Multiple Lyapunov
다중 랴푸노프

Multiple Lyapunov(다중 랴푸노프)란 무엇입니까?

Multiple Lyapunov 다중 랴푸노프 - Multiple Lyapunov-like functions and graph-theoretic tools are the primary apparatuses for our analysis. ^[1] Especially, we further improve the applicability of this synchronization criterion by using multiple Lyapunov-type functions. ^[2] Sufficient conditions are achieved by using a multiple Lyapunov–Krasovskii functional. ^[3] Furthermore, stringent transient and steady-state performance and a guaranteed non-weighted H ∞ performance are achieved with the aid of multiple Lyapunov-like functions. ^[4] Proofs in detail are accomplished by using multiple Lyapunov–Krasovskii functions. ^[5] Multiple Lyapunov-Krasovskii function with multiple integral functions allows us to obtain less conservative conditions for a networked control system to satisfy the disturbance attenuation criterion. ^[6] By employing the method of multiple Lyapunov-Krasovskii functionals and the uniformly exponentially stable function, some relaxed Krasovskii-type sufficient conditions ensuring the pISS/piISS of the addressed systems are developed. ^[7]
다중 Lyapunov 유사 함수 및 그래프 이론 도구는 우리의 분석을 위한 기본 장치입니다. ^[1] 특히, 다중 Lyapunov-type 함수를 사용하여 이 동기화 기준의 적용성을 더욱 향상시킵니다. ^[2] 다중 Lyapunov-Krasovskii 기능을 사용하여 충분한 조건을 달성합니다. ^[3] 더욱이, 엄격한 과도 및 정상 상태 성능과 보장된 비가중 H ∞ 성능은 다중 Lyapunov 유사 함수의 도움으로 달성됩니다. ^[4] 자세한 증명은 여러 Lyapunov-Krasovskii 함수를 사용하여 수행됩니다. ^[5] 다중 적분 함수가 있는 다중 Lyapunov-Krasovskii 함수를 사용하면 방해 감쇠 기준을 충족하기 위해 네트워크 제어 시스템에 대해 덜 보수적인 조건을 얻을 수 있습니다. ^[6] 다중 Lyapunov-Krasovskii 함수와 균일하게 지수적으로 안정적인 함수의 방법을 사용하여, 주소 시스템의 pISS/piISS를 보장하는 일부 완화된 Krasovskii 유형 충분 조건이 개발되었습니다. ^[7]

