Categories: Mathematics.

Grönwall-Bellman inequality

Suppose we have a first-order ordinary differential equation for some function $$u(t)$$, and that it can be shown from this equation that the derivative $$u'(t)$$ is bounded as follows:

\begin{aligned} u'(t) \le \beta(t) \: u(t) \end{aligned}

Where $$\beta(t)$$ is known. Then Grönwall’s inequality states that the solution $$u(t)$$ is bounded:

\begin{aligned} \boxed{ u(t) \le u(0) \exp\!\bigg( \int_0^t \beta(s) \dd{s} \bigg) } \end{aligned}

Grönwall’s inequality can be generalized to non-differentiable functions. Suppose we know:

\begin{aligned} u(t) \le \alpha(t) + \int_0^t \beta(s) \: u(s) \dd{s} \end{aligned}

Where $$\alpha(t)$$ and $$\beta(t)$$ are known. Then the Grönwall-Bellman inequality states that:

\begin{aligned} \boxed{ u(t) \le \alpha(t) + \int_0^t \alpha(s) \: \beta(s) \exp\!\bigg( \int_s^t \beta(r) \dd{r} \bigg) \dd{s} } \end{aligned}

In the special case where $$\alpha(t)$$ is non-decreasing with $$t$$, the inequality reduces to:

\begin{aligned} \boxed{ u(t) \le \alpha(t) \exp\!\bigg( \int_0^t \beta(s) \dd{s} \bigg) } \end{aligned}

1. U.H. Thygesen, Lecture notes on diffusions and stochastic differential equations, 2021, Polyteknisk Kompendie.

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