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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 1997 • Question 5

Question

Math 6410 • ODE Fall 1997 • Question 5

Assume that \(A\) is a real matrix whose eigenvalues have negative real part \((\Reals \,\sigma (A)<0)\) and \(B(t)\) is continuous for all \(t\in \mathbb {R}\). Show that there is an \(\epsilon >0\) depending on \(A\) so that if \(\norm {B}\le \epsilon \) then the zero solution of (2) with \(f(x,t)=Ax+B(t)x\) is uniformly Liapunov stable.

Raw LaTeX

Assume that \(A\) is a real matrix whose eigenvalues have negative real part
\((\Reals\,\sigma(A)<0)\) and \(B(t)\) is continuous for all \(t\in\R\). Show that there is
an \(\epsilon>0\) depending on \(A\) so that if \(\norm{B}\le\epsilon\) then the zero
solution of (2) with \(f(x,t)=Ax+B(t)x\) is uniformly Liapunov stable.

Topic Tags

Gronwall Inequality and Continuous Dependence Poincare Maps and Periodic Solutions Spectral Theory of Operators Stability of Linear Systems