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Let $A$ be an $n \times n$ matrix. Then the solution of the initial value problem

\begin{align*}

\dot{x}(t) = A x(t), \quad x(0) = x_0

\end{align*}

is given by $x(t) = \mathrm{e}^{At} x_0$.

I am interested in the following matrix

\begin{align*}

\int_{0}^T \mathrm{e}^{At}\, dt

\end{align*}

for some $T>0$. Can one write down a general solution to this without distinguishing cases (e.g. $A$ nonsingular)?

Is this matrix always invertible?

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**Case I.** If $A$ is nonsingular, then

$$

\int_0^T\mathrm{e}^{tA}\,dt=A^{-1}\big(\mathrm{e}^{TA}-I\big),

$$

where $I$ is the identity matrix.

**Case II.** If $A$ is singular, then using the Jordan form we can write $A$ as

$$

A=U^{-1}\left(\begin{matrix}B&0\\0&C\end{matrix}\right)U,

$$

where $C$ is nonsingular, and $B$ is strictly upper triangular. Then

$$

\mathrm{e}^{tA}=U^{-1}\left(\begin{matrix}\mathrm{e}^{tB}&0\\0&\mathrm{e}^{tC}

\end{matrix}\right)U,

$$

and

$$

\int_0^T\mathrm{e}^{tA}\,dt=U^{-1}\left(\begin{matrix}\int_0^T\mathrm{e}^{tB}dt&0\\0&C^{-1}\big(\mathrm{e}^{TC}-I\big)

\end{matrix}\right)U

$$

But $\int_0^T\mathrm{e}^{tB}dt$ may have different expressions. For example if

$$

B_1=\left(\begin{matrix}0&0\\0&0\end{matrix}\right), \quad

B_2=\left(\begin{matrix}0&1\\0&0\end{matrix}\right),

$$

then

$$

\int_0^T\mathrm{e}^{tB_1}dt=\left(\begin{matrix}T&0\\0&T\end{matrix}\right), \quad

\int_0^T\mathrm{e}^{tB_2}dt=\left(\begin{matrix}T&T^2/2\\0&T\end{matrix}\right).

$$

The general formula is the power series

$$ \int_0^T e^{At} dt = T \left( I + \frac{AT}{2!} + \frac{(AT)^2}{3!} + \dots + \frac{(AT)^{n-1}}{n!} + \dots \right) $$

Note that also

$$ \left(\int_0^T e^{At} dt \right) A + I = e^{AT} $$

is always satisfied.

A sufficient condition for this matrix to be non-singular is the so-called Kalman-Ho-Narendra Theorem, which states that the matrix $\int_0^T e^{At} dt$ is invertible if

$$ T(\mu – \lambda) \neq 2k \pi i $$

for any nonzero integer $k$, where $\lambda$ and $\mu$ are any pair of eigenvalues of $A$.

Note to the interested: This matrix also comes from the discretization of a continuous linear time invariant system. It can also be said that controllability is preserved under discretization if and only if this matrix has an inverse.

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