# Finite-Time Control of Singular Linear Semi-Markov Jump Systems
## 1 Introduction  
During the past few years, the research of Markov jump systems (MJSs) has gotten a lot of attention due to the fact that those systems can effectively describe the dynamics of physical systems suffering from some changes in structure and parameters, random sudden variations and unexpected environment disturbances [1]. Various fundamental problems, including stability [2], state feedback stabilization [3], output feedback stabilization [4] and filter design [5], have been fully addressed within this topic. The sojourn times of continuous-time MJSs obey a unique exponential distribution, whereas those of discrete-time MJSs follow the geometric distribution, which results in constant transition rates and certain limitations on the MJSs in practical applications. Compared with the standard Markov process, the semi-Markov jump process [6] is subject to a nonexponential distribution, such as the Weibull distribution [7] or phase type distribution [8], and has time-varying transition rates. For this reason, the semi-Markov system model can describe more general practical systems and has a wider range of applications in many real-life situations [9]--epidemic infectious systems [10], social networks [11] and so on. In addition to the above stochastic processes, there are more general stochastic processes, such as regenerative processes and Markovian regenerative processes. The regenerative switched linear system was studied in [12]. The mean stability of this system was investigated and represented by the spectral radius of a expectation matrix. In [13],a Markovian regenerative process was applied to a RAID storage system to obtain more accurate performance prediction. Sufficient conditions for the stochastic stability of linear SMJSs were given in [6], where the information for the upper and lower bounds of the transition rates was required. Moreover, for positive SMJSs, conditions to verify the exponential and stochastic mean stability were given in [14], where the spectral radius of a matrix and a discretization method were introduced. Numerically testable criteria on the basis of a semi-Markov kernel and the emission probability of hidden SMJSs were obtained in [15], with the aid of certain techniques that can eliminate the terms containing products of matrices. Singular systems, also known as descriptor systems and differential-algebraic systems, have wide applications in actual systems, including power systems [16], singular perturbation systems [17], chemical processes [18] and nuclear energy reaction processes [19],since such systems can contain both dynamic and algebraic relations between state variables and can better describe the complex dynamic systems than the normal state-space ones. By using the fast-slow subsystems decomposition approach, singular systems can be decomposed to fast subsystems depicted by algebraic equations and slow subsystems depicted by differential equations or difference equations [19]. The structural characteristics of singular systems were analyzed in [19], and some criteria for the regularity, absence of impulse and asymptotic stability were given in [20]. For singular MJSs, sufficient and necessary conditions for stability and stabilization of continuous-time singular MJSs were given in [21], and the dynamic output feedback control for singular MJSs was given in [22] by using linear matrix inequality technology. When the system was subject to amplitude bounded disturbance, the PD state feedback control method was extended to the reachable set synthesis problem [23]. The dissipative control problem for T-S fuzzy singular MJSs was studied in [24], where the state-dependent noise and asynchronous modes were considered. Based on the extended passivity theory, a non-fragile filter was designed for singular MJSs with time-varying delay and missing measurements in [25].