Direct Adaptive Control for Infinite-Dimensional Symmetric Hyperbolic Systems with Application to Controlled Wave-like Behavior

摘要

Symmetric Hyperbolic Systems of partial differential equations describe many physical phenomena such as wave behavior, electromagnetic fields, and quantum fields. We consider the controlled general symmetric hyperbolic system with coercive boundary conditions as a linear continuous-time infinite-dimensional plant on a Hilbert space with persistent disturbances of known waveform, and show that there exists a stabilizing direct model reference adaptive control law with certain disturbance rejection properties. The symmetric hyperbolic plant is described by a closed densely defined linear operator that generates a continuous semigroup of bounded operators on the Hilbert space of states. The closed loop system is shown to be asymptotically regulated to zero with bounded adaptive gains in the presence of persistent disturbances. We apply these results to the control of the wave equation in two space dimensions formulated in symmetric hyperbolic form.