For any matrix $A \in \mathscr { M } _ { n } ( \mathbf { R } )$, we denote by $u _ { A }$ the endomorphism canonically associated with $A$ in $\mathbf { R } ^ { n }$ and $v _ { A }$ the endomorphism of $\mathbf { C } ^ { n }$ canonically associated with $A$, viewed as a matrix in $\mathscr { M } _ { n } ( \mathbf { C } )$. We keep the notation $\| . \| _ { c }$ for the norm introduced in part A on $\mathcal { L } \left( \mathbf { C } ^ { n } \right)$ and we use $\| . \| _ { r }$ on $\mathcal { L } \left( \mathbf { R } ^ { n } \right)$. Show that there exists $C > 0$ such that: $$\forall A \in \mathscr { M } _ { n } ( \mathbf { R } ) , \quad \forall t \in \mathbf { R } , \quad \left\| e ^ { t u _ { A } } \right\| _ { r } \leqslant C \left\| e ^ { t v _ { A } } \right\| _ { c }$$

We consider $u$ an endomorphism of $\mathbf { R } ^ { n }$, and $A \in \mathscr { M } _ { n } ( \mathbf { R } )$ its matrix in the canonical basis. We denote by $S p ( A )$ the complex spectrum of $A$. Let $g _ { x _ { 0 } }$ be the unique $\mathcal { C } ^ { 1 }$ solution on $\mathbf { R } _ { + }$ of: $$\left\{ \begin{aligned} y ^ { \prime } & = u ( y ) \\ y ( 0 ) & = x _ { 0 } \end{aligned} \right.$$
Show that: $$\forall x _ { 0 } \in \mathbf { R } ^ { n } , \quad \lim _ { t \rightarrow + \infty } \left\| g _ { x _ { 0 } } ( t ) \right\| = 0 \Longleftrightarrow S p ( A ) \subset \mathbf { R } _ { - } ^ { * } + i \mathbf { R } .$$

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 $t \in \mathbf { R } _ { + }$.

In this part we consider a map $\varphi$ from $\mathbf { R } ^ { n }$ to $\mathbf { R } ^ { n }$ of class $\mathcal { C } ^ { 1 }$ such that $\varphi ( 0 ) = 0$, and denoting $a = d \varphi ( 0 )$, such that all eigenvalues of $a$ have strictly negative real part.
Show that the function $$b : \left\lvert \, \begin{array} { r l l } \mathbf { R } ^ { n } \times \mathbf { R } ^ { n } & \rightarrow & \mathbf { R } \\ ( x , y ) & \mapsto & \int _ { 0 } ^ { + \infty } \left\langle e ^ { t a } ( x ) \mid e ^ { t a } ( y ) \right\rangle d t \end{array} \right.$$ is well-defined and that it defines an inner product on $\mathbf { R } ^ { n }$.

We consider the Euclidean space $E = \mathscr{M}_{N,1}(\mathbf{R})$ equipped with the inner product $\langle X, Y \rangle = \sum_{i=1}^{N} X[i] Y[i] \pi[i]$, and the matrix $H_t$ defined by $\forall (i,j) \in \llbracket 1;N \rrbracket^2, H_t[i,j] = e^{-t} \sum_{n=0}^{+\infty} \frac{t^n K^n[i,j]}{n!}$. Let $X \in E$. We denote by $\psi_X$ the function defined from $\mathbf{R}$ to $E$ by $\psi_X : t \mapsto H_t X$ and $\varphi_X$ the function defined from $\mathbf{R}$ to $\mathbf{R}$ by $\varphi_X : t \mapsto \|H_t X\|^2$. Justify that $\psi_X$ is differentiable and that for all $t$ in $\mathbf{R}$, $$\psi_X'(t) = -(I_N - K) H_t X$$

Show that there exists a unique constant solution of equation $\left( E _ { \ell } \right)$, denoted $\gamma \in \mathbf { R }$, and verify that the solution $u$ found in question 1 satisfies
$$\lim _ { x \rightarrow + \infty } u ( x ) = \gamma .$$
where $\left( E _ { \ell } \right) : \quad u ^ { \prime } ( x ) + u ( x ) + 1 = \frac { 1 } { 2 } ( 1 + u ( x ) )$.