Exercise 8, § 2

Let $G$ be a complete normed algebra with unit $\mathbf{e}$ over the field $\mathbf{R}$.

a) Let $t \mapsto \mathbf{a}(t)$ be a regulated function on an interval $J \subset \mathbf{R}$ with values in $G$. Show that the integral $\mathbf{u}$ of the linear equation $d\mathbf{x}/dt = \mathbf{a}(t)\mathbf{x}$ which takes the value $\mathbf{e}$ at a point $t_0 \in J$ is invertible on $J$, and that its inverse is the solution of the equation $d\mathbf{x}/dt = -\mathbf{x}\mathbf{a}(t)$ (if $\mathbf{v}$ is the integral of this last equation which takes the value $\mathbf{e}$ at the point $t_0$, consider the linear equations satisfied by $\mathbf{uv}$ and $\mathbf{vu}$). Deduce that, for every $\mathbf{x}_0 \in G$, the integral of $d\mathbf{x}/dt = \mathbf{a}(t)\mathbf{x}$ which takes the value $\mathbf{x}_0$ at the point $t_0$ is equal to $\mathbf{u}(t)\mathbf{x}_0$.

b) Let $\mathbf{a}(t), \mathbf{b}(t)$ be two regulated functions on $J$, and $\mathbf{u}$ and $\mathbf{v}$ the integrals of the equations $d\mathbf{x}/dt = \mathbf{a}(t)\mathbf{x}, d\mathbf{x}/dt = \mathbf{x}\mathbf{b}(t)$ taking the value $\mathbf{e}$ at the point $t_0$. Show that the integral of the equation $d\mathbf{x}/dt = \mathbf{a}(t)\mathbf{x} + \mathbf{x}\mathbf{b}(t)$ which takes the value $\mathbf{x}_0$ at the point $t_0$ is equal to $\mathbf{u}(t)\mathbf{x}_0 \mathbf{v}(t)$.

c) Let $\mathbf{a}(t), \mathbf{b}(t), \mathbf{c}(t), \mathbf{d}(t)$ be four regulated functions on $J$, and $(\mathbf{u}, \mathbf{v})$ a solution of the system of two linear equations

Show that, if $v$ is invertible on $J$, then $w = uv^{-1}$ is an integral of the equation $d\mathbf{z}/dt = \mathbf{b}(t) + \mathbf{a}(t)\mathbf{z} - \mathbf{z}\mathbf{d}(t) - \mathbf{z}\mathbf{c}(t)\mathbf{z}$ (the "Ricatti equation"); and conversely. Deduce that every integral of this last equation on $J$, taking the value $x_0$ at the point $t_0$, is of the form $(A(t)x_0 + B(t)e)(C(t)x_0 + D(t)e)^{-1}$, where $A(t)$, $B(t)$, $C(t)$ and $D(t)$ are continuous maps from $J$ into $\mathscr{L}(E)$ satisfying the identity $U.(xy) = (U.x)y$.

d) Let $w_1$ be an integral of $d\mathbf{z}/dt = \mathbf{b}(t) + \mathbf{a}(t)\mathbf{z} - \mathbf{z}\mathbf{d}(t) - \mathbf{z}\mathbf{c}(t)\mathbf{z}$ on $J$; show that if $w$ is another integral of this equation such that $w - w_1$ is invertible on $J$ then $w - w_1$ can be expressed in terms of integrals of the equations $d\mathbf{x}/dt = -(d + cw_1)\mathbf{x}$ and $d\mathbf{x}/dt = \mathbf{x}(\mathbf{a} - w_1\mathbf{c})$ (consider the equation satisfied by $(w - w_1)^{-1}$, and use $b$) ).