average dwell time 평균 체류 시간

In view of the transient behavior caused by controller switching, the global stability of the switched system is analyzed using the multiple Lyapunov function approach and average dwell time condition. ^[1] Under the average impulsive interval (AII) with the average dwell time (ADT) and the mode-dependent average impulsive interval (MDAII) with the mode-dependent average dwell time, by adopting multiple Lyapunov functional approach, the inequality analysis technique and pinning control strategy, the global exponential cluster synchronization conditions are achieved. ^[2] Considering asynchronous switching phenomenon, the mean-square asymptotic stability is analyzed by the aid of multiple Lyapunov functional method and average dwell time (ADT) method. ^[3] The multiple Lyapunov function method is adopted to guarantee the tracking performance with designed average dwell time. ^[4] Then, by using the average dwell time switching approach and constructing multiple Lyapunov functions, a sufficient condition of IO-FTS for the considered system is presented in the form of linear matrix inequalities (LMIs). ^[5] Firstly, we propose the exponential stability criterion for a type of SPNSs when all subsystems succumb to average dwell time (ADT) switching by employing multiple Lyapunov functions (MLFs). ^[6] Taking the asynchronous switching into account, we give a switching law with the average dwell time (ADT) guaranteeing the exponential stability with the $H_{\infty }$ performance of the event-triggered switched filtering error system (SFES) in the absence of attacks by employing the multiple Lyapunov function (MLF) method. ^[7] First, by using the multiple Lyapunov stability theory, we show that node-to-node synchronization in two-layer networks can be achieved by choosing sufficiently large coupling strength as well as suitable feedback gain matrix if the average dwell time is bounded below by some positive constants. ^[8] By adopting the average dwell time (ADT) technique and multiple Lyapunov functions, new conditions are obtained to guarantee that the filtering error system is finite-time bounded with a prescribed disturbance attenuation performance. ^[9] Different from multiple Lyapunov function and average dwell time in the previous work, a general multiple Lyapunov-like function and a modified admissible edge-dependent average dwell time switching scheme based on a directed graph are implemented in this paper. ^[10] By constructing proper multiple Lyapunov–Krasovskii functions and applying average dwell time methods, a switching law and the relevant asynchronous resilient controller are designed to guarantee the finite-time boundedness of the closed-loop system with a specified ${H} _{\infty }$ performance index. ^[11] Based on a new bounded maximum average dwell time (BMADT) switching approach, an exponential stability condition is derived with a novel time-varying piecewise multiple Lyapunov function approach. ^[12] By constructing a novel piecewise multiple Lyapunov function approach, an exponential stability condition of switched T-S fuzzy systems is first established with the bounded maximum average dwell time. ^[13] For unforced SPCH systems, via energy-based multiple Lyapunov functions methods, sufficient conditions of both exponential stability and asymptotical stability are obtained under arbitrary switching signals with an average dwell time scheme. ^[14] By skillfully constructing multiple Lyapunov–Krasovskii functionals and successfully solving several troublesome obstacles, such as time-varying delay and switching signals and nonlinearity in the design procedure, the switched linear output-feedback controllers designed can render the resulting closed-loop switched system semi-globally stabilizable under a class of switching signals with average dwell time. ^[15] First, by employing the average dwell time switching technique, together with the introduction of multiple Lyapunov functions, some sufficient conditions of the finite-time stability and finite-time boundedness are derived for the considered system. ^[16] Then, by applying results from the multiple Lyapunov functional theory and average dwell time technique, sufficient conditions in terms of linear matrix inequalities and a lower bound on the switching signal are established to ensure asymptotic stability for the augmented closed-loop system with a prescribed ℓ2 − ℓ∞ disturbance attenuation performance level γ. ^[17] This paper studies stability of a general class of impulsive switched systems under time delays and random disturbances using multiple Lyapunov functions and average dwell-time. ^[18]
컨트롤러 스위칭으로 인한 일시적인 동작의 관점에서 스위칭 시스템의 전체 안정성은 다중 Lyapunov 함수 접근과 평균 체류 시간 조건을 사용하여 분석됩니다. ^[1] 평균 지속 시간(ADT)이 있는 평균 충격 간격(AII)과 모드 종속 평균 지속 시간이 있는 모드 종속 평균 충격 간격(MDAII)에서 다중 Lyapunov 함수 접근법, 불평등 분석 기법 및 고정 제어를 채택하여 전략, 글로벌 지수 클러스터 동기화 조건이 달성됩니다. ^[2] nan ^[3] nan ^[4] nan ^[5] 첫째, 우리는 모든 하위 시스템이 다중 리아푸노프 함수(MLF)를 사용하여 평균 체류 시간(ADT) 전환에 굴복할 때 SPNS 유형에 대한 지수 안정성 기준을 제안합니다. ^[6] nan ^[7] 첫째, 다중 Lyapunov 안정성 이론을 사용하여 평균 드웰 시간이 몇 가지 이하로 제한되는 경우 충분히 큰 결합 강도와 적절한 피드백 이득 행렬을 선택하여 2계층 네트워크에서 노드 간 동기화를 달성할 수 있음을 보여줍니다. 양의 상수. ^[8] ADT(Average Dwell Time) 기술과 다중 Lyapunov 기능을 채택하여 필터링 오류 시스템이 규정된 외란 감쇠 성능으로 제한되는 유한 시간을 보장하는 새로운 조건을 얻습니다. ^[9] nan ^[10] 적절한 다중 Lyapunov-Krasovskii 함수를 구성하고 평균 체류 시간 방법을 적용하여 스위칭 법칙과 관련 비동기 탄력성 컨트롤러는 지정된 <inline-formula> <tex- math notation="LaTeX">${H} _{\infty }$ </tex-math></inline-formula> 성능 지수. ^[11] 새로운 BMADT(Bounded Maximum Average Dwell Time) 전환 접근 방식을 기반으로 하는 새로운 시간 변화 조각별 다중 Lyapunov 함수 접근 방식으로 지수 안정성 조건이 파생됩니다. ^[12] 새로운 조각별 다중 Lyapunov 함수 접근 방식을 구성하여 제한된 최대 평균 체류 시간으로 전환된 T-S 퍼지 시스템의 지수 안정성 조건이 먼저 설정됩니다. ^[13] nan ^[14] nan ^[15] nan ^[16] nan ^[17] nan ^[18]

