## On the stability of reproducing kernel Hilbert spaces of discrete-time impulse responses.(English)Zbl 1402.93091

Summary: Reproducing Kernel Hilbert Spaces (RKHSs) have proved themselves to be key tools for the development of powerful machine learning algorithms, the so-called regularized kernel-based approaches. Recently, they have also inspired the design of new linear system identification techniques able to challenge classical parametric prediction error methods. These facts motivate the study of the RKHS theory within the control community. In this note, we focus on the characterization of stable RKHSs, i.e. RKHSs of functions representing stable impulse responses. Related to this, working in an abstract functional analysis framework, C. Carmeli et al. [Anal. Appl., Singap. 4, No. 4, 377–408 (2006; Zbl 1116.46019)] has provided conditions for an RKHS to be contained in the classical Lebesgue spaces $$\mathcal{L}^p$$. In particular, we specialize this analysis to the discrete-time case with $$p = 1$$. Necessary and sufficient conditions for the stability of an RKHS are worked out by a quite simple proof, more easily accessible to the control community.

### MSC:

 93B30 System identification 93C55 Discrete-time control/observation systems 93C25 Control/observation systems in abstract spaces 93C05 Linear systems in control theory 93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, $$L^p, l^p$$, etc.) in control theory 68T05 Learning and adaptive systems in artificial intelligence

Zbl 1116.46019
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### References:

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