Stability and boundary stabilization of 1-D hyperbolic systems
This very nice book concerns initial-boundary value problems for 1-D hyperbolic systems of first-order PDEs, in particular for balance laws and for conservation laws. The main topic is the boundary control problem, when the control input, the controlled output and the measured output are located at the boundary, and the goal is to identify and to characterize so-called dissipative boundary conditions and dissipative source terms, i.e. those which guarantee bounded smooth solutions converging to an equilibrium. Different criteria are derived for stability in different function space norms. Moreover, it is presented a list of mathematical models (navigable rivers, chemotaxis, networks of electrical lines, lasers, chromatography, epitemiology, road traffic, gas dynamics, shallow water equations, telegrapher equations), for which the criteria are applicable. One of the main difficulties of stability theory for hyperbolic PDEs is nicely described already in the case of linear systems of conservation laws: Necessary and sufficient for stability is the negativity of the real parts of all roots of the characteristic equation. But this criterion is not robust with respect to small perturbations of the system, in general (because of small denominator effects)! That's why the authors mainly present sufficient stability criteria by means of the Lyapunov function approach.
scientific article; zbMATH DE number 6567166