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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