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. Let $b(x,y) = \int_0^{+\infty} \langle e^{ta}(x) \mid e^{ta}(y) \rangle\, dt$ be the inner product on $\mathbf{R}^n$, and $q$ the associated quadratic form, i.e., $q(x) = b(x,x)$ for all $x \in \mathbf{R}^n$.
Prove then that: $$\forall x \in \mathbf { R } ^ { n } , \quad d q ( x ) ( a ( x ) ) = 2 b ( x , a ( x ) ) = - \| x \| ^ { 2 }$$

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. Let $b(x,y) = \int_0^{+\infty} \langle e^{ta}(x) \mid e^{ta}(y) \rangle\, dt$ and $q(x) = b(x,x)$. Let $x_0 \in \mathbf{R}^n$ and $f_{x_0}$ the solution of $y' = \varphi(y),\ y(0) = x_0$. For any function $y$ defined on $\mathbf{R}_+$, define: $$\varepsilon ( y ) : \mathbf { R } _ { + } \rightarrow \mathbf { R } ^ { n }, \quad t \mapsto \varphi ( y ( t ) ) - a ( y ( t ) )$$
Verify the equality $$\forall t \in \mathbf { R } _ { + } , \quad q \left( f _ { x _ { 0 } } \right) ^ { \prime } ( t ) = - \left\| f _ { x _ { 0 } } ( t ) \right\| ^ { 2 } + 2 b \left( f _ { x _ { 0 } } ( t ) , \varepsilon \left( f _ { x _ { 0 } } ( t ) \right) \right)$$

Let $q : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous application, periodic with period $T > 0$. We consider the differential equation $$y'' + qy = 0 \tag{1}$$ Justify the existence of two solutions $y_1$ and $y_2$ in $\mathscr{C}^2(\mathbb{R}, \mathbb{C})$ to (1) such that: $$\left\{ \begin{array}{l} y_1(0) = 1 \\ y_1'(0) = 0 \end{array} \quad \text{and} \quad \left\{ \begin{array}{l} y_2(0) = 0 \\ y_2'(0) = 1 \end{array} \right. \right.$$ Justify that $\operatorname{Vect}(y_1, y_2)$ is the set of solutions of (1) in $\mathscr{C}^2(\mathbb{R}, \mathbb{C})$.

Let $q : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous application, periodic with period $T > 0$. We consider the differential equation $$y'' + qy = 0 \tag{1}$$ Let $y_1$ and $y_2$ be the solutions in $\mathscr{C}^2(\mathbb{R}, \mathbb{C})$ to (1) satisfying $$\left\{ \begin{array}{l} y_1(0) = 1 \\ y_1'(0) = 0 \end{array} \quad \text{and} \quad \left\{ \begin{array}{l} y_2(0) = 0 \\ y_2'(0) = 1 \end{array} \right. \right.$$ Show that: $$\forall t \in \mathbb{R}, \quad y_1(t) y_2'(t) - y_1'(t) y_2(t) = 1$$

Let $q : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous application, periodic with period $T > 0$. We consider the differential equation $$y'' + qy = 0. \tag{1}$$ Justify the existence of two solutions $y_1$ and $y_2$ in $\mathscr{C}^2(\mathbb{R}, \mathbb{C})$ to (1) such that: $$\left\{ \begin{array}{l} y_1(0) = 1 \\ y_1'(0) = 0 \end{array} \text{ and } \left\{ \begin{array}{l} y_2(0) = 0 \\ y_2'(0) = 1. \end{array} \right. \right.$$ Justify that $\operatorname{Vect}(y_1, y_2)$ is the set of solutions of (1) in $\mathscr{C}^2(\mathbb{R}, \mathbb{C})$.

