Let $A \in \mathcal{M}_{n}(\mathbb{R})$ be a positively stable matrix. For all real $t$, we set $V(t) = \exp\left(-tA^{\top}\right) \exp(-tA)$ and $W(t) = \int_{0}^{t} V(s) \mathrm{d}s$. a) Show that, for all real $t, V(t) \in \mathcal{S}_{n}^{++}(\mathbb{R})$ and that, if $t > 0, W(t) \in \mathcal{S}_{n}^{++}(\mathbb{R})$. b) Show that, for all real $t, A^{\top} W(t) + W(t) A = I_{n} - V(t)$. c) What do we obtain by letting $t$ tend to $+\infty$ in the previous equality? Deduce that the matrix $B$ of question III.C.2 is positive definite.
Let $A \in \mathcal{M}_{n}(\mathbb{R})$ be a positively stable matrix. For all real $t$, we set $V(t) = \exp\left(-tA^{\top}\right) \exp(-tA)$ and $W(t) = \int_{0}^{t} V(s) \mathrm{d}s$.
a) Show that, for all real $t, V(t) \in \mathcal{S}_{n}^{++}(\mathbb{R})$ and that, if $t > 0, W(t) \in \mathcal{S}_{n}^{++}(\mathbb{R})$.
b) Show that, for all real $t, A^{\top} W(t) + W(t) A = I_{n} - V(t)$.
c) What do we obtain by letting $t$ tend to $+\infty$ in the previous equality? Deduce that the matrix $B$ of question III.C.2 is positive definite.