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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 2004 • Question 4

Question

Math 6410 • ODE Fall 2004 • Question 4

Linear Systems. Suppose \(A\) is a real \(n\times n\) matrix whose eigenvalues \(\lambda _i\) satisfy \(\operatorname {Re}\lambda _i\le 0\). Show that there are constants \(C<\infty \) and \(k\le n-1\) so that any solution \(x(t)\in \mathbb {R}^n\) of the initial value problem \[ \begin {aligned} \frac {dx}{dt} &= Ax,\\ x(0) &= x_0, \end {aligned} \] satisfies \[ \lvert x(t)\rvert \le C(1+\lvert t\rvert ^k)\lvert x_0\rvert \] for all \(t\in \mathbb {R}\) and \(x_0\in \mathbb {R}^n\).

Raw LaTeX

\textbf{Linear Systems.}
Suppose \(A\) is a real \(n\times n\) matrix whose eigenvalues \(\lambda_i\) satisfy
\(\operatorname{Re}\lambda_i\le0\). Show that there are constants \(C<\infty\) and
\(k\le n-1\) so that any solution \(x(t)\in\R^n\) of the initial value problem
\[
\begin{aligned}
  \frac{dx}{dt} &= Ax,\\
  x(0) &= x_0,
\end{aligned}
\]
satisfies
\[
  \lvert x(t)\rvert\le C(1+\lvert t\rvert^k)\lvert x_0\rvert
\]
for all \(t\in\R\) and \(x_0\in\R^n\).

Topic Tags

Existence and Uniqueness Theorems Linear Systems and Matrix Exponentials Spectral Theory of Operators Stability of Linear Systems