Stability, boundedness and persistence are three important aspects for an ecological model. In this paper, a further analysis of a class of anaerobic digestion ecological models is performed. Based on the Liupunov Met...Stability, boundedness and persistence are three important aspects for an ecological model. In this paper, a further analysis of a class of anaerobic digestion ecological models is performed. Based on the Liupunov Method, the local stability of all equilibria in the system is got. According to the vector fields described by the system, the proof of the boundedness of the solution on the anaerobic digestion processes is completed in three steps. The method proposed in the discussion on the boundedness can be generalized to the similar problems. Results in this paper give information on how to run the ecological system well by adjusting the system parameters.展开更多
We consider a system of neutral equations with unbounded delay, and derive conditions on Liapunov functionals to ensure that the solutions