Given a linear discrete-time infinite-dimensional plant on a Hilbert space and disturbances of known waveform but unknown amplitude and phase, we show that there exists a stabilizing direct model reference adaptive control law with certain disturbance rejection and robustness properties. The central result is a discrete-time version of Barbalat-Lyapunov result for infinite dimensional Hilbert spaces. This is used to determine conditions under which a linear Infinite-dimensional system can be directly adaptively regulated. Our results are applied to adaptive control of general linear diffusion systems.

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