The Unapologetic Mathematician

Mathematics for the interested outsider

Gronwall’s Inequality

We’re going to need another analytic lemma, this one called “Gronwall’s inequality”. If v:[0,\alpha]\to\mathbb{R} is a continuous, nonnegative function, and if C and K are nonnegative constants such that

\displaystyle v(t)\leq C+\int\limits_0^tKv(s)\,ds

for all t\in[0,\alpha] then for all t in this interval we have

\displaystyle v(t)\leq Ce^{Kt}

That is, we can conclude that v grows no faster than an exponential function. Exponential growth may seem fast, but at least it doesn’t blow up to an infinite singularity in finite time, no matter what Kurzweil seems to think.

Anyway, first let’s deal with strictly positive C. If we define

\displaystyle V(t)=C+\int\limits_0^tKv(s)\,ds>0

then by assumption we have v(t)\leq V(t). Differentiating, we find V'(t)=Kv(t), and thus

\displaystyle\frac{d}{dt}\left(\log(V(t))\right)=\frac{V'(t)}{V(t)}=\frac{Kv(t)}{V(t)}\leq K

Integrating, we find

\displaystyle\log(V(t))\leq\log(V(0))+Kt=\log(C)+Kt

Finally we can exponentiate to find

\displaystyle v(t)\leq V(t)\leq Ce^{Kt}

proving Gronwall’s inequality.

If C=0, in our hypothesis, the hypothesis is true for any \bar{C}>0 in its place, and so we see that v(t)\leq\bar{C}e^{Kt} for any positive \bar{C}, which means that v(t) must be zero, as required by Gronwall’s inequality in this case.

About these ads

May 11, 2011 - Posted by | Analysis, Differential Equations

3 Comments »

  1. [...] Gronwall’s inequality tells us that , which is exactly the inequality we asserted [...]

    Pingback by Control on the Divergence of Solutions « The Unapologetic Mathematician | May 13, 2011 | Reply

  2. [...] turns out to be a lot more messy, but essentially doable by similar methods and a generalization of Gronwall’s inequality. For the sake of getting back to differential equations I’m going to just assert that not [...]

    Pingback by Smooth Dependence on Initial Conditions « The Unapologetic Mathematician | May 16, 2011 | Reply

  3. [...] Armstrong: Another Existence Proof (of the convergence of the Picard iteration), Gronwall’s Inequality, Lie [...]

    Pingback by Sixth Linkfest | May 25, 2011 | Reply


Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

Follow

Get every new post delivered to your Inbox.

Join 391 other followers

%d bloggers like this: