Dual Representations and Proxy Characterization of H2 Norm for Partial Integral Equations
Abstract
Duality and dual representations are essential tools for systems theory and controller synthesis. For any finite-dimensional causal linear time-invariant (CLTI) system, there exists a dual CLTI system with identical H∞ and H2 norm. These relations are found by simply taking the adjoint of the transfer function. For nonlinear, time-varying or infinite-dimensional systems, however, such a transfer function may not exist or be difficult to obtain. For duality in the H∞ norm, this obstacle is overcome by using the equivalent time-domain, induced L2-norm characterization. For H2 norms there is no such equivalent characterization and hence proxy norms such as induced energy to peak and impulse to energy norms are used. These proxy norms bound the H2 norm in the CLTI case. However, neither of these H2 proxy norms equal the same H2 proxy of the canonical dual representation. In this paper, we overcome this by proposing a dual representation whose induced energy to peak proxy equals the impulse to energy proxy of the primal and vice-versa. This relation is developed in the context of Partial Integral Equation (PIE) representations. The result is then used to establish primal-dual operator inequalities for both proxy norms similar in form to the primal-dual LMIs used to characterize the H2 norm.
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BibTeX
@misc{braghini2026dual,
title = {{Dual Representations and Proxy Characterization of $H_2$ Norm for Partial Integral Equations}},
author = {Danilo Braghini and Tristan Thomas and Sachin Shivakumar and Javad M. Velni and Matthew M. Peet},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21355},
eprint = {2610.21355},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21355},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Dual Representations and Proxy Characterization of H_2 Norm for Partial Integral Equations
AU - Danilo Braghini
AU - Tristan Thomas
AU - Sachin Shivakumar
AU - Javad M. Velni
AU - Matthew M. Peet
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21355
UR - https://arxiv.org/abs/2610.21355
AB - Duality and dual representations are essential tools for systems theory and controller synthesis. For any finite-dimensional causal linear time-invariant (CLTI) system, there exists a dual CLTI system with identical H_∞ and H_2 norm. These relations are found by simply taking the adjoint of the transfer function. For nonlinear, time-varying or infinite-dimensional systems, however, such a transfer function may not exist or be difficult to obtain. For duality in the H_∞ norm, this obstacle is overcome by using the equivalent time-domain, induced L_2-norm characterization. For H_2 norms there is no such equivalent characterization and hence proxy norms such as induced energy to peak and impulse to energy norms are used. These proxy norms bound the H_2 norm in the CLTI case. However, neither of these H_2 proxy norms equal the same H_2 proxy of the canonical dual representation. In this paper, we overcome this by proposing a dual representation whose induced energy to peak proxy equals the impulse to energy proxy of the primal and vice-versa. This relation is developed in the context of Partial Integral Equation (PIE) representations. The result is then used to establish primal-dual operator inequalities for both proxy norms similar in form to the primal-dual LMIs used to characterize the H_2 norm.
ER -