# 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}}\mathit{d}s{Z}^{\prime}(s)\parallel $ | |||

$\le C\parallel I\parallel +C{\displaystyle {\int}_{0}^{t}}\mathit{d}s\parallel {Z}^{\prime}(s)\parallel $ | |||

$\le C+C{\displaystyle {\int}_{0}^{t}}\mathit{d}s\parallel {Z}^{\prime}(s)\parallel $ |

For convenience, let us define $f(t)={\int}_{0}^{t}\mathit{d}s\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

$${\int}_{0}^{t}\mathit{d}s\parallel {Z}^{\prime}(s)\parallel \le C\left({e}^{Ct}-1\right).$$ |

Finally, by the triangle inequality,

$$\parallel Z(t)-I\parallel =\parallel {\int}_{0}^{t}\mathit{d}sZ(s)\parallel \le {\int}_{0}^{t}\mathit{d}s\parallel Z(s)\parallel .$$ |

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 |