bound on matrix differential equation

Suppose that $A$ and $Z$ are two square matrices dependent on a parameter which satisfy the differential equation

$$Z^{\prime }(t)=A(t)Z(t)$$

withh initial condition $Z(0)=I$. Letting $\parallel \cdot \parallel$ denote the matrix operator norm, we will show that, if $\parallel A(t)\parallel \le C$ for some constant $C$ when $0\le t\le R$, then

$$\parallel Z(t)-I\parallel \le C\left(e^{Ct}-1\right)$$

when $0\le t\le R$.

We begin by applying the product inequality for the norm, then employing the triangle inequality (both in the sum and integral forms) after expressing $Z$ as the integral of its derivative:

$\parallel Z^{\prime }(t)\parallel$ $\le \parallel A(t)\parallel \parallel Z(t)\parallel$

$\le C\parallel Z(t)\parallel$

$=C\parallel I+{\displaystyle {\int }_0^t}𝑑s{Z}^{\prime }(s)\parallel$

$\le C\parallel I\parallel +C{\displaystyle {\int }_0^t}𝑑s\parallel Z^{\prime }(s)\parallel$

$\le C+C{\displaystyle {\int }_0^t}𝑑s\parallel Z^{\prime }(s)\parallel$

For convenience, let us define $f(t)={\int }_0^t𝑑s\parallel Z^{\prime }(s)\parallel$. Then we have $f^{\prime }(t)\le C+Cf(t)$ according to the foregoing derivation. By the product rule,

$$\frac{d}{dt}\left(e^{-Ct}f(t)\right)=e^{-Ct}(f^{\prime }(t)-Cf(t)).$$

Since $f^{\prime }(t)-Cf(t)\le C$, we have

$$\frac{d}{dt}\left(e^{-Ct}f(t)\right)\le C{e}^{-Ct}.$$

Taking the integral from $0$ to $t$ of both sides and noting that $f(0)=0$, we have

$$e^{-Ct}f(t)\le C\left(1-e^{-Ct}\right).$$

Multiplying both sides by $e^{Ct}$ and recalling the definition of $f$, we conclude

$${\int }_0^t𝑑s\parallel Z^{\prime }(s)\parallel \le C\left(e^{Ct}-1\right).$$

Finally, by the triangle inequality,

$$\parallel Z(t)-I\parallel =\parallel {\int }_0^t𝑑sZ(s)\parallel \le {\int }_0^t𝑑s\parallel Z(s)\parallel .$$

Combining this with the inequality derived in the last paragraph produces the answer:

$$\parallel Z(t)-I\parallel \le C\left(e^{Ct}-1\right).$$

Title	bound on matrix differential equation
Canonical name	BoundOnMatrixDifferentialEquation
Date of creation	2013-03-22 18:59:00
Last modified on	2013-03-22 18:59:00
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	4
Author	rspuzio (6075)
Entry type	Theorem
Classification	msc 34A30