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

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

This document was uploaded on 03/06/2014 for the course MA 5623 at City University of Hong Kong.

Ask a homework question - tutors are online