## What is/are Discontinuous Lyapunov?

Discontinuous Lyapunov - Technically, the work integrates with the new type of discontinuous Lyapunov-Krasovskii functional candidate with integral terms, several delay-dependent stability conditions, and Wirtinger-based integral inequality.^{[1]}The closed-loop system is formulated through a hybrid systems framework, within which stability is proven using a discontinuous Lyapunov-like function and a meagre-limsup invariance argument.

^{[2]}Using the discontinuous Lyapunov-Krasoskii functional (LKF) approach and the free-matrix-based integral inequality bounds processing technique, a stability condition with less conservativeness has been obtained, and the controller of the sampled-data T-S fuzzy system with the quantized state has been designed.

^{[3]}In this paper, the discontinuous Lyapunov-based method is proposed to reject time variant and bounded disturbances.

^{[4]}

## linear switched singular

By using a multiple discontinuous Lyapunov function approach and exploring the properties of mode-dependent average dwell time(MDADT) switching signal, new sufficient conditions of E-exponential stability for linear switched singular systems are presented.^{[1]}By using a multiple discontinuous Lyapunov function approach and adopting the mode-dependent average dwell time (MDADT) switching signals, new sufficient conditions of E-exponential stability and l 2 − gain analysis for linear switched singular systems are presented.

^{[2]}First, by constructing an appropriate multiple discontinuous Lyapunov function, new sufficient conditions of E-exponential stability for linear switched singular systems are established.

^{[3]}

## Multiple Discontinuous Lyapunov

To solve the problem, a general multiple discontinuous Lyapunov functions (MDLFs) analysi.^{[1]}Moreover, multiple discontinuous Lyapunov function (MDLF) approach, which is less conservative than the traditional multiple Lyapunov function (MLF) method, is used to analyse the closed-loop stability and performance by incorporating the idea of AED–ADT.

^{[2]}By integrating the new strategy with multiple discontinuous Lyapunov function approach, one gets some stability results of switched singular systems with stable and unstable subsystems.

^{[3]}To enlarge the room for switching scheme design, a new multiple discontinuous Lyapunov functions (MDLFs) method is developed, and a novel mode-dependent average dwell time (MDADT) tradeoff strategy is explored.

^{[4]}By dividing the dwell time into several segments, and constructing a reverse timer which starts timing at the end of each segment, we propose a new reverse-timer-dependent multiple discontinuous Lyapunov function (RTDMDLF), which is more general than the multiple Lyapunov function (MLF) and the multiple discontinuous Lyapunov function (MDLF).

^{[5]}By using a multiple discontinuous Lyapunov function approach and exploring the properties of mode-dependent average dwell time(MDADT) switching signal, new sufficient conditions of E-exponential stability for linear switched singular systems are presented.

^{[6]}By using a multiple discontinuous Lyapunov function approach and adopting the mode-dependent average dwell time (MDADT) switching signals, new sufficient conditions of E-exponential stability and l 2 − gain analysis for linear switched singular systems are presented.

^{[7]}The discrete-time multiple discontinuous Lyapunov function is also utilized for the analysis.

^{[8]}It is pointed out in the paper that the multiple Lyapunov function (MLF) and the multiple discontinuous Lyapunov function (MDLF) can be regarded as special cases of the proposed MCLF and MPCLF, respectively.

^{[9]}First, by constructing an appropriate multiple discontinuous Lyapunov function, new sufficient conditions of E-exponential stability for linear switched singular systems are established.

^{[10]}

## Constructing Discontinuous Lyapunov

By constructing discontinuous Lyapunov functions, it is proved that the proposed ET controllers guarantee the stability and^{[1]}By constructing discontinuous Lyapunov functions, it is proved that the proposed ET controllers guarantee the stability and H∞ performance of the closed-loop systems.

^{[2]}

## Dependent Discontinuous Lyapunov

To reflect more realistic the information on both the intervals e(t) to $$e(t_{k})$$ and e(t) to $$e(t_{k+1})$$ , a novel two-side sampling-interval-dependent discontinuous Lyapunov functional (DLF) is constructed, which can fully utilizes the available characteristics of actual sampling information.^{[1]}By taking advantage of characteristic information on the whole sampling interval, a new two-sided sampling-interval-dependent discontinuous Lyapunov functional is first constructed, which depends on the available information of both the intervals from tk to t and from t to t k + 1.

^{[2]}

## Improved Discontinuous Lyapunov

This paper is devoted to proposing improved discontinuous Lyapunov functionals for the stability analysis of sampled-data systems.^{[1]}Firstly, constructing an improved discontinuous Lyapunov-Krasovskii function (LKF), which is fully considered the characteristics of sampled-data to reduce the conservativeness.

^{[2]}

## Designing Discontinuous Lyapunov

By designing discontinuous Lyapunov function with time-varying Lyapunov matrix, sufficient conditions in terms of linear matrix inequalities (LMIs) are obtained to ensure the stability of the closed-loop system.^{[1]}By designing discontinuous Lyapunov function with time-varying Lyapunov matrix, sufficient conditions in terms of linear matrix inequalities (LMIs) are obtained to ensure the stability of the closed-loop system.

