A Riccati Approach to Mixed H2/H∞ Closed-Loop Games for Infinite-Dimensional Stochastic Systems
Abstract
This paper studies a finite-horizon mixed H2/H∞ feedback Nash game for stochastic evolution equations on a separable Hilbert space. The drift generator is unbounded, the remaining coefficients are bounded, and the one-dimensional Brownian diffusion depends on the state, control, and disturbance. The H2 channel is an LQ state--control energy. For the disturbance channel, the stochastic LQ uniform-convexity characterization yields equivalence between strict induced L2 attenuation and unique strongly regular mild Riccati solvability. Simultaneous bounded-generator approximation of the Lyapunov equation and the state justifies quadratic identities for strongly continuous mild operator solutions. These identities verify both Nash inequalities and full-output strict attenuation from a strongly regular coupled Riccati pair. An invertible feedback block and a contraction argument establish locally unique coupled solutions on a sufficiently short terminal interval for every positive attenuation level. For a specified stochastic heat-equation model at γ=0.09, a uniform invariant rectangle further proves full-horizon existence, bounded infinite-dimensional feedbacks, and operator-norm spectral convergence. Numerical computations reproduce the projected gains and compare selected best responses. Global coupled solvability for general coefficients remains an explicit hypothesis.
Related
- Sub-Infinite Horizon Stochastic Linear-Quadratic Optimal Control Problems and Delayed Backward Riccati Equations
- Dynamic programming principle for controlled systems driven by fractional Brownian motions and related backward stochastic Riccati equations
- Parametric estimation for stochastic wave equations driven by an infinite dimensional fractional Brownian motion
- SCMO: Stochastic Control for Optimization over Probability Measures on Infinite-Dimensional Spaces
- On the Stability Analysis of Infinite-Dimensional Port-Hamiltonian Systems Within the Variational Framework
BibTeX
@misc{shen2026riccati,
title = {{A Riccati Approach to Mixed $H_2/H_\infty$ Closed-Loop Games for Infinite-Dimensional Stochastic Systems}},
author = {Mingyang Shen and Weihai Zhang and Qingxin Meng and Maoning Tang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00756},
eprint = {2610.00756},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00756},
}
RIS
TY - UNPB
TI - A Riccati Approach to Mixed H_2/H_∞ Closed-Loop Games for Infinite-Dimensional Stochastic Systems
AU - Mingyang Shen
AU - Weihai Zhang
AU - Qingxin Meng
AU - Maoning Tang
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00756
UR - https://arxiv.org/abs/2610.00756
AB - This paper studies a finite-horizon mixed H_2/H_∞ feedback Nash game for stochastic evolution equations on a separable Hilbert space. The drift generator is unbounded, the remaining coefficients are bounded, and the one-dimensional Brownian diffusion depends on the state, control, and disturbance. The H_2 channel is an LQ state--control energy. For the disturbance channel, the stochastic LQ uniform-convexity characterization yields equivalence between strict induced L^2 attenuation and unique strongly regular mild Riccati solvability. Simultaneous bounded-generator approximation of the Lyapunov equation and the state justifies quadratic identities for strongly continuous mild operator solutions. These identities verify both Nash inequalities and full-output strict attenuation from a strongly regular coupled Riccati pair. An invertible feedback block and a contraction argument establish locally unique coupled solutions on a sufficiently short terminal interval for every positive attenuation level. For a specified stochastic heat-equation model at γ=0.09, a uniform invariant rectangle further proves full-horizon existence, bounded infinite-dimensional feedbacks, and operator-norm spectral convergence. Numerical computations reproduce the projected gains and compare selected best responses. Global coupled solvability for general coefficients remains an explicit hypothesis.
ER -