Let $M \in M_{n}(\mathbf{R})$. We denote by (S) the differential system: $$(\mathrm{S}) \quad X' = MX$$ where $X$ is a function from the variable $t$ in $\mathbf{R}$ to $\mathbf{R}^{n}$, differentiable on $\mathbf{R}$.
We assume $n = 2$, we then denote $X = (x; y)$ where $x$ and $y$ are two functions differentiable from $\mathbf{R}$ to $\mathbf{R}$ and we set $z = x + iy$.
Let $M \in M_{2}(\mathbf{R})$ be semi-simple. Give a necessary and sufficient condition, concerning the real and imaginary parts of the eigenvalues of $M$, for every solution of (S) to have each of its coordinates tend to 0 as $+\infty$.

Let $M \in M_{n}(\mathbf{R})$. We denote by (S) the differential system: $$(\mathrm{S}) \quad X' = MX$$ where $X$ is a function from the variable $t$ in $\mathbf{R}$ to $\mathbf{R}^{n}$, differentiable on $\mathbf{R}$.
Let $T \in M_{n}(\mathbf{R})$. We assume that $M$ is similar to $T$ in $M_{n}(\mathbf{R})$ and we denote by $(\mathrm{S}^{*})$ the differential system $$(\mathrm{S}^{*}) \quad Y' = TY$$
We consider the following assertions:

• [$\mathbf{A}_{1}$] $\chi_{M}$ is a Hurwitz polynomial;
• [$\mathbf{A}_{2}$] The solutions of (S) tend to $0_{\mathbf{R}^{n}}$ as $t$ tends to $+\infty$;
• [$\mathbf{A}_{3}$] There exist $\alpha > 0$ and $k > 0$ such that for every solution $\Phi$ of (S), $$\forall t \geq 0 \quad : \quad \|\Phi(t)\| \leq k e^{-\alpha t} \|\Phi(0)\|.$$

Let $T \in M_{n}(\mathbf{R})$. We assume that $T$ satisfies the following condition: $$(\mathrm{C}) \quad \exists \beta \in \mathbf{R}_{+}^{*}, \forall X \in \mathbf{R}^{n} : \langle TX, X \rangle \leq -\beta \|X\|^{2}.$$
Prove that $\mathrm{A}_{3}$ is true with $k = 1$ for every solution $\Phi$ of $(\mathrm{S}^{*})$.
Hint: one may introduce the function $t \mapsto e^{2\beta t} \|\Phi(t)\|^{2}$.

Let $M \in M_{n}(\mathbf{R})$. We denote by (S) the differential system: $$(\mathrm{S}) \quad X' = MX$$ where $X$ is a function from the variable $t$ in $\mathbf{R}$ to $\mathbf{R}^{n}$, differentiable on $\mathbf{R}$.
We consider the following assertions:

• [$\mathbf{A}_{1}$] $\chi_{M}$ is a Hurwitz polynomial;
• [$\mathbf{A}_{2}$] The solutions of (S) tend to $0_{\mathbf{R}^{n}}$ as $t$ tends to $+\infty$;
• [$\mathbf{A}_{3}$] There exist $\alpha > 0$ and $k > 0$ such that for every solution $\Phi$ of (S), $$\forall t \geq 0 \quad : \quad \|\Phi(t)\| \leq k e^{-\alpha t} \|\Phi(0)\|.$$

Assume that $M \in M_{n}(\mathbf{R})$ is semi-simple. Prove that the assertions $\mathrm{A}_{1}$, $\mathrm{A}_{2}$ and $\mathrm{A}_{3}$ are equivalent.
Hint: one may start with $A_{3}$ implies $A_{2}$.

In this part, $a$ denotes an endomorphism of $\mathbf { C } ^ { n }$. We use the decomposition $\mathbf { C } ^ { n } = \bigoplus _ { i = 1 } ^ { r } E _ { i }$ where $E _ { i } = \operatorname { Ker } \left( a - \lambda _ { i } id _ { \mathbf { C } ^ { n } } \right) ^ { m _ { i } }$, with $a_i = p_i a q_i$ the endomorphism of $E_i$ and $\|.\|_i$ the norm on $\mathcal{L}(E_i)$.
Show moreover that: $$\forall i \in \llbracket 1 ; r \rrbracket , \quad \forall t \in \mathbf { R } , \quad \left\| e ^ { t a _ { i } } \right\| _ { i } \leqslant \left| e ^ { t \lambda _ { i } } \right| \sum _ { k = 0 } ^ { m _ { i } - 1 } \frac { | t | ^ { k } } { k ! } \left\| a _ { i } - \lambda _ { i } id _ { E _ { i } } \right\| _ { i } ^ { k }$$

In this part, $a$ denotes an endomorphism of $\mathbf { C } ^ { n }$. We use the decomposition $\mathbf { C } ^ { n } = \bigoplus _ { i = 1 } ^ { r } E _ { i }$ where $E _ { i } = \operatorname { Ker } \left( a - \lambda _ { i } id _ { \mathbf { C } ^ { n } } \right) ^ { m _ { i } }$, with $\|.\|_c$ the norm on $\mathcal{L}(\mathbf{C}^n)$.
Deduce the existence of a polynomial $P$ with real coefficients such that: $$\forall t \in \mathbf { R } , \quad \left\| e ^ { t a } \right\| _ { c } \leqslant P ( | t | ) \sum _ { i = 1 } ^ { r } e ^ { t \operatorname { Re } \left( \lambda _ { i } \right) }$$ where $\operatorname { Re } ( z )$ denotes the real part of a complex number $z$.

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 $\mathbf{R}$, $$\psi_X'(t) = -(I_N - K) H_t X$$

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 a continuous application 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 (in particular, $Z_k(t) = e^{\lambda_k t}\left(\sum_{i=0}^{n-1} t^i R_{i,k}(t)\right)$ where $R_{i,k}$ are continuous and $T$-periodic), 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 $$\lim_{t \rightarrow +\infty} Y(t) = 0$$

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 $\left( E _ { \ell } \right) : \quad u ^ { \prime } ( x ) + u ( x ) + 1 = \frac { 1 } { 2 } ( 1 + u ( x ) )$.

Consider that complex-valued functions $p(x)$ and $q(x)$ satisfy the simultaneous ordinary differential equations below: $$\frac{\mathrm{d}p(x)}{\mathrm{d}x} = -ib\, q(x)\exp(-2iax),$$ $$\frac{\mathrm{d}q(x)}{\mathrm{d}x} = -ib\, p(x)\exp(2iax).$$ Here, $i$ is the imaginary unit, and $a$ and $b$ are real constants. Let $f(x) = p(x)\exp(iax)$ and $g(x) = q(x)\exp(-iax)$.
Show that the value of $|f(x)|^{2} + |g(x)|^{2}$ is independent of $x$, where $|A|$ denotes the absolute value of a complex number $A$.