Let $q : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous application, periodic with period $T > 0$. We consider the differential equation $$y'' + qy = 0. \tag{1}$$ Let $y_1$ and $y_2$ be the solutions of (1) with initial conditions $y_1(0)=1, y_1'(0)=0$ and $y_2(0)=0, y_2'(0)=1$. Show that: $$\forall t \in \mathbb{R}, \quad y_1(t) y_2'(t) - y_1'(t) y_2(t) = 1.$$

Let $q : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous application, periodic with period $T > 0$. We consider the differential equation $$y'' + qy = 0 \tag{1}$$ Let $y_1$ and $y_2$ be the solutions in $\mathscr{C}^2(\mathbb{R}, \mathbb{C})$ to (1) satisfying $$\left\{ \begin{array}{l} y_1(0) = 1 \\ y_1'(0) = 0 \end{array} \quad \text{and} \quad \left\{ \begin{array}{l} y_2(0) = 0 \\ y_2'(0) = 1 \end{array} \right. \right.$$ Show that if $y \in \mathscr{C}^2(\mathbb{R}, \mathbb{C})$ is a solution of (1), then the function $t \mapsto y(t+T)$ is also one. Deduce that for all $t \in \mathbb{R}$: $$y(t+T) = y(T) y_1(t) + y'(T) y_2(t)$$

Let $q : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous application, periodic with period $T > 0$. We consider the differential equation $$y'' + qy = 0. \tag{1}$$ Let $y_1$ and $y_2$ be the solutions of (1) with initial conditions $y_1(0)=1, y_1'(0)=0$ and $y_2(0)=0, y_2'(0)=1$. Show that if $y \in \mathscr{C}^2(\mathbb{R}, \mathbb{C})$ is a solution of (1), then the function $t \mapsto y(t+T)$ is also one. Deduce that for all $t \in \mathbb{R}$: $$y(t+T) = y(T) y_1(t) + y'(T) y_2(t).$$

Let $q : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous application, periodic with period $T > 0$. We consider the differential equation $$y'' + qy = 0 \tag{1}$$ Let $y_1$ and $y_2$ be the solutions in $\mathscr{C}^2(\mathbb{R}, \mathbb{C})$ to (1) satisfying $$\left\{ \begin{array}{l} y_1(0) = 1 \\ y_1'(0) = 0 \end{array} \quad \text{and} \quad \left\{ \begin{array}{l} y_2(0) = 0 \\ y_2'(0) = 1 \end{array} \right. \right.$$ Let $\mu \in \mathbb{C}^*$, and let $\lambda \in \mathbb{C}$ such that $\mu = e^{\lambda T}$. Show that the following three assertions are equivalent.
(a) Equation (1) has a non-zero solution $y \in \mathscr{C}^2(\mathbb{R}, \mathbb{C})$ that satisfies: $$\forall t \in \mathbb{R}, \quad y(t+T) = \mu y(t)$$
(b) The complex number $\mu$ is a solution of the equation with unknown $x$: $$x^2 - \left(y_1(T) + y_2'(T)\right) x + 1 = 0$$
(c) The differential equation (1) has a non-zero solution $y \in \mathscr{C}^2(\mathbb{R}, \mathbb{C})$ such that: $$\forall t \in \mathbb{R}, \quad y(t) = e^{\lambda t} u(t)$$ where $u \in \mathscr{C}^2(\mathbb{R}, \mathbb{C})$ is a $T$-periodic function.

Let $q : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous application, periodic with period $T > 0$. We consider the differential equation $$y'' + qy = 0. \tag{1}$$ Let $y_1$ and $y_2$ be the solutions of (1) with initial conditions $y_1(0)=1, y_1'(0)=0$ and $y_2(0)=0, y_2'(0)=1$. Let $\mu \in \mathbb{C}^*$, and let $\lambda \in \mathbb{C}$ such that $\mu = e^{\lambda T}$. Show that the following three assertions are equivalent.
(a) Equation (1) has a non-zero solution $y \in \mathscr{C}^2(\mathbb{R}, \mathbb{C})$ that satisfies: $$\forall t \in \mathbb{R}, \quad y(t+T) = \mu y(t).$$
(b) The complex number $\mu$ is a solution of the equation with unknown $x$: $$x^2 - \left(y_1(T) + y_2'(T)\right) x + 1 = 0.$$
(c) The differential equation (1) has a non-zero solution $y \in \mathscr{C}^2(\mathbb{R}, \mathbb{C})$ such that: $$\forall t \in \mathbb{R}, \quad y(t) = e^{\lambda t} u(t),$$ where $u \in \mathscr{C}^2(\mathbb{R}, \mathbb{C})$ is a $T$-periodic function.