mode dependent average 모드 종속 평균

For each case, stability conditions are derived using multiple Lyapunov functions and mode-dependent average dwell-time, and thus relax the conservatisms caused by common Lyapunov function and average dwell-time. ^[1] By constructing appropriate multiple Lyapunov functions and employing mode-dependent average dwell time (MDADT) approach, some sufficient conditions are given to ensure that the system is FTB and finite-time H ∞ controllable. ^[2] By using multiple Lyapunov functions and mode-dependent average dwell time, we provide a sufficient condition for the p th moment exponential stability with fast convergence rate for deterministic switched discrete-time stochastic systems with stable and unstable subsystems. ^[3] By using the multiple Lyapunov–Krasovskii functionals (MLKFs) method combined with the mode-dependent average dwell time of stable subsystems (MDADTSS) and the mode-dependent average dwell time of unstable subsystems (MDADTUS) method, SSDSs with unstable subsystems are analyzed, and several novel sufficient conditions for finite-time stability and p th moment exponential stability are given. ^[4] For each case, stability conditions are derived using multiple Lyapunov functions and mode-dependent average dwell-time, and thus relax the conservatisms caused by common Lyapunov function and average dwell-time. ^[5] Secondly, based on the Lie derivative and iteration technique, some sufficient conditions, which can guarantee that switched affine non-linear systems are finite-time bounded with a prescribed H ∞ performance, are derived by applying the mode-dependent average dwell time and the multiple Lyapunov functions. ^[6] By using multiple Lyapunov functionals (MLFs) method, mode-dependent average dwell times, and the total activating time length of MLFs, some stability criteria are explicitly obtained for SNDSs with SDIs. ^[7] By adopting extended mode-dependent average dwell-time switching laws and semi-time-dependent multiple Lyapunov-like functions, the improved stability criterion is deduced for switched time-delay systems comprising of both stable and unstable subsystems. ^[8] The delay-dependent stability criterion of the STDSs with mode-dependent average dwell time (MDADT) switching is developed via the multiple Lyapunov–Krasovskii functional approach. ^[9]
각 경우에 대해 안정성 조건은 다중 Lyapunov 함수와 모드 종속 평균 체류 시간을 사용하여 파생되므로 일반적인 Lyapunov 함수와 평균 체류 시간으로 인한 보수성을 완화합니다. ^[1] 적절한 다중 Lyapunov 함수를 구성하고 모드 종속 평균 체류 시간(MDADT) 접근 방식을 사용하여 시스템이 FTB 및 유한 시간 H ∞ 제어 가능함을 보장하기 위해 몇 가지 충분한 조건이 제공됩니다. ^[2] nan ^[3] nan ^[4] nan ^[5] 둘째, Lie 미분 및 반복 기술을 기반으로, 교환된 아핀 비선형 시스템이 규정된 H로 유한 시간 제한됨을 보장할 수 있는 몇 가지 충분한 조건 ∞ 성능은 모드 종속 평균 체류 시간과 다중 Lyapunov 함수를 적용하여 파생됩니다. ^[6] 다중 Lyapunov 기능(MLF) 방법, 모드 종속 평균 체류 시간 및 MLF의 총 활성화 시간 길이를 사용하여 SDI가 있는 SNDS에 대해 일부 안정성 기준을 명시적으로 얻습니다. ^[7] 확장된 모드 종속 평균 체류 시간 스위칭 법칙과 반시간 종속 다중 Lyapunov 유사 기능을 채택함으로써 안정 및 불안정 서브시스템으로 구성된 스위칭 시간 지연 시스템에 대해 개선된 안정성 기준이 추론됩니다. ^[8] nan ^[9]

