Consider A to be a matrix that has all eigenvalues $\lambda$ with negative real part, that is, $Re(\lambda) < 0$.
a) Show that the integral $\int_{0}^\infty (e^{As})^T e^{As}\,ds$ is convergent, where $(e^{As})^T$ is the transpose of $e^{As}$.
b) If $C=\int_{0}^\infty (e^{As})^T e^{As}\,ds$, show that $CA +A^tC=-I$
Hint: Integrate both sides of $\frac{d}{ds} ((e^{As})^T e^{As})= (e^{As})^T e^{As}A+A^T(e^{As})^T e^{As}$ for $s$ varying from $0$ to $\infty$.
Could someone help with letter a, please? Thanks!!
The second part is easier, we'll start with that. We begin by noting that $$ \frac{d}{ds} ((e^{As})^T e^{As})= (e^{As})^T e^{As}A+A^T(e^{As})^T e^{As} $$ Integrating the left side, we have $$ \int_0^\infty \frac{d}{ds} ((e^{As})^T e^{As})\,ds = \left. (e^{As})^T e^{As} \right|_0^\infty = \lim_{s \to \infty} (e^{As})^T e^{As} - (e^{A(0)})^T e^{A(0)} $$ However, by the continuity of the exponential map, we know that $\lim_{s \to \infty} (e^{As})^T e^{As} = 0$. So, the left side is simply $-(e^{A(0)})^T e^{A(0)} = -I$.
Integrating the other side, we have $$ \int_0^\infty \left[(e^{As})^T e^{As}A+A^T(e^{As})^T e^{As}\right]ds =\\ \left[\int_0^\infty(e^{As})^T e^{As}ds\right]A+A^T\left[ \int_0^\infty (e^{As})^T e^{As}ds\right] $$ Setting the two integrated sides equal, we have the desired result.
For part (a), let $\|\cdot\|$ denote any matrix norm (such as the spectral norm). It suffices for us to show that $$ \int_0^\infty \left\|(e^{As})^Te^{(As)}\right\|\,ds < \infty $$ let $a > 0$ denote the smallest value $-\text{Re}(\lambda)$, where $\lambda$ is an eigenvalue of $A$. Note that $$ \lim_{n \to \infty} \left\|(e^A)^n \right\|^{1/n} = e^{-a} < 1 $$ The same applies to $A^T$. Fix $\epsilon \in (e^{-a},1)$. Let $n_0$ be such that $\left\|(e^A)^n \right\|^{1/n}, \left\|([e^A]^T)^n \right\|^{1/n} < \epsilon$ for $n > n_0$.
We then note that
$$ \int_0^\infty \left\|(e^{As})^Te^{(As)}\right\|\,ds =\\ \sum_{k=1}^\infty \int_{(k-1)n_0}^{kn_0} \left\|(e^{As})^Te^{(As)}\right\|\,ds \leq \\ \left(\sum_{k=1}^\infty \left(\left\|e^{(An_0)}\right\| \left\|(e^{(An_0)})^T\right\|\right)^{(k-1)}\right) \int_{0}^{n_0} \left\|(e^{A(s)})^T\right\| \left\|e^{(A(s))}\right\|\,ds $$