Numerical boundary control for semilinear hyperbolic systems

Abstract

This work is devoted to the design of boundary controls of physical systems that are described by semilinear hyperbolic balance laws. A computational framework is presented that yields sufficient conditions for a boundary control to steer the system towards a desired state. The presented approach is based on a Lyapunov stability analysis and a CWENO-type reconstruction.Introduction. Systems of hyperbolic partial differential equations model fluid flow, chemotaxis and viscoplastic material dynamics [2,23,26,34]. Boundary stabilization of these problems has been studied intensively in the past years [2,5]. An underlying tool for the study of these problems are Lyapunov functions that yield upper bounds on the deviation from steady states in suitable norms. The virtue of this approach are control rules that do not require the solution of the whole system, but take only measurements at the boundaries into account. So-called dissipative boundary conditions [6][7][8][9] ensure exponential decay of a continuous Lyapunov function, which in turn guarantees that the solution converges exponentially fast to a desired steady state.More precisely, a general theory for the stabilization of linear conservation laws with respect to the L 2 -norm is available [2, Sec. 3]. For nonlinear systems, however, results are still partial. A problem is posed by the fact that Lyapunov's indirect method [24] does not necessarily hold for hyperbolic systems. Furthermore, solutions to systems of conservation laws exist in the classical sense only for a finite time due to formation of shocks [31]. To this end, stability results are typically stated in terms of the Sobolev H 2 -norm or in the C 1 -norm [6,7,19,21] and restrictive smoothness assumptions on the H 2 -norm of the initial data may be needed [2, Sec. 4]. Furthermore, most analytical results are based on the assumption that the influence of the source term is small or in intuitive terms, the considered balance laws are viewed as perturbations of conservation laws [8]. On the other