Special Session 97: Analysis and control of nonlinear partial differential equation evolution systems
Contents
Sobolev regularity of the solutions to hyperbolic boundary value problems is a persistent issue in many questions concerning the control and stabilization of evolution equations. In the context of shape optimization non-traditional boundary conditions may arise. These boundary conditions satisfy in most cases the Lopatinskii condition but not the Kreiss-Sakamoto (uniform Lopatinskii) condition. Hence, the hyperbolic boundary problem is not well-posed in the usual $L_2$ or Sobolev spaces and a loss of derivatives occurs. We present a survey of the existing literature and show that in most cases the loss of derivatives occurs only in the boundary terms.