A − λi I)j Gi . j! ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ , (7.9.18) ⎟ ⎟ ⎟ ⎟ ⎠ T H (7.9.19) Either of these can be used to show that the three properties (7.4.3)–(7.4.5) on p. 541 still hold. In particular, d eAt /dt = AeAt = eAt A, so, just as in the diagonalizable case, u(t) = eAt c is the unique solution of (7.9.17) (the uniqueness argument given in §7.4 remains valid). In the diagonalizable case, the solution of (7.9.17) involves only the eigenvalues and eigenvectors of A as described in (7.4.7) on p. 542, but generalized eigenvectors are needed for the nondiagonalizable case. Using (7.9.19) yields the solution to (7.9.17) as s u(t) = eAt c = ki −1 j λi t IG R te vj (λi ), where vj (λi ) = (A − λi I)j Gi c. (7.9.20) j! Y P i=1 j =0 Each vki −1 (λi ) is an eigenvector associated with λi because (A − λi I)ki Gi = 0, and {vki −2 (λi ), . . . , v1 (λi ), v0 (λi )} is an associated chain of generalized eigenvectors. The behavior of the solution (7.9.20) as t → ∞ is similar but not identical to that discussed on p. 544 because for λ = x + iy and t > 0, ⎧ if x < 0, ⎪0 ⎪ ⎪ unbounded ⎨ if x ≥ 0 and j > 0, tj eλt = tj ext (cos yt + i sin yt) → oscillates indefinitely if x = j = 0 and y = 0, ⎪ ⎪1 ⎪ if x = y = j = 0. ⎩ O C In particular, if Re (λi ) < 0 for every λi ∈ σ (A) , then u(t) → 0 for every initial vector c, in which case the system is said to be stable . • Nonhomogeneous Systems. It can be verified by direct manipulation that the solution of u (t) = Au(t) + f (t) with u(t0 ) = c is given by u(t) = eA(t−t0 ) c + t eA(t−τ ) f (τ )dτ...
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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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