## 1 Introduction  
During the last few decades, much attention has been devoted to singular systems, thanks to their capacity to represent the dynamic structure of physical systems involving algebraic constraints. Many findings on stability and controller/filter design for singular systems in discrete and continuous frameworks have been reported in the literature [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11]. For example, stability and stabilization of continuous descriptor systems using an LMI approach has been studied in [12]. Robust admissibility of descriptor systems by static output feedback has been established in [13]. A solid criterion based on strict LMI without invoking equality constraint for stabilization of continuous singular systems has been proposed in [14]. On the other hand, two-dimensional (2D) systems have also attracted considerable attention from researchers since they have many applications in areas such as water stream heating, seismographic data processing, thermal processes, multidimensional digital filtering, process control, image and signal processing, etc. [15; 16; 17; 18; 19]. There are several works on 2D continuous systems [15; 20; 21; 22; 23; 24]. For instance, the stability and stabilization of 2D continuous state-delayed systems have been established in [25]. The problem of \(H_{\infty}\) performance analysis for 2D continuous time-varying delay systems has been addressed in [24]. The robust \(H_{\infty}\) filtering for uncertain 2D continuous systems based on a polynomial parameter-dependent Lyapunov function has been studied in [26]. Furthermore, 2D singular systems have received great interest from the academic community due to their wide applications to describe physical systems in several practical areas [27; 28]. However, a great number of fundamental results on 1D singular systems have been extended to 2D singular systems, but the majority of existing results have been devoted to the discrete case [27; 29; 30; 31; 32; 33; 34; 35; 36]. It should be pointed out that in [36], we established necessary and sufficient conditions for the admissibility and the admissibility using strict LMIs for 2D singular systems described by the Roesser model in the discrete form. Recently, based on this result, we proposed sufficient conditions for the regularity, causality, and stability for 2D discrete singular systems in the stochastic case in [37]. Furthermore, we have derived a sufficient mean square asymptotic stability condition with a prescribed \(H_{\infty}\) disturbance attenuationlevel bound. The majority of papers addressing 2D singular systems thus far have focused on the discrete case, with the exception of [31]. In [31], nonstrict LMI conditions for the admissibility of 2D singular continuous systems, derived from existing results for 1D singular systems, have been established. The authors proposed then necessary and sufficient conditions for admissibility and admissibility expressed in terms of strict LMIs. To the best of our knowledge, there is no significant progress reported on the \(H_{\infty}\) control of 2D singular continuous systems which motivates the investigations here. This paper deals with admissibility, admissibility, and \(H_{\infty}\) control of 2D singular continuous systems. The main contributions of this paper can be summarized as follows:  
1. The regularity and impulse-freeness of 2D singular continuous systems is investigated. 2. Necessary and sufficient conditions dealing with the admissibility and admissibility of 2D singular continuous systems are established in terms of a strict LMI condition. 3. A state feedback controller is designed to ensure the admissibility of the closed-loop system and guarantee a prescribed \(H_{\infty}\) disturbance attenuation level bound. This paper is structured as follows: Section 2 presents the problem formulation. Section 3 studies the regularity and impulse-freeness of 2D singular continuous systems. Section 4 introduces necessary and sufficient admissibility condition in terms of a strict LMI.