Stability and Boundary Stabilization of 1-D Hyperbolic Systems

An optimisation approach for stability analysis and controller synthesis of linear hyperbolic systems

“…Let us compute the time-derivative of the candidate Lyapunov function along the solutions to system (1), (2). Using the commutativity condition (20), we havė…”

“…Though most of the physical devices are controlled from the boundaries (see, for instance, [2] or [5]), there exist physical applications embedding such distributed controller (32). For instance, the evolution of a population of bacteria or animals in cultivation conditions may be described by a system of the form (1).…”

Numerical Feedback Stabilization with Applications to Networks

“…During the last few years, a huge amount of literature has emerged that deals with theoretical stability results for boundary control of conservation laws; see, for example, [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. Further work also includes hyperbolic systems with source terms (so-called balance laws) for special applications such as gas dynamics [16][17][18] or water flow in open canals [19][20][21][22][23].…”

“…For the analytical case it can be proven that feedback boundary values, designed under certain conditions, yield an exponential decay of a continuous Lyapunov function [3,9,31] also in the context of networks.…”

“…Boundary conditions can be designed to ensure that the analytical solution for these systems converges towards a desired steady-state. To prove the convergence of the approximate solution to a desired steady-state, a technique based on Lyapunov functions [3,4,8,28] is used. The required boundary conditions are called feedback boundary conditions and the method is the boundary feedback stabilization of hyperbolic systems.…”