Exponential boundary feedback stabilization of a shock steady state for the inviscid Burgers equation

In this paper, we study the exponential stabilization of a shock steady state for the inviscid Burgers equation on a bounded interval. Our analysis relies on the construction of an explicit strict control Lyapunov function. We prove that by appropriately choosing the feedback boundary conditions, we can stabilize the state as well as the shock location to the desired steady state in [Formula: see text]-norm, with an arbitrary decay rate.

Exact null controllability, complete stabilizability and continuous final observability of neutral type systems

For abstract linear systems in Hilbert spaces we revisit the problems of exact controllability and complete stabilizability (stabilizability with an arbitrary decay rate), the latter property being related to exact null controllability. We also consider the case when the feedback is not bounded. We obtain a characterization of complete stabilizability for neutral type systems. Conditions for exact null controllability of neutral type systems are discussed. By duality, we obtain a result about continuous final observability. Illustrative examples are given.