Topics on potential theory on Lipschitz domains and boundary control problems

Abstract

The trace theorem of the Sobolev space H s(Sl) on Lipschitz domain ti has not been proved completely before. In chapter II, a proof of trace theorem for the range < s < | and s > f is first given. In Chapter III, the regularities of S in L p and the relations between ( ± | / + /C) -1 and ( ± | I + £ * ) -1 are obtained. Furtherm ore, new regularities of 1C and 1C* on sm ooth domains in aft3 are found, which reveal that 5, rC and JC* have the same regularities on smooth boundaries. In C hapter IV, a new approach based on the potential theory and variational m ethod is proposed to study the linear quadratic regulator problems (LQR) governed by the elliptic equation on Lipschitz domains with point observations on boundary. The LQR problems with or without control constraints are completely solved in this work. The regularities of optim al controls and states and the explicit expressions of optimal controls are derived. The singularities in optim al controls are displayed explicitely through decomposition formulas. Finally in C hapter V, a gradient-truncation m ethod and an iterative truncation m ethod have been developed to com pute optimal controls of (LQR) problems. The test problems show th at both m ethods are insensitive to the partition of boundary. It is found th a t the classical Lagrangian m ultiplier method may fail to provide reliable numerical algorithm on our (LQR) problems.