Finite-dimensional control of the heat equation: Dirichlet actuation and point measurement

Recently finite-dimensional observer-based controllers were introduced for the 1D heat equation, where at least one of the observation or control operators was bounded. In this paper, for the first time, we manage with such controllers for the 1D heat equation with both operators being unbounded. We consider Dirichlet actuation and point measurement and use a modal decomposition approach via dynamic extension. We suggest a direct Lyapunov approach to the full-order closed-loop system, where the finite-dimensional state is coupled with the infinite-dimensional tail of the state Fourier expansion, and provide Linear Matrix Inequalities (LMIs) for finding the controller dimension and resulting exponential decay rate. A numerical example demonstrates the efficiency of the proposed method. In the discussion section, we show that the suggested controller design is well suited for the 1D heat equation with various boundary conditions.