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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 2004 • Question 2

Question

Math 6410 • ODE Fall 2004 • Question 2

Global Existence. Let \(\varphi _t(x_0)\) be the flow of the differential equation \[ \begin {aligned} \dot {x} &= f(x),\\ x(0) &= x_0, \end {aligned} \] where \(f\in C^1(E)\), \(E\subset \mathbb {R}^n\) is open, and \(x_0\in E\). Prove that if the system has a Liapunov function \(V\) with \(\dot {V}(x)=\left .\frac {d}{dt}V(\varphi _t(x))\right |_{t=0}\le 0\) for all \(x\in E\), and the set \(S=\{x\in E:V(x)\le 1\}\) is compact, then solutions \(\varphi _t(x_0)\) with \(x_0\in S\) exist for all positive times \(t>0\).

Raw LaTeX

\textbf{Global Existence.}
Let \(\varphi_t(x_0)\) be the flow of the differential equation
\[
\begin{aligned}
  \dot{x} &= f(x),\\
  x(0) &= x_0,
\end{aligned}
\]
where \(f\in C^1(E)\), \(E\subset\R^n\) is open, and \(x_0\in E\). Prove that if the system
has a Liapunov function \(V\) with
\(\dot{V}(x)=\left.\frac{d}{dt}V(\varphi_t(x))\right|_{t=0}\le0\) for all \(x\in E\), and the
set \(S=\{x\in E:V(x)\le1\}\) is compact, then solutions \(\varphi_t(x_0)\) with \(x_0\in S\)
exist for all positive times \(t>0\).

Topic Tags

Distributions and Weak Solutions Existence and Uniqueness Theorems Global Existence and Finite-Time Blow-Up Poincare Maps and Periodic Solutions