Let $q : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous application, periodic with period $T > 0$. We consider the differential equation $$y'' + qy = 0 \tag{1}$$ Let $y_1$ and $y_2$ be the solutions in $\mathscr{C}^2(\mathbb{R}, \mathbb{C})$ to (1) satisfying $$\left\{ \begin{array}{l} y_1(0) = 1 \\ y_1'(0) = 0 \end{array} \quad \text{and} \quad \left\{ \begin{array}{l} y_2(0) = 0 \\ y_2'(0) = 1 \end{array} \right. \right.$$ Let $\mu_1, \mu_2$ be the complex roots of the equation with unknown $x$: $$x^2 - \left(y_1(T) + y_2'(T)\right) x + 1 = 0$$ Show that if $\mu_1 \neq \mu_2$ and if $\lambda$ is a complex number such that $\mu_1 = e^{\lambda T}$, then for any solution $y$ of (1), there exist two $T$-periodic functions $w_1$ and $w_2$, as well as two complex numbers $\alpha$ and $\beta$ such that $$\forall t \in \mathbb{R}, \quad y(t) = \alpha e^{\lambda t} w_1(t) + \beta e^{-\lambda t} w_2(t)$$

Let $q : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous application, periodic with period $T > 0$. We consider the differential equation $$y'' + qy = 0 \tag{1}$$ Let $y_1$ and $y_2$ be the solutions in $\mathscr{C}^2(\mathbb{R}, \mathbb{C})$ to (1) satisfying $$\left\{ \begin{array}{l} y_1(0) = 1 \\ y_1'(0) = 0 \end{array} \quad \text{and} \quad \left\{ \begin{array}{l} y_2(0) = 0 \\ y_2'(0) = 1 \end{array} \right. \right.$$ Let $\mu_1, \mu_2$ be the complex roots of the equation with unknown $x$: $$x^2 - \left(y_1(T) + y_2'(T)\right) x + 1 = 0$$ Suppose that $\mu_1 = \mu_2$. Show that $\mu_1 = \mu_2 = \pm 1$ and that equation (1) admits a periodic solution in $\mathscr{C}^2(\mathbb{R}, \mathbb{C})$.

Let $q : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous application, periodic with period $T > 0$. We consider the differential equation $$y'' + qy = 0. \tag{1}$$ Let $y_1$ and $y_2$ be the solutions of (1) with initial conditions $y_1(0)=1, y_1'(0)=0$ and $y_2(0)=0, y_2'(0)=1$. Let $\mu_1, \mu_2$ be the complex roots of the equation with unknown $x$: $$x^2 - \left(y_1(T) + y_2'(T)\right) x + 1 = 0.$$ Show that if $\mu_1 \neq \mu_2$ and if $\lambda$ is a complex number such that $\mu_1 = e^{\lambda T}$, then for any solution $y$ of (1), there exist two $T$-periodic functions $w_1$ and $w_2$, as well as two complex numbers $\alpha$ and $\beta$ such that $$\forall t \in \mathbb{R}, \quad y(t) = \alpha e^{\lambda t} w_1(t) + \beta e^{-\lambda t} w_2(t).$$

Let $q : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous application, periodic with period $T > 0$. We consider the differential equation $$y'' + qy = 0. \tag{1}$$ Let $y_1$ and $y_2$ be the solutions of (1) with initial conditions $y_1(0)=1, y_1'(0)=0$ and $y_2(0)=0, y_2'(0)=1$. Let $\mu_1, \mu_2$ be the complex roots of the equation with unknown $x$: $$x^2 - \left(y_1(T) + y_2'(T)\right) x + 1 = 0.$$ Suppose that $\mu_1 = \mu_2$. Show that $\mu_1 = \mu_2 = \pm 1$ and that equation (1) admits a periodic solution in $\mathscr{C}^2(\mathbb{R}, \mathbb{C})$.

