Book review: Input-to-State Stability for PDEs, Iasson Karafyllis, Miroslav Krstic. Communications and Control Engineering, Springer, (2018). ISBN: 978-3-319-91011-6

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BOUNDARY FEEDBACK STABILIZATION FOR THE INTRINSIC GEOMETRICALLY EXACT BEAM MODEL

The geometrically exact beam (GEB) model is a 1-D second-order non-linear system of six equations which gives the position of a beam in R 3. The beam may undergo large deflections and rotations, as well as shear deformation. A closely related model, the intrinsic formulation of GEB (IGEB), is a 1-D first-order semilin-ear hyperbolic system of twelve equations which has for states velocities and strains. Here, we consider a freely vibrating slender beam made of an isotropic linear-elastic material. Applying a feedback boundary control at one end of the beam, while the other end is clamped, we show that the steady state 0 of IGEB is locally exponentially stable for the H 1 and H 2 norms. The strategy employed is to choose the control so that the energy of the beam is nonincreasing and find appropriate quadratic Lyapunov functions, relying on the energy of the beam, the relationship between GEB and IGEB, and properties of the system's coefficients

Local Exponential Stabilization of Semi-Linear Hyperbolic Systems by Means of a Boundary Feedback Control

International audienceThis paper investigates the boundary feedback control for a class of semi-linear hyperbolic partial differential equations with nonlinear relaxation, which is local Lipschitz continuous with a stable matrix structure. A sufficient condition in terms of linear inequalities is developed for the existence of global Cauchy solutions and the exponential stability by seeking a balance between the relaxation term and the boundary condition. These results are illustrated with an application to the boundary feedback control for a class of hyperbolic Lotka-Volterra models