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The persistence of excitation (PE) condition is sufficient to ensure parameter convergence in adaptive estimation problems. Recent results on adaptive estimation in reproducing kernel Hilbert spaces (RKHS) introduce PE conditions for RKHS. This paper presents sufficient conditions for PE for the particular class of uniformly embedded reproducing kernel Hilbert spaces (RKHS) defined over smooth Riemannian manifolds. This paper also studies the implications of the sufficient condition in the case when the RKHS is finite or infinite-dimensional. When the RKHS is finite-dimensional, the sufficient condition implies parameter convergence as in the conventional analysis. On the other hand, when the RKHS is infinite-dimensional, the same condition implies that the function estimate error is ultimately bounded by a constant that depends on the approximation error in the infinite-dimensional RKHS. We illustrate the effectiveness of the sufficient condition in a practical example.

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It has been shown theoretically that by prescribing the mass and stiffness distributions of a subordinate oscillator array (SOA) that is attached to a host structure, significant vibration attenuation of a host can be obtained over a finite frequency range. This case stands in stark contrast to classical vibration isolator designs for two degree of freedom systems that achieve exact vibration cancellation at a single isolated frequency. Despite the attractiveness of SOAs for the design of broader band vibration suppression, the theoretically desired result can deteriorate rapidly due to small fabrication imperfections in the SOA. This paper introduces and compares variational thermodynamic formulations of composite piezoelectric SOA that are designed to be adjustable in real-time to ameliorate the effects of disorder due to fabrication in a SOA.