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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 2013 • Question 2

Question

Math 6410 • ODE Fall 2013 • Question 2

Let \(A\) be an \(n\times n\) real matrix. Consider the linear initial value problem \[ \begin {cases} \dfrac {dx}{dt}=Ax,\\ x(0)=x_0. \end {cases} \] Give an algebraic condition on the matrix \(A\) which holds if and only if all solutions of the initial value problem are bounded for all \(t\in \mathbb {R}\). Prove the equivalence.

Raw LaTeX

Let \(A\) be an \(n\times n\) real matrix. Consider the linear initial value problem
\[
  \begin{cases}
    \dfrac{dx}{dt}=Ax,\\
    x(0)=x_0.
  \end{cases}
\]
Give an algebraic condition on the matrix \(A\) which holds if and only if all solutions
of the initial value problem are bounded for all \(t\in\R\). Prove the equivalence.

Topic Tags

Bounded Linear Operators Dual Spaces and Bounded Linear Functionals Existence and Uniqueness Theorems Linear Systems and Matrix Exponentials