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 $$X'(t) = A(t) X(t) \tag{2}$$ where $X$ is a function from $\mathbb{R}$ to $\mathbb{C}^n$, of class $\mathscr{C}^1$ on $\mathbb{R}$.
Show that there exists $\mu \in \mathbb{C}^*$ and a non-zero solution $Y \in \mathscr{C}^1(\mathbb{R}, \mathbb{C}^n)$ of (2) such that $$\forall t \in \mathbb{R}, \quad Y(t+T) = \mu Y(t)$$

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 $$X'(t) = A(t) X(t) \tag{2}$$ where $X$ is a function from $\mathbb{R}$ to $\mathbb{C}^n$, of class $\mathscr{C}^1$ on $\mathbb{R}$.
Show that there exists $\mu \in \mathbb{C}^*$ and a non-zero solution $Y \in \mathscr{C}^1(\mathbb{R}, \mathbb{C}^n)$ of (2) such that $$\forall t \in \mathbb{R}, \quad Y(t+T) = \mu Y(t).$$

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 $$X'(t) = A(t) X(t) \tag{2}$$ where $X$ is a function from $\mathbb{R}$ to $\mathbb{C}^n$, of class $\mathscr{C}^1$ on $\mathbb{R}$. Let $\mathscr{S}$ be the space of solutions in $\mathscr{C}^1(\mathbb{R}, \mathbb{C}^n)$ of (2). Let $(Y_1, Y_2, \ldots, Y_n)$ be a basis of $\mathscr{S}$. For $t \in \mathbb{R}$, we denote by $M(t)$ the matrix whose columns are $Y_1(t), \ldots, Y_n(t)$.
Show that for every real number $t$, $M(t) \in \mathrm{GL}_n(\mathbb{C})$ and $M'(t) = A(t) M(t)$.

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 $$X'(t) = A(t) X(t) \tag{2}$$ where $X$ is a function from $\mathbb{R}$ to $\mathbb{C}^n$, of class $\mathscr{C}^1$ on $\mathbb{R}$. Let $\mathscr{S}$ be the space of solutions in $\mathscr{C}^1(\mathbb{R}, \mathbb{C}^n)$ of (2). Let $(Y_1, Y_2, \ldots, Y_n)$ be a basis of $\mathscr{S}$. For $t \in \mathbb{R}$, we denote by $M(t)$ the matrix whose columns are $Y_1(t), \ldots, Y_n(t)$.
Show that the matrix $(M(t))^{-1} M(t+T)$ is independent of $t \in \mathbb{R}$.

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 $$X'(t) = A(t) X(t) \tag{2}$$ where $X$ is a function from $\mathbb{R}$ to $\mathbb{C}^n$, of class $\mathscr{C}^1$ on $\mathbb{R}$. Let $\mathscr{S}$ be the space of solutions in $\mathscr{C}^1(\mathbb{R}, \mathbb{C}^n)$ of (2). Let $(Y_1, Y_2, \ldots, Y_n)$ be a basis of $\mathscr{S}$. For $t \in \mathbb{R}$, we denote by $M(t)$ the matrix whose columns are $Y_1(t), \ldots, Y_n(t)$.
Deduce that there exists $B \in \mathscr{M}_n(\mathbb{C})$ such that: $$\forall t \in \mathbb{R}, \quad M(t+T) = M(t) \exp(TB)$$

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 $$X'(t) = A(t) X(t) \tag{2}$$ where $X$ is a function from $\mathbb{R}$ to $\mathbb{C}^n$, of class $\mathscr{C}^1$ on $\mathbb{R}$. Let $\mathscr{S}$ be the space of solutions in $\mathscr{C}^1(\mathbb{R}, \mathbb{C}^n)$ of (2). Let $(Y_1, Y_2, \ldots, Y_n)$ be a basis of $\mathscr{S}$. For $t \in \mathbb{R}$, we denote by $M(t)$ the matrix whose columns are $Y_1(t), \ldots, Y_n(t)$. There exists $B \in \mathscr{M}_n(\mathbb{C})$ such that $M(t+T) = M(t)\exp(TB)$ for all $t \in \mathbb{R}$.
Deduce that there exists an application $Q : \mathbb{R} \rightarrow \mathrm{GL}_n(\mathbb{C})$ continuous on $\mathbb{R}$ and $T$-periodic such that $$\forall t \in \mathbb{R}, \quad M(t) = Q(t) \exp(tB)$$ (This identity is called the normal form of the matrix $M$).

