EDB — 1T1

[1T1] Prerequisites:[118]↺↻,[11K]↺↻.Difficulty:*.

Let \(V=ℂ^{n× n}\) a matrix space, we equip it with a submultiplicative norm \(\| C\| _ V\). Let \(C∈ V\) and let \(A,B:ℝ→ V\) be continuous curves in space of matrices.

• We recursively define \(Q_ 0=C\), and

  \[ Q_{n+1}(s)=∫_ 0^ s A(𝜏) Q_ n(𝜏)B(𝜏)\, {\mathbb {d}}𝜏\quad ; \]

  show that the series

  \[ Y(t)=∑_{n=0}^∞ Q_ n(t) \]

  is well defined, showing that, for every \(T{\gt}0\), it converges totally in the space of continuous functions \(C^ 0=C^ 0([-T,T]→ V)\), endowed with the norm

  \[ \| Q\| _{C^ 0}{\stackrel{.}{=}}\max _{|t|≤ T} \| Q(t)\| _ V \quad . \]

• Show that the function just defined is the solution of the differential equation

  \[ \frac{d\hskip5.5pt}{d{t}} Y(t) = A(t) Y(t) B(t)~ ~ ~ ,~ ~ ~ Y(0)=C~ ~ . \]

• If \(A,B\) are constant, note that

  \[ Y(t)=∑_{n=0}^∞ t^ n \frac{A^ n C B^ n}{n!}\quad . \]