# Finite-Time Control of Singular Linear Semi-Markov Jump Systems
## 1 Introduction
For singular semi-Markov jump systems (SSMJSs), the problem of a sliding mode control for nonlinear uncertain SSMJSs with stochastic disturbance was studied by using the weak infinitesimal operator from the viewpoints of probability distribution in [26]. For dynamic systems, stability is an important concept for the control system. Lyapunov stability focuses on the asymptotic behavior in an infinite time domain, which is a concept relating to qualitative research in fact. Sometimes, a system which is asymptotically stable may have bad steady-state performance. In practical engineering, the attention may be paid to the system's performance in a prescribed finite time interval [27], which may be considered as a quantitative concept. Motivated by this, the concept of finite-time stability was given to analyze the transient behavior within a finite time interval. Given some initial conditions, the dynamic system is finite-time-stable if the trajectory of the state does not exceed a given threshold in a certain time interval [28]. Annular finite-time stability not only defines that the state trajectory of the system is limited to a certain upper threshold within a finite time interval, but also considers the lower limit of the state trajectory, which is of great significance under specific requirements. The annular finite-time boundedness for networked switched systems was investigated with an adaptive event-triggered scheme [29]. A finite-time guaranteed cost control algorithm for SMJSs was given by a set of linear matrix inequalities [30]. In that paper, the event-triggered scheme and quantization input were applied. For singular MJSs with time-varying delays and quantization, the finite-time non-fragile filtering problem was addressed in [31] by constructing a Lyapunov-Krasovskii functional. In [32], the asynchronous filter for singular SMJS was designed to guarantee that the filtering error system was finite-time bounded under the dynamic event-based scheme, but there was a equality constraint, which led to some difficulty in solving numerical values. Recently, based on stochastic system theory and the singular value decomposition approach, the finite-time stability and stabilization problem for SSMJSs has been studied, but not enough. One reason is that the SSMJS has a complicated structure; the other is that the research is not mature. For SSMJSs, in order to design a controller and deal with nonlinear items, the singular matrix was decomposed into a left matrix and right matrix, and it should be pointed out that new matrices were introduced to represent the inverse of the matrices in Lyapunov function in [26], which might not be good. In [33], only under a specified expression, could the gain of the controller be obtained in terms of the linear matrix inequality, which implies that the parameters of the controller were constrained. Motivated by this, the problem of finite-time control for SSMJSs is considered. By designing a new semi-positive definite Lyapunov-like function with different perspectives in mind, a sufficient condition for the design of the desired state feedback control law is given to guarantee the closed-loop system is regular, impulse-free and finite-time-stable. By some matrix decomposition and construction, this criterion is further simplified to a set of strict matrix inequalities. When the integrally bounded disturbance is introduced, a sufficient condition is also given for the closed-loop SSMJSs to be finite-time bounded. It is noted that the obtained criteria are strict matrix inequalities and can be simplified to be linear matrix inequalities by using a one-dimensional search for a scalar. This gives some computational simplicity when those criteria are verified numerically. Two numerical examples are given to verify the applicability and effectiveness of the proposed method. The main contributions can be organized as  
1.