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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Spring 2014 • Question 2

Question

Math 6410 • ODE Spring 2014 • Question 2

Let \(A\) be an \(n\times n\) real matrix such that \(\Re \lambda <0\) for all eigenvalues \(\lambda \) of \(A\). Show that there are positive constants \(c\) and \(\mu \) such that any solution \(x(t)\) of \[ \begin {cases} \dfrac {dx}{dt}=Ax,\\ x(0)=x_0. \end {cases} \] satisfies \[ \norm {x(t)}\le ce^{-\mu t}\norm {x_0}\qquad \text {for all }t\ge 0. \]

Raw LaTeX

Let \(A\) be an \(n\times n\) real matrix such that \(\Re\lambda<0\) for all eigenvalues
\(\lambda\) of \(A\). Show that there are positive constants \(c\) and \(\mu\) such that
any solution \(x(t)\) of
\[
  \begin{cases}
    \dfrac{dx}{dt}=Ax,\\
    x(0)=x_0.
  \end{cases}
\]
satisfies
\[
  \norm{x(t)}\le ce^{-\mu t}\norm{x_0}\qquad\text{for all }t\ge0.
\]

Topic Tags

Distributions and Weak Solutions Poincare Maps and Periodic Solutions Spectral Theory of Operators Variation of Constants Formula