Salvato in:
| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2601.05460 |
| Tags: |
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Sommario:
- 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.