^{[2]}

## discontinuous lyapunov function

The estimation error is analyzed via a discontinuous Lyapunov function, and an e -independent observer gain is designed.^{[1]}By constructing discontinuous Lyapunov functions, it is proved that the proposed ET controllers guarantee the stability and

^{[2]}To solve the problem, a general multiple discontinuous Lyapunov functions (MDLFs) analysi.

^{[3]}According to a discontinuous Lyapunov function, sufficient conditions are given to ensure that the closed-loop system is asymptotically stable and satisfies H ∞ performance.

^{[4]}By establishing discontinuous Lyapunov function (LF), it is proved that the constructed controllers can ensure the global uniform ultimate boundedness (uubs) of the systems.

^{[5]}The stability analysis is carried out using a specially constructed discontinuous Lyapunov function.

^{[6]}Moreover, multiple discontinuous Lyapunov function (MDLF) approach, which is less conservative than the traditional multiple Lyapunov function (MLF) method, is used to analyse the closed-loop stability and performance by incorporating the idea of AED–ADT.

^{[7]}By integrating the new strategy with multiple discontinuous Lyapunov function approach, one gets some stability results of switched singular systems with stable and unstable subsystems.

^{[8]}By designing discontinuous Lyapunov function with time-varying Lyapunov matrix, sufficient conditions in terms of linear matrix inequalities (LMIs) are obtained to ensure the stability of the closed-loop system.

^{[9]}A method for constructing a discontinuous Lyapunov function is proposed, which is applied to obtain sufficient conditions of the asymptotic stability of the zero equilibrium position of the equation studied.

^{[10]}To enlarge the room for switching scheme design, a new multiple discontinuous Lyapunov functions (MDLFs) method is developed, and a novel mode-dependent average dwell time (MDADT) tradeoff strategy is explored.

^{[11]}By dividing the dwell time into several segments, and constructing a reverse timer which starts timing at the end of each segment, we propose a new reverse-timer-dependent multiple discontinuous Lyapunov function (RTDMDLF), which is more general than the multiple Lyapunov function (MLF) and the multiple discontinuous Lyapunov function (MDLF).

^{[12]}By constructing discontinuous Lyapunov functions, it is proved that the proposed ET controllers guarantee the stability and H∞ performance of the closed-loop systems.

^{[13]}By using a multiple discontinuous Lyapunov function approach and exploring the properties of mode-dependent average dwell time(MDADT) switching signal, new sufficient conditions of E-exponential stability for linear switched singular systems are presented.

^{[14]}By using a multiple discontinuous Lyapunov function approach and adopting the mode-dependent average dwell time (MDADT) switching signals, new sufficient conditions of E-exponential stability and l 2 − gain analysis for linear switched singular systems are presented.

^{[15]}The discrete-time multiple discontinuous Lyapunov function is also utilized for the analysis.

^{[16]}Based on the discontinuous Lyapunov function, the stability analysis helps the optimal gain selecting.

^{[17]}By designing discontinuous Lyapunov function with time-varying Lyapunov matrix, sufficient conditions in terms of linear matrix inequalities (LMIs) are obtained to ensure the stability of the closed-loop system.

^{[18]}It is pointed out in the paper that the multiple Lyapunov function (MLF) and the multiple discontinuous Lyapunov function (MDLF) can be regarded as special cases of the proposed MCLF and MPCLF, respectively.

^{[19]}First, by constructing an appropriate multiple discontinuous Lyapunov function, new sufficient conditions of E-exponential stability for linear switched singular systems are established.

^{[20]}

## discontinuous lyapunov functional

To reflect more realistic the information on both the intervals e(t) to $$e(t_{k})$$ and e(t) to $$e(t_{k+1})$$ , a novel two-side sampling-interval-dependent discontinuous Lyapunov functional (DLF) is constructed, which can fully utilizes the available characteristics of actual sampling information.^{[1]}This paper is devoted to proposing improved discontinuous Lyapunov functionals for the stability analysis of sampled-data systems.

^{[2]}By taking advantage of characteristic information on the whole sampling interval, a new two-sided sampling-interval-dependent discontinuous Lyapunov functional is first constructed, which depends on the available information of both the intervals from tk to t and from t to t k + 1.

^{[3]}The sampled-data with stochastic sampling is used to design the controller by a discontinuous Lyapunov functional.

^{[4]}Some discontinuous Lyapunov functionals and zero equation are employed to deal with a sampled-data pattern.

^{[5]}For a special case that the sampled-data controller suffers constant input delay, a discontinuous Lyapunov functional is presented based on the vector extension of Wirtinger’s inequality.

^{[6]}In order to make full use of the sawtooth structure characteristic of the sampling input delay, a discontinuous Lyapunov functional is proposed.

^{[7]}