dorsal/arxiv
View SchemaStochastic Bounded Real Lemma and $H_{\infty}$ Control of Difference Systems in Hilbert Spaces
| Authors | Cheng'ao Li, Ting Hou, Weihai Zhang, Feiqi Deng |
|---|---|
| Categories | |
| ArXiv ID | 2601.05460vv1 |
| URL | https://arxiv.org/abs/2601.05460 |
| License | http://arxiv.org/licenses/nonexclusive-distrib/1.0/ |
Abstract
This paper mainly establishes the finite-horizon stochastic bounded real lemma, and then solves the $H_{\infty}$ control problem for discrete-time stochastic linear systems defined on the separable Hilbert spaces, thereby unifying the relevant theoretical results previously confined to the Euclidean space $\mathbb{R}^n$. To achieve these goals, the indefinite linear quadratic (LQ)-optimal control problem is firstly discussed. By employing the bounded linear operator theory and the inner product, a sufficient and necessary condition for the existence of a linear state feedback LQ-optimal control law is derived, which is closely linked with the solvability of the backward Riccati operator equation with a sign condition. Based on this, stochastic bounded real lemma is set up to facilitate the $H_{\infty}$ performance of the disturbed system in Hilbert spaces. Furthermore, the Nash equilibrium problem associated with two parameterized quadratic performance indices is worked out, which enables a uniform treatment of the $H_{\infty}$ and $H_2/H_{\infty}$ control designs by selecting specific values for the parameters. Several examples are supplied to illustrate the effectiveness of the obtained results, especially the practical significance in engineering applications.
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"abstract": "This paper mainly establishes the finite-horizon stochastic bounded real lemma, and then solves the $H_{\\infty}$ control problem for discrete-time stochastic linear systems defined on the separable Hilbert spaces, thereby unifying the relevant theoretical results previously confined to the Euclidean space $\\mathbb{R}^n$. To achieve these goals, the indefinite linear quadratic (LQ)-optimal control problem is firstly discussed. By employing the bounded linear operator theory and the inner product, a sufficient and necessary condition for the existence of a linear state feedback LQ-optimal control law is derived, which is closely linked with the solvability of the backward Riccati operator equation with a sign condition. Based on this, stochastic bounded real lemma is set up to facilitate the $H_{\\infty}$ performance of the disturbed system in Hilbert spaces. Furthermore, the Nash equilibrium problem associated with two parameterized quadratic performance indices is worked out, which enables a uniform treatment of the $H_{\\infty}$ and $H_2/H_{\\infty}$ control designs by selecting specific values for the parameters. Several examples are supplied to illustrate the effectiveness of the obtained results, especially the practical significance in engineering applications.",
"arxiv_id": "2601.05460",
"authors": [
"Cheng\u0027ao Li",
"Ting Hou",
"Weihai Zhang",
"Feiqi Deng"
],
"categories": [
"math.OC"
],
"license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
"title": "Stochastic Bounded Real Lemma and $H_{\\infty}$ Control of Difference Systems in Hilbert Spaces",
"url": "https://arxiv.org/abs/2601.05460",
"version": "v1"
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