Stochastic Bounded Real Lemma and $H_{\infty}$ Control of Difference Systems in Hilbert Spaces

TL;DR

The work extends finite-horizon stochastic control theory to separable Hilbert spaces, introducing a backward Riccati operator framework for indefinite LQ-optimal control and a finite-horizon stochastic bounded real lemma to characterize $H_{\infty}$ performance in infinite dimensions. It unifies $H_{\infty}$ and $H_2/H_{\infty}$ designs through a Nash-game formulation with coupled backward Riccati equations, providing necessary and sufficient conditions for linear state-feedback controls and explicit gain formulas. The theory is illustrated via examples and demonstrates practical relevance for distributed-parameter systems, signal processing, and heat-transport problems. Overall, the paper builds a comprehensive operator-theoretic foundation for finite-horizon control and robust performance in Hilbert-space settings, with potential extensions to infinite-horizon problems.

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.

Figures (3)

• Figure 1: $(a)$ Initial signal. $(b)$ Final signal.
• Figure 2: The temperature at time $k$ in $x$.
• Figure 3: $J^N_1(x_0, u^*, v^*)$ and $J^N_2(x_0, u^*, v^*)$.