closed loop system 폐쇄 루프 시스템

Based on the novel multiple Lyapunov-like function, the sufficient conditions for the closed loop system to be globally uniformly exponentially stable (GUES) are obtained with admissible edge-dependent switching signals. ^[1] By using multiple Lyapunov function and defuzzifying for MFDM, a new mode-dependent fuzzy controller is proposed to assure the boundedness of the closed-loop system and the desired tracking performance. ^[2] Then, by means of the multiple Lyapunov-Krasovskii functional method, the delay-dependent bounded real lemma is derived to guarantee the stability of closed-loop system. ^[3] Secondly, some sufficient conditions are derived to guarantee that the corresponding closed-loop system is finite-time stable with a certain cost upper bound, using multiple Lyapunov functions and average dwell time approach. ^[4] Then, by means of constructing SPP-dependent multiple Lyapunov-like functions, some sufficient conditions with the ability to ensure the stability and an expected H∞ performance of the closed-loop system are deduced. ^[5] By utilizing the multiple Lyapunov function method and the backstepping technique together with the prescribed performance bounds, an adaptive NN controller is established which can ensure that all the signals in the closed-loop system are bounded under a class of switching signals with average dwell time and the tracking error converges to the predefined bounds. ^[6] Furthermore, by using multiple Lyapunov functions, a sufficient condition is put forward to make sure the state bumpless transfer performance, input bumpless transfer performance and asymptotical stability for the closed-loop systems. ^[7] By proposing a state-dependent switching law and utilizing the multiple Lyapunov function technology, the criteria are presented to ensure the local asymptotic stability with an $H_{\infty }$ performance level for the closed-loop system. ^[8]
새로운 다중 Lyapunov 유사 함수를 기반으로 하여 허용 가능한 에지 종속 스위칭 신호를 사용하여 폐쇄 루프 시스템이 전역적으로 균일하게 지수적으로 안정(GUES)되기 위한 충분한 조건을 얻습니다. ^[1] 다중 Lyapunov 기능을 사용하고 MFDM에 대해 역퍼지화함으로써 폐쇄 루프 시스템의 경계성과 원하는 추적 성능을 보장하기 위해 새로운 모드 종속 퍼지 컨트롤러가 제안됩니다. ^[2] nan ^[3] nan ^[4] nan ^[5] nan ^[6] nan ^[7] 상태 종속 스위칭 법칙을 제안하고 다중 Lyapunov 함수 기술을 활용하여 <inline-formula> <tex-math notation="LaTeX">$H_{\infty }$로 국소 점근 안정성을 보장하는 기준을 제시합니다. </tex-math></inline-formula> 폐쇄 루프 시스템의 성능 수준. ^[8]

finite time stability 유한 시간 안정성

Based on the multiple Lyapunov function method, the closed-loop teleoperation system with these two control methods is proved to be bounded and finite-time stability. ^[1] Some new conditions via multiple Lyapunov functions are given for stochastic finite-time stability of nonlinear time-varying SDEs. ^[2] Based on the multiple Lyapunov function method, the closed-loop teleoperation system with these two control methods is proved to be bounded and finite-time stability. ^[3] In addition, multiple Lyapunov functions-based criteria on stochastic finite-time stability are presented, which further relax the constraint of the infinitesimal generator $\mathcal{L}V$. ^[4]
다중 Lyapunov 함수 방법을 기반으로 이 두 가지 제어 방법을 사용하는 폐루프 원격 조작 시스템은 제한적이며 유한 시간 안정성이 입증되었습니다. ^[1] 다중 Lyapunov 함수를 통한 몇 가지 새로운 조건은 비선형 시변 SDE의 확률적 유한 시간 안정성에 대해 제공됩니다. ^[2] 다중 Lyapunov 함수 방법을 기반으로 이러한 두 가지 제어 방법을 사용하는 폐루프 원격 조작 시스템은 제한적이며 유한 시간 안정성이 입증되었습니다. ^[3] 또한 확률론적 유한 시간 안정성에 대한 여러 Lyapunov 함수 기반 기준이 제시되어 극소 생성기 $\mathcal{L}V$의 제약을 더욱 완화합니다. ^[4]

