14. Bellman-Gronwall lemma

Introduction

We will continue to look at the existence and uniqueness theorem, following the previous module, particularly on uniqueness. We will discuss Bellman-Gronwall lemma, a tool that we use to prove a uniqueness of a function, when we have some ideas that a function is a solution to an ODE.

Bellman-Gronwall lemma

Suppose we have function \(u(\cdot), K(\cdot)\) that are real-valued, \(PC\) and positive on \(R_+\). Let us also have some constant \(c_1 \ge 0, t_0 \ge 0\), the Bellman-Gronwall states the following:

Bellman-Gronwall lemma

If \(u(t) \le c_1 + \int_{t_0}^t K(\tau)u(\tau)d\tau\) then:

\[ u(t) \le c_1 e^{\int_{t_0}^t K(\tau)d\tau} \]

It’s worth noticing that in the resulting inequality, \(u(t)\) only appears on the left hand side.

Proof

Denotes \(U(t) = c_1 + \int_{t_0}^t K(\tau)u(\tau)d\tau\), we have

\[ u(t) \le U(t) \]

Multiply both sides by the function \(K(t)e^{-\int_{t_0}^t K(\tau)d\tau} \ge 0\), we have:

\[ u(t)K(t)e^{-\int_{t_0}^t K(\tau)d\tau} \le U(t)K(t)e^{-\int_{t_0}^t K(\tau)d\tau} \]

proof continue: https://people.math.wisc.edu/~jwrobbin/angelic/gronwall.pdf

Using Bellman-Gronwall lemma in proof of uniqueness of solution to differential equation

Suppose we have \(\dot{x} = f(x, t)\), \(x(t_0) = x_0\) with solution \(\phi(t)\). How do we know that there is no other solution? One way to prove this is to assume that it is not unique, and the find a contradiction.

Proving uniqueness

In proving uniqueness, we often do contradiction.

Assume that \(\phi(t)\) is not the unique solution, let us have \(\chi(t)\) also the solution. We know that:

\[\begin{split} \begin{align*} \dot{\phi}(t) &= f(\phi, t), \quad \phi(t_0) = x_0 \rightarrow \phi(t) = x_0 + \int_{t_0}^t f(\phi, t)dt\\ \dot{\chi}(t) &= f(\chi, t), \quad \chi(t_0) = x_0 \rightarrow \chi(t) = x_0 + \int_{t_0}^t f(\chi, t)dt\\ \end{align*} \end{split}\]

If we take the difference between these two solutions:

\[ \begin{align*} ||\phi(t) - \chi(t)|| \le \int_{t_0}^t ||f(\phi, \tau) - f(\chi, \tau)|| d\tau \end{align*} \]

Because \(f(\cdot)\) is Lipschitz continuous, there there exists a PC function \(K(\cdot)\) such that \(||f(x_1) - f(x_2)|| \le K(t)||x_1 - x_2||\). Also because \(K(\cdot)\) is PC, there exists a supremum value of \(K(\cdot)\) called \(\bar{K}\). Then we have:

\[ ||\phi(t) - \chi(t)|| \le \bar{K} \int_{t_0}^t||\phi(\tau) - \chi(\tau)|| d\tau \]

Now this looks like the form that we can use the Bellman-Gronwall lemma on. For this, we will have \(c_1=0\), and so with Bellman-Gronwall lemma, we can further bound this to:

\[ ||\phi(t) - \chi(t)|| \le \bar{K} \int_{t_0}^t||\phi(\tau) - \chi(\tau)|| d\tau \rightarrow c_1 e^{\int_{t_0}^t \bar{K} d\tau} = 0 \]

Then we conclude that \(\phi(t) = \chi(t)\).