Stability for linear systems

Harald Hanche-Olsen

                       hanche@math.ntnu.no

The purpose of this text is to correct an error in Perko’s book (see the remarks and reference at the end).

Theorem 1 Given a real n×n matrix A, the following are equivalent:

(a) lim_{t→∞} e^{At}x_{0} = 0 for every x_{0} ∈ ℝ^{n}

(b) lim_{t→∞} ∥e^{At}∥ = 0

(c) There exist constants a > 0 and M so that |e^{At}x_{0}| ≤ Me^{−at}|x_{0}| for all t ≥ 0 and x_{0} ∈ ℝ^{n}

(d) Every eigenvalue of A har real part < 0

(e) lim_{t→−∞} |e^{At}x_{0}| = ∞ for every nonzero x_{0} ∈ ℝ^{n}

(f) There exist constants a > 0 and m > 0 so that |e^{At}x_{0}| ≥ me^{−at}|x_{0}| for all t ≤ 0 and x_{0} ∈ ℝ^{n}

Proof: We prove (c) ⇒ (b) ⇒ (a) ⇒ (d) ⇒ (c) ⇒ (f) ⇒ (e) ⇒ (d). The proof is arranged to keep related parts together, and to leave the hardest part to the end.

(c) ⇒ (b): If (c) holds then ∥e^{At}∥ ≤ Me^{−at} → 0 as t → ∞.

(b) ⇒ (a): If (b) holds then |e^{At}x_{0}| ≤ ∥e^{At}∥ · |x_{0}| → 0 as t → ∞.

(a) ⇒ (d): Assume (d) does not hold. Then there exist a nonzero vector z_{0} ∈ ℂ^{n} and λ ∈ ℂ with Az_{0} = λz_{0} where λ ∈ ℂ with Re λ ≥ 0. Then e^{At}z_{0} = e^{λt}z_{0} ↛ 0 when t → ∞. This non-convergence must hold for at least either the real part or the imaginary part of z_{0}, so (a) does not hold.

(e) ⇒ (d): Assume (d) does not hold. Choose z_{0} and λ as above. Then |e^{At}z_{0}| = |e^{λt}z_{0}| = e^{t Re λ}|z_{0}| ↛ ∞ as t → −∞, so (e) does not hold (again, split z_{0} into real and imaginary parts to see this).

(f) ⇒ (e): This is obvious.

(c) ⇒ (f): Replace x_{0} in (c) by e^{−At}x_{0}, to conclude that |x_{0}| ≤ Me^{−at}|e^{−At}x_{0}| when t ≥ 0. Now replace t with −t, and conclude |x_{0}| ≤ Me^{at}|e^{At}x_{0}| when t ≤ 0. But this is exactly the inequality of (f), with m = 1/M.

(d) ⇒ (c): Write A = V JV^{−1} where J is the complex Jordan form of A, and furthermore write J = M + N where M is a diagonal matrix whose diagonal entries are the same as those of J. The entries of N will be zero except possibly for some ones just above the diagonal. In particular N is nilpotent, i.e. N^{q+1} = 0 for some q. We may also note that MN = NM, so that e^{Jt} = e^{Mt}e^{Nt}.

Let β be the largest real part of any eigenvalue of A. By assumption, β < 0. The eigenvalues of A are also the diagonal entries of J, and therefore of M, and since M is diagonal, we get ∥e^{Mt}∥ = e^{βt} when t ≥ 0. Also

e^{Nt} = I + N + \frac{t^{2}}{2!} N^{2} + ⋯ + \frac{t^{k}}{k!} N^{k}

so we have ∥e^{Nt}∥ ≤ p(t) for t ≥ 0, where p is a polynomial. We end up, therefore, with

∥e^{At}∥ = ∥V e^{Jt}V^{−1}∥ = ∥V e^{Mt}e^{Nt}V^{−1}∥
≤ ∥V ∥ · ∥e^{Mt}∥ · ∥e^{Nt}∥ · ∥V^{−1}∥ ≤ ∥V ∥ · ∥V^{−1}∥ · e^{βt}p(t)

from which we conclude

e^{at}∥e^{At}∥ ≤ ∥V ∥ · ∥V^{−1}∥ · e^{(a+β)t}p(t) ≤ M

for all t ≥ 0, provided a + β < 0 and M is large enough. But this inequality is ∥e^{At}∥ ≤ Me^{−at}, which is (c). (We note that because β < 0, we can in fact satisfy both requirements a > 0 and a + β < 0 simultaneously.)

The next theorem follows immediately from the above theorem by replacing A with −A:

Theorem 2 Given a real n×n matrix A, the following are equivalent:

(a) lim_{t→−∞} e^{At}x_{0} = 0 for every x_{0} ∈ ℝ^{n}

(b) lim_{t→−∞} ∥e^{At}∥ = 0

(c) There exist constants a > 0 and M so that |e^{At}x_{0}| ≤ Me^{at}|x_{0}| for all t ≤ 0 and x_{0} ∈ ℝ^{n}

(d) Every eigenvalue of A har real part > 0

(e) lim_{t→∞} |e^{At}x_{0}| = ∞ for every nonzero x_{0} ∈ ℝ^{n} (f) There exist constants a > 0 and m > 0 so that |e^{At}x_{0}| ≥

me^{at}|x_{0}| for all t ≥ 0 and x_{0} ∈ ℝ^{n}

Remark. The theorems above are suitable replacements for Theorem 2 and 3 in section 1.9 of Perko’s book, which are wrong as stated.

To see that Perko’s Theorem 2 is wrong, consider the matrix

A = [\begin{matrix}−1 & 0 \\ 0 & −2 \end{matrix}]

which clearly satisfies (b).^{1} We shall see that it does not satisfy (c): We find

e^{At} = [\begin{matrix}e^{−t} & 0 \\ 0 & e^{−2t} \end{matrix}] , ∥e^{At}∥ = max(e^{−t}, e^{−2t}) = \begin{cases}e^{−t} & t ≥ 0, \\ e^{−2t} & t ≤ 0. \end{cases}

But the righthand inequality of (c) states that e^{ct}∥e^{At}∥ is bounded, which means that e^{(c−1)t} is bounded for t > 0, and e^{(c−2)t} is bounded for t < 0. These statements imply both c − 1 ≤ 0 and c − 2 ≥ 0, which is impossible.

Reference. Lawrence Perko, Differential equations and dynamical systems, Second edition, Springer-Verlag 1996 (ISBN 0–387–94778–7) or First edition, Springer-Verlag 1991 (ISBN 0–387–96443–1)

1(b) and (c) here refer to Perko, of course.