dependent average dwell 종속 평균 거주

The passivity conditions of such systems are studied by adopting the transition-dependent average dwell time (TDADT) and multiple Lyapunov functions (MLFs). ^[1] To effectively guarantee the stability of the considered system with unstable subsystems and reduce conservatism of the stability criteria, admissible edge-dependent average dwell time (AED-ADT) is first utilized to restrict switching signals for the continuous-time SNNs, and multiple Lyapunov–Kravosikii functionals (LKFs) combining relaxed integral inequalities are employed to develop two novel less-conservative stability conditions. ^[2]
이러한 시스템의 수동성 조건은 TDADT(Transition-Dependent Average Dwell Time) 및 MLF(Multiple Lyapunov Functions)를 채택하여 연구됩니다. ^[1] 불안정한 하위 시스템이 있는 고려된 시스템의 안정성을 효과적으로 보장하고 안정성 기준의 보수성을 줄이기 위해 AED-ADT(허용 가능한 에지 종속 평균 체류 시간)를 먼저 사용하여 연속 시간 SNN에 대한 스위칭 신호를 제한하고 다중 Lyapunov– 완화된 적분 부등식을 결합한 LKF(Kravosikii functions)를 사용하여 두 가지 덜 보수적인 안정성 조건을 개발했습니다. ^[2]

dwell time switching

Combining the multiple Lyapunov functions (MLFs) approach and the persistent dwell time switching technique, new sufficient conditions are obtained for the globally uniformly asymptotical stability of filtering error systems. ^[1]
다중 리아푸노프 함수(MLF) 접근 방식과 지속적인 체류 시간 전환 기술을 결합하여 필터링 오류 시스템의 전 세계적으로 균일한 점근적 안정성을 위한 새로운 충분 조건을 얻습니다. ^[1]

Dependent Multiple Lyapunov

Then, by means of constructing SPP-dependent multiple Lyapunov-like functions, some sufficient conditions with the ability to ensure the stability and an expected H∞ performance of the closed-loop system are deduced. ^[1] By adopting extended mode-dependent average dwell-time switching laws and semi-time-dependent multiple Lyapunov-like functions, the improved stability criterion is deduced for switched time-delay systems comprising of both stable and unstable subsystems. ^[2] By using the delay- and parameter-dependent multiple Lyapunov–Krasovskii functional approach, sufficient criteria on uniform H∞ finite-time stabilization via observer-based state feedback are presented for the solvability of the problem, which can be tackled by a feasibility problem in terms of linear matrix inequalities. ^[3]
nan ^[1] 확장된 모드 종속 평균 체류 시간 스위칭 법칙과 반시간 종속 다중 Lyapunov 유사 기능을 채택함으로써 안정 및 불안정 서브시스템으로 구성된 스위칭 시간 지연 시스템에 대해 개선된 안정성 기준이 추론됩니다. ^[2] 지연 및 매개변수 종속 다중 Lyapunov-Krasovskii 기능 접근 방식을 사용하여 관찰자 기반 상태 피드백을 통한 균일한 H∞ 유한 시간 안정화에 대한 충분한 기준이 문제의 해결 가능성에 대해 제시되며, 이는 다음에서 실행 가능성 문제로 해결할 수 있습니다. 선형 행렬 부등식의 항. ^[3]

Employing Multiple Lyapunov

Firstly, we propose the exponential stability criterion for a type of SPNSs when all subsystems succumb to average dwell time (ADT) switching by employing multiple Lyapunov functions (MLFs). ^[1] By employing multiple Lyapunov-like functions, criteria on GUAS and event-GUAS are established. ^[2]
첫째, 우리는 모든 하위 시스템이 다중 리아푸노프 함수(MLF)를 사용하여 평균 체류 시간(ADT) 전환에 굴복할 때 SPNS 유형에 대한 지수 안정성 기준을 제안합니다. ^[1] nan ^[2]

