Let $T > 0$ be a real number and $n \in \mathbb{N}^*$ be a natural integer. Let $A : \mathbb{R} \rightarrow \mathscr{M}_n(\mathbb{C})$ be an application continuous on $\mathbb{R}$ and $T$-periodic. We consider the differential system $$X'(t) = A(t) X(t) \tag{2}$$ Let $\mathscr{S}$ be the space of solutions in $\mathscr{C}^1(\mathbb{R}, \mathbb{C}^n)$ of (2). Using the notation and results of question 17b, deduce that if the real parts of the $\lambda_i$ for $1 \leqslant i \leqslant n$ are strictly negative and if $Y$ is any solution of (2), then $$\lim_{t \rightarrow +\infty} Y(t) = 0.$$
Let $T > 0$ be a real number and $n \in \mathbb{N}^*$ be a natural integer. Let $A : \mathbb{R} \rightarrow \mathscr{M}_n(\mathbb{C})$ be an application continuous on $\mathbb{R}$ and $T$-periodic. We consider the differential system
$$X'(t) = A(t) X(t) \tag{2}$$
Let $\mathscr{S}$ be the space of solutions in $\mathscr{C}^1(\mathbb{R}, \mathbb{C}^n)$ of (2). Using the notation and results of question 17b, deduce that if the real parts of the $\lambda_i$ for $1 \leqslant i \leqslant n$ are strictly negative and if $Y$ is any solution of (2), then
$$\lim_{t \rightarrow +\infty} Y(t) = 0.$$