Delayed finite-dimensional observer-based control of 1D heat equation under Neumann actuation

Recently a constructive method was introduced for finite-dimensional observer-based control of the 1D heat equation under Dirichlet actuation and non-local measurement. In this paper, we extend this method to Neumann actuation, where modal decomposition is applied to the original system (without dynamic extension) and L² exponential stability is proved by a direct Lyapunov method. We provide reduced-order LMI conditions for finding the observer dimension N and resulting decay rate. The LMI dimension is defined by N0 +1 unstable modes and does not grow with N (which is larger than N0 +1). The obtained LMI is always feasible for large N, and feasibility for N implies feasibility for N +1. Differently from Dirichlet actuation, here we manage with delayed implementation of the controller in the presence of fast-varying (without constraints on the delay-derivative) input and output delays. We consider the case of interval input delay which is lower-bounded by r > 0. By employing Lyapunov functionals and Halanay’s inequality, we derive LMIs for finding N, upper bounds on delays and the decay rate. A numerical example demonstrates the efficiency of our method.