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 $$X'(t) = A(t) X(t) \tag{2}$$ where $X$ is a function from $\mathbb{R}$ to $\mathbb{C}^n$, of class $\mathscr{C}^1$ on $\mathbb{R}$. Let $\mathscr{S}$ be the space of solutions in $\mathscr{C}^1(\mathbb{R}, \mathbb{C}^n)$ of (2). Let $(Y_1, Y_2, \ldots, Y_n)$ be a basis of $\mathscr{S}$. For $t \in \mathbb{R}$, we denote by $M(t)$ the matrix whose columns are $Y_1(t), \ldots, Y_n(t)$.
Show that for every real number $t$, $M(t) \in \mathrm{GL}_n(\mathbb{C})$ and $M'(t) = A(t) M(t)$.

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 $$X'(t) = A(t) X(t) \tag{2}$$ where $X$ is a function from $\mathbb{R}$ to $\mathbb{C}^n$, of class $\mathscr{C}^1$ on $\mathbb{R}$. Let $\mathscr{S}$ be the space of solutions in $\mathscr{C}^1(\mathbb{R}, \mathbb{C}^n)$ of (2). Let $(Y_1, Y_2, \ldots, Y_n)$ be a basis of $\mathscr{S}$. For $t \in \mathbb{R}$, we denote by $M(t)$ the matrix whose columns are $Y_1(t), \ldots, Y_n(t)$.
Show that the matrix $(M(t))^{-1} M(t+T)$ is independent of $t \in \mathbb{R}$.

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 $$X'(t) = A(t) X(t) \tag{2}$$ where $X$ is a function from $\mathbb{R}$ to $\mathbb{C}^n$, of class $\mathscr{C}^1$ on $\mathbb{R}$. Let $\mathscr{S}$ be the space of solutions in $\mathscr{C}^1(\mathbb{R}, \mathbb{C}^n)$ of (2). Let $(Y_1, Y_2, \ldots, Y_n)$ be a basis of $\mathscr{S}$. For $t \in \mathbb{R}$, we denote by $M(t)$ the matrix whose columns are $Y_1(t), \ldots, Y_n(t)$.
Deduce that there exists $B \in \mathscr{M}_n(\mathbb{C})$ such that: $$\forall t \in \mathbb{R}, \quad M(t+T) = M(t) \exp(TB).$$

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 $$X'(t) = A(t) X(t) \tag{2}$$ where $X$ is a function from $\mathbb{R}$ to $\mathbb{C}^n$, of class $\mathscr{C}^1$ on $\mathbb{R}$. Let $\mathscr{S}$ be the space of solutions in $\mathscr{C}^1(\mathbb{R}, \mathbb{C}^n)$ of (2). Let $(Y_1, Y_2, \ldots, Y_n)$ be a basis of $\mathscr{S}$. For $t \in \mathbb{R}$, we denote by $M(t)$ the matrix whose columns are $Y_1(t), \ldots, Y_n(t)$. There exists $B \in \mathscr{M}_n(\mathbb{C})$ such that $M(t+T) = M(t)\exp(TB)$ for all $t \in \mathbb{R}$.
Deduce that there exists an application $Q : \mathbb{R} \rightarrow \mathrm{GL}_n(\mathbb{C})$ continuous on $\mathbb{R}$ and $T$-periodic such that $$\forall t \in \mathbb{R}, \quad M(t) = Q(t) \exp(tB).$$ (This identity is called the normal form of the matrix $M$).