Finite-Dimensional Observer-Based Boundary Control of 1-D Linear Parabolic-Elliptic Systems

This letter investigates the finite-dimensional observer-based boundary control for 1D linear parabolic-elliptic systems via the modal decomposition method. To address the potential multiple eigenvalues arising from the elliptic equation, we implement bilateral actuations (one Dirichlet and one Neumann) on the boundary of the parabolic equation with two point measurements. When the eigenvalues are simple, one boundary actuation and one point measurement are sufficient, but the second input and output may reduce the observer dimension. We present efficient LMI conditions for finding observer dimension, as well as controller and observer gains, ensuring the ${\mathrm { H}}^{1}$ exponential stability with any desirable decay rate. We show that the LMIs are always feasible for large enough values of the observer dimension. Numerical examples demonstrate the efficiency of the method.