We consider $u$ an endomorphism of $\mathbf { R } ^ { n }$, and $A \in \mathscr { M } _ { n } ( \mathbf { R } )$ its matrix in the canonical basis. Let $g _ { x _ { 0 } }$ be the unique $\mathcal { C } ^ { 1 }$ solution on $\mathbf { R } _ { + }$ of: $$\left\{ \begin{aligned} y ^ { \prime } & = u ( y ) \\ y ( 0 ) & = x _ { 0 } \end{aligned} \right.$$ In this question we assume that all eigenvalues of $A$ have strictly negative real part. Show then that there exist two strictly positive constants $C _ { 2 }$ and $\alpha$ such that: $$\forall t \in \mathbf { R } _ { + } , \quad \left\| e ^ { t u } \right\| _ { r } \leqslant C _ { 2 } e ^ { - \alpha t }$$ and deduce a bound on $\left\| g _ { x _ { 0 } } ( t ) \right\|$ for $t \in \mathbf { R } _ { + }$.
We consider $u$ an endomorphism of $\mathbf { R } ^ { n }$, and $A \in \mathscr { M } _ { n } ( \mathbf { R } )$ its matrix in the canonical basis. Let $g _ { x _ { 0 } }$ be the unique $\mathcal { C } ^ { 1 }$ solution on $\mathbf { R } _ { + }$ of:
$$\left\{ \begin{aligned} y ^ { \prime } & = u ( y ) \\ y ( 0 ) & = x _ { 0 } \end{aligned} \right.$$
In this question we assume that all eigenvalues of $A$ have strictly negative real part. Show then that there exist two strictly positive constants $C _ { 2 }$ and $\alpha$ such that:
$$\forall t \in \mathbf { R } _ { + } , \quad \left\| e ^ { t u } \right\| _ { r } \leqslant C _ { 2 } e ^ { - \alpha t }$$
and deduce a bound on $\left\| g _ { x _ { 0 } } ( t ) \right\|$ for $t \in \mathbf { R } _ { + }$.