Adaptive observers with projection operator and L1 adaptive controllers for infinite dimensional systems

Adaptive observers with projection operator based parameter update laws are considered for a class of linear infinite dimensional systems with bounded input operator and full state measurement under constant and time-varying matched uncertainties. The L
₁
adaptive control architecture, introduced recently for finite dimensional plants to provide guaranteed transient performance via fast adaptation, is then extended to this class. Existence and uniqueness of solution to the closed loop system and asymptotic decay of the observation error are established. Under certain assumptions on the transfer function and on the solution to the Lyapunov inequality, the L
₁
architecture is analyzed and uniform bounds on the state and control signal are established. Two examples, a heat equation and a wave equation, satisfying the assumptions are presented.