My current research purpose is the design of numerical methods for the analysis, control and observation of infinite-dimensional systems. These numerical tools are dedicated to interconnections between a linear ordinary differential equation (ODE) and a linear partial differential equations (PDE) which traduces a propagation, diffusion or transport phenomena. During my PhD thesis, I worked on two specific cases in this wide class of infinite-dimensional systems: 

  • Time-delay systems with a single delay (h), seen as an interconnection between an ODE and a transport equation (of velocity 1/h),
  • Systems interconnected with a reaction-diffusion equation.

From one part, the stability results are based on Lyapunov analysis. By extension of the finite-dimensional state with Legendre polynomials coefficients of the infinite-dimensional part, we present sufficient conditions of stability in linear matrix inequalities (LMIs) framework. Thanks to Fourier-Legendre remainder convergence, we actually work on the structure of these LMIs and the necessity of the LMI criteria for large enough orders.

From the other part, the control and observation is based on Hinf synthesis in the linear matrix inequality framework. Our stability results lead us to the construction of finite-dimensional output dynamical controllers for interconnected ODE-PDEs.