Proper Multiple Lyapunov

By constructing proper multiple Lyapunov–Krasovskii functions and applying average dwell time methods, a switching law and the relevant asynchronous resilient controller are designed to guarantee the finite-time boundedness of the closed-loop system with a specified ${H} _{\infty }$ performance index. ^[1] Sufficient conditions are initially established on the existence of the finite-time distributed controller by using proper multiple Lyapunov functions and the design criteria are presented in the form of linear matrix inequalities. ^[2]
적절한 다중 Lyapunov-Krasovskii 함수를 구성하고 평균 체류 시간 방법을 적용하여 스위칭 법칙과 관련 비동기 탄력성 컨트롤러는 지정된 <inline-formula> <tex- math notation="LaTeX">${H} _{\infty }$ </tex-math></inline-formula> 성능 지수. ^[1] 적절한 다중 Lyapunov 함수를 사용하여 유한 시간 분산 제어기의 존재에 대한 충분한 조건을 초기에 설정하고 설계 기준을 선형 행렬 부등식의 형태로 제시합니다. ^[2]

Piecewise Multiple Lyapunov

Based on a new bounded maximum average dwell time (BMADT) switching approach, an exponential stability condition is derived with a novel time-varying piecewise multiple Lyapunov function approach. ^[1] By constructing a novel piecewise multiple Lyapunov function approach, an exponential stability condition of switched T-S fuzzy systems is first established with the bounded maximum average dwell time. ^[2]
새로운 BMADT(Bounded Maximum Average Dwell Time) 전환 접근 방식을 기반으로 하는 새로운 시간 변화 조각별 다중 Lyapunov 함수 접근 방식으로 지수 안정성 조건이 파생됩니다. ^[1] 새로운 조각별 다중 Lyapunov 함수 접근 방식을 구성하여 제한된 최대 평균 체류 시간으로 전환된 T-S 퍼지 시스템의 지수 안정성 조건이 먼저 설정됩니다. ^[2]

Applying Multiple Lyapunov

First, several algebraic conditions are presented to guarantee asymptotic stability by applying multiple Lyapunov function (MLF) method, dwell time technique and fast-slow switching mechanism. ^[1] Furthermore, by applying multiple Lyapunov functions and S-procedure theory, the observer design problem can be converted into the existence issue of the solutions to the linear matrix inequality. ^[2]

Constructing Multiple Lyapunov

Then, by using the average dwell time switching approach and constructing multiple Lyapunov functions, a sufficient condition of IO-FTS for the considered system is presented in the form of linear matrix inequalities (LMIs). ^[1] By skillfully constructing multiple Lyapunov–Krasovskii functionals and successfully solving several troublesome obstacles, such as time-varying delay and switching signals and nonlinearity in the design procedure, the switched linear output-feedback controllers designed can render the resulting closed-loop switched system semi-globally stabilizable under a class of switching signals with average dwell time. ^[2]

Traditional Multiple Lyapunov

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. ^[1] This is not a simple extension of traditional multiple Lyapunov functions because of the introduction of external parameter and dwell time. ^[2]

Novel Multiple Lyapunov 소설 다중 랴푸노프

Based on the novel multiple Lyapunov-like function, the sufficient conditions for the closed loop system to be globally uniformly exponentially stable (GUES) are obtained with admissible edge-dependent switching signals. ^[1] By constructing a novel multiple Lyapunov functional (MLF), a sufficient criterion formulated by linear programming (LP) is established for the FTBS and the estimation of the settling time. ^[2]
새로운 다중 Lyapunov 유사 함수를 기반으로 하여 허용 가능한 에지 종속 스위칭 신호를 사용하여 폐쇄 루프 시스템이 전역적으로 균일하게 지수적으로 안정(GUES)되기 위한 충분한 조건을 얻습니다. ^[1] 새로운 다중 Lyapunov 기능(MLF)을 구성함으로써 선형 계획법(LP)에 의해 공식화된 충분한 기준이 FTBS 및 안정화 시간 추정에 대해 설정됩니다. ^[2]

