arXiv subjects

Ting Hou

Publications and source records attributed to Ting Hou.

5 recordsLinked to original sources

arXiv2026

Multi-Objective Submodular Maximization with Differential Privacy

In this paper, we study multi-objective submodular maximization (MOSM) subject to a cardinality constraint under differential privacy (DP). Specifically, we aim to select a set of at most $k \in \mathbb{Z}_{+}$ elements to maximize the minimum of $d > 1$ monotone submodular functions while satisfying $\varepsilon$-DP. Although extensive studies have been conducted on both differentially private single-objective submodular maximization on sensitive data and non-private MOSM, to the best of our knowledge, there has not yet been any prior work on MOSM with DP. We propose two novel algorithms: the first extends the classic greedy algorithm and the second employs a truncation technique, both of which are integrated with DP mechanisms for privacy protection and achieve approximation guarantees for MOSM. Finally, we conduct numerical experiments on two submodular maximization applications, namely maximum coverage and facility location, in multi-objective settings to validate the efficacy and efficiency of our proposed algorithms.

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arXiv2026

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

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.

math.OC↗

arXiv2025

Small Gain Theorem-Based Robustness Analysis of Discrete-Time MJLSs with the Markov Chain on a Borel Space and Its Application to NCSs

This paper is concerned with the robustness of discrete-time Markov jump linear systems (MJLSs) with the Markov chain on a Borel space. For this general class of MJLSs, a small gain theorem is first established and subsequently applied to derive a lower bound of the stability radius. On this basis, with the aid of the extended bounded real lemma and Schur complements, the robust stability problems for the MJLSs are tackled via linear matrix inequality (LMI) techniques. The novel contribution, primarily founded on the scenario where the state space of the Markov chain is restricted in a continuous set, lies in the formulation of a griding approach. The approach converts the existence problem of solutions of an inequality related to $H_{\infty}$ analysis, which is an infinite-dimensional challenge, into a finite-dimensional LMI feasibility problem. As an application, within the framework of MJLSs, a robustness issue of the sampled-data systems is addressed by using a Markov chain, which is determined by the initial distribution and the stochastic kernel, to model transmission delays existing in networked control systems (NCSs). Finally, the feasibility of the results is verified through two examples.

math.OC↗

arXiv2025

Detectability, Riccati Equations, and the Game-Based Control of Discrete-Time MJLSs with the Markov Chain on a Borel Space

In this paper, detectability is first put forward for discrete-time Markov jump linear systems with the Markov chain on a Borel space ($\Theta$, $\mathcal{B}(\Theta)$). Under the assumption that the unforced system is detectable, a stability criterion is established relying on the existence of the positive semi-definite solution to the generalized Lyapunov equation. It plays a key role in seeking the conditions that guarantee the existence and uniqueness of the maximal solution and the stabilizing solution for a class of general coupled algebraic Riccati equations (coupled-AREs). Then the nonzero-sum game-based control problem is tackled, and Nash equilibrium strategies are achieved by solving four integral coupled-AREs. As an application of the Nash game approach, the infinite horizon mixed $H_{2}/H_{\infty}$ control problem is studied, along with its solvability conditions. These works unify and generalize those set up in the case where the state space of the Markov chain is restricted to a finite or countably infinite set. Finally, some examples are included to validate the developed results, involving a practical example of the solar thermal receiver.

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arXiv2024

Stability and Bounded Real Lemmas of Discrete-Time MJLSs with the Markov Chain on a Borel Space

In this paper, exponential stability of discrete-time Markov jump linear systems (MJLSs) with the Markov chain on a Borel space $(\Theta, \mathcal{B}(\Theta))$ is studied, and bounded real lemmas (BRLs) are given. The work generalizes the results from the previous literature that considered only the Markov chain taking values in a countable set to the scenario of an uncountable set and provides unified approaches for describing exponential stability and $H_{\infty}$ performance of MJLSs. This paper covers two kinds of exponential stabilities: one is exponential mean-square stability with conditioning (EMSSy-C), and the other is exponential mean-square stability (EMSSy). First, based on the infinite-dimensional operator theory, the equivalent conditions for determining these two kinds of stabilities are shown respectively by the exponentially stable evolutions generated by the corresponding bounded linear operators on different Banach spaces, which turn out to present the spectral criteria of EMSSy-C and EMSSy. Furthermore, the relationship between these two kinds of stabilities is discussed. Moreover, some easier-to-check criteria are established for EMSSy-C of MJLSs in terms of the existence of uniformly positive definite solutions of Lyapunov-type equations or inequalities. In addition, BRLs are given separately in terms of the existence of solutions of the $\Theta$-coupled difference Riccati equation for the finite horizon case and algebraic Riccati equation for the infinite horizon case, which facilitates the $H_{\infty}$ analysis of MJLSs with the Markov chain on a Borel space.

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