### Concours Mines-Ponts: mathématiques 1 MP

An analysis-algebra problem that studies the stability conditions of the following differential equation \[ \left\{ \begin{array}{ll} y'=\varphi(y)& \\ y(0)=x_0& \end{array} \right. \] where \(\varphi\) is a \(C^1\) map from \(\mathbb{R}^n\) to \(\mathbb{R}^n\).

The first part allows the study of the linearized system around the equilibrium state \(\varphi(0)=0\): \[ \left\{ \begin{array}{ll} y'=\mathrm{d}\varphi_0(y)& \\ y(0)=x_0& \end{array} \right. \] whose solution is well known: \begin{alignat*}{1} \mathbb{R}^{+}&\to\mathbb{R}^{n}\\ t&\mapsto e^{t\mathrm{d}\varphi_0}x_0 \end{alignat*}

The last part uses analysis to apply the results obtained for the linear differential equation to the study of the solution of the general problem.

Lyapunov analysis methods are important for the study of nonlinear systems in robotics, for example. It is rarely possible to obtain an explicit solution for a nonlinear differential equation, and the real world is rarely linear; instead, we seek to prove asymptotic stability properties.

Solution to the Maths 1 test on May 2 in the MP concours Mines-Ponts 2023:

Problem set: