A Pseudodifferential Approach to Distributed Parameter Systems and Stabilization: Neumann Feedback Control Problems

The recent developments in microlocal analysis and pdeudodifferential boundary calculus are well suited tools in the investigation of a large number of problems occurring in control theory for partial differential equations. We explain some of the basic ideas of a pseudodifferential model in the case of a distributed system with feedback acting on the boundary of a bounded domain in Rn and appearing in the Neumann boundary condition. We establish the pseudodifferental setting for the Neumann feedback control problem previously established for the corresponding Dirichlet problem by Pederson (SIAM J. Control Optim.29 (1991 )). The pseudo-differential techniques apply easily in the proof of existence of a feedback semigroup for the parabolic and hyperbolic evolution problems, and we reprove in this new setting some of the stabilization results of Lasiecka and Triggiani (see, e.g., J. Differential Equations47 (1983); Appl. Math. Optim.10 (1983)). So far, this work seems to have simplified or unified many of the previous works cited above. We hope that in the future it will even provide stronger and newer results in the boundary control of distributed parameter systems.