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

Abstract

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.

Boundary feedback stabilization of hydraulic jumps

“…Fundamentally, the stabilization of shock steady states for hyperbolic systems, while being very interesting, has rarely been studied. One can refer to [5] and [21] for the scalar case and to our knowledge, no such result exists for systems. By a Lyapunov approach we prove the exponential H 2 -stability of the steady state, with an arbitrary decay rate and with an exact exponential stabilization of the desired location of the hydraulic jump.…”

“…, where H * is given by (5). Thus the steady states are not isolated and therefore not asymptotically stable in open loop.…”

“…Remark 4. For the proof of Lemma 2.1, we refer to [5,Appendix], where the well-posedness of a 2×2 nonlinear hyperbolic system coupled with an ODE was studied. But the proof there can be easily adapted to the 4 × 4 nonlinear hyperbolic system coupled with an ODE.…”

“…Thus, (30) is indeed strictly hyperbolic provided that |u| C 0 ([0,T ];H 2 ((0,x * s );R 4 )) is small enough and can be diagonalized in a neighbourhood of u = 0. Then we can perform similar fixed point argument as in [5,Appendix] by carefully estimating the related norms of the solution along the characteristic curves. The C 1 regularity of x s is then obtained directly from (32).…”

PI Controllers for 1-D Nonlinear Transport Equation

“…solution of the system (1)-(3) (we could get the result for C 0 ([0, T ], H 2 × R) later on by density as in [4,Section 4], this will not be done in this section as it is only a sketch proof). Let us denote V 0 (z(x, •), X(t)) by V 0 (t).…”