multiple lyapunov function 다중 랴푸노프 함수

In view of the transient behavior caused by controller switching, the global stability of the switched system is analyzed using the multiple Lyapunov function approach and average dwell time condition. ^[1] By using multiple Lyapunov function and defuzzifying for MFDM, a new mode-dependent fuzzy controller is proposed to assure the boundedness of the closed-loop system and the desired tracking performance. ^[2] We present sufficient conditions in terms of multiple Lyapunov functions for the origin of a class of hybrid systems to be finite-time stable. ^[3] For each case, stability conditions are derived using multiple Lyapunov functions and mode-dependent average dwell-time, and thus relax the conservatisms caused by common Lyapunov function and average dwell-time. ^[4] This is accomplished by generalizing Common Lyapunov Functions (CLF) and Multiple Lyapunov Functions (MLF) methods, the latter when applied to fractional switching systems (FSS) in the resetting. ^[5] In terms of multiple Lyapunov functions, a state-dependent switching law is designed to realize fault detection performance even if the robustness and fault sensitivity of each subsystem are local. ^[6] The sufficient conditions for the stochastic state-feedback incremental H ∞ control problem are obtained by solving a set of coupled iHJIs and by resorting to multiple Lyapunov functions. ^[7] For the longitudinal maneuver mission of air-breathing hypersonic vehicles, this paper develops an adaptive control strategy based on multiple Lyapunov function method. ^[8] The passivity conditions of such systems are studied by adopting the transition-dependent average dwell time (TDADT) and multiple Lyapunov functions (MLFs). ^[9] Based on the multiple Lyapunov function method, the closed-loop teleoperation system with these two control methods is proved to be bounded and finite-time stability. ^[10] Moreover, based on the method of multiple Lyapunov functions (MLFs) and the technique of adding a power integrator, a sufficient condition guaranteeing the solvability of the finite-time control problem for the system under consideration is derived via a designed switching law, where there is no subsystem whose corresponding control problem must be solvable. ^[11] When the cascade structure contains multiple controlled converter systems, the stability of the entire cascaded system is guaranteed by the superposition of multiple Lyapunov functions. ^[12] Then the switched PID adaptive controller is designed via error transformation and multiple Lyapunov function methods. ^[13] By the ergodic property of the semi-Markov process and the method of multiple Lyapunov functions, some sufficient conditions for almost sure exponential stability of stochastic systems are formulated, which cover some existing results in the literature. ^[14] The novel observer is designed using multiple Lyapunov functions based on L1-norm, reducing the estimation noise while increasing the accuracy. ^[15] Then, the stability of linear–nonlinear switching extended state observers (ESOs) is proved by multiple Lyapunov functions. ^[16] First, several algebraic conditions are presented to guarantee asymptotic stability by applying multiple Lyapunov function (MLF) method, dwell time technique and fast-slow switching mechanism. ^[17] Some new conditions via multiple Lyapunov functions are given for stochastic finite-time stability of nonlinear time-varying SDEs. ^[18] This paper presents the development of a new set of switched velocity controllers of a swarm of unmanned ground vehicles (UGVs) from multiple Lyapunov functions, which are invoked according to a switching rule. ^[19] This function can be understood as a multiple Lyapunov function, consisting of an infinite number of partial Lyapunov functions, each of which is used for a certain time interval. ^[20] Then, by combining multiple Lyapunov functions with the initial state and acceptable state of B chi automata, a sufficient condition for the global exponential stability of the system with step decreasing Lyapunov functions is given. ^[21] Secondly, the stability criteria for cyclic switched linear (or nonlinear) systems with ACDT or both S-ACDT and U-ACDT are derived by resorting to a technique that uses multiple Lyapunov functions. ^[22] Closed-loop stability is shown to be guaranteed using the multiple Lyapunov functions theorem and the Barbalat’s lemma. ^[23] A common finite-time barrier function, multiple barrier functions and multiple Lyapunov functions are merged as a tool for analyzing the asymptotic stability of a switched nonlinear system under spatio-temporal specifications. ^[24] The main contribution of the study is the synthesis of a continuous-phase adaptive robust tracking control law for hybrid models of bipedal robotic walking by incorporating the construction of multiple Lyapunov functions into the control Lyapunov function. ^[25] Secondly, some sufficient conditions are derived to guarantee that the corresponding closed-loop system is finite-time stable with a certain cost upper bound, using multiple Lyapunov functions and average dwell time approach. ^[26] By using the tools of multiple Lyapunov function and linear matrix inequality method, a neighbor-based distributed controller is designed for each following-agent so as to achieve (average) consensus tracking, even if the underlying switching communication network contains disconnected digraphs. ^[27] By applying the multiple Lyapunov function approach, sufficient conditions of exponential stability a. ^[28] 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. ^[29] The multiple Lyapunov function method is adopted to guarantee the tracking performance with designed average dwell time. ^[30] Then, by using the average dwell time switching approach and constructing multiple Lyapunov functions, a sufficient condition of IO-FTS for the considered system is presented in the form of linear matrix inequalities (LMIs). ^[31] Next, by utilizing the state-feedback controller and multiple Lyapunov functions method, some criteria incorporating the restriction of DoS jamming attacks are proposed to guarantee the L∞ control performance of the event-triggered Markov closed-loop jump system. ^[32] These two sufficient conditions can be viewed as generalizations of the clock-dependent Lyapunov and multiple Lyapunov function methods, respectively. ^[33] With the aid of the multiple Lyapunov function method, constraints on switching signals are derived under which global fixed-time stability of zero solutions of considered systems can be guaranteed. ^[34] Firstly, we propose the exponential stability criterion for a type of SPNSs when all subsystems succumb to average dwell time (ADT) switching by employing multiple Lyapunov functions (MLFs). ^[35] On the basis of the switched model, a multiple Lyapunov function method is utilized and a set of sufficient conditions incorporating the event-triggering scheme (ETS) and restriction of DoS attacks are provided to preserve performance. ^[36] By utilizing the multiple Lyapunov function method and the backstepping technique together with the prescribed performance bounds, an adaptive NN controller is established which can ensure that all the signals in the closed-loop system are bounded under a class of switching signals with average dwell time and the tracking error converges to the predefined bounds. ^[37] By virtue of the multiple Lyapunov function technology and Newton-Leibniz formula, a sufficient condition is established to guarantee the exponential synchronization of switched cellular neural networks. ^[38] Taking the asynchronous switching into account, we give a switching law with the average dwell time (ADT) guaranteeing the exponential stability with the $H_{\infty }$ performance of the event-triggered switched filtering error system (SFES) in the absence of attacks by employing the multiple Lyapunov function (MLF) method. ^[39] By constructing appropriate multiple Lyapunov functions and employing mode-dependent average dwell time (MDADT) approach, some sufficient conditions are given to ensure that the system is FTB and finite-time H ∞ controllable. ^[40] By using multiple Lyapunov functions and mode-dependent average dwell time, we provide a sufficient condition for the p th moment exponential stability with fast convergence rate for deterministic switched discrete-time stochastic systems with stable and unstable subsystems. ^[41] 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). ^[42] For each case, stability conditions are derived using multiple Lyapunov functions and mode-dependent average dwell-time, and thus relax the conservatisms caused by common Lyapunov function and average dwell-time. ^[43] Based on multiple Lyapunov functions (MLFs), the sufficient conditions for controller gains that guarantee the leader-following consensus of MASs are proposed when the ADT of switching topologies is larger than the allowable value. ^[44] Based on the multiple Lyapunov functions approach, some sufficient conditions for the existence of guaranteed cost controllers are obtained. ^[45] The stability of the controller is proved step by step through the decreasing of the multiple Lyapunov functions along the state trajectory. ^[46] Furthermore, by using multiple Lyapunov functions, a sufficient condition is put forward to make sure the state bumpless transfer performance, input bumpless transfer performance and asymptotical stability for the closed-loop systems. ^[47] According to mode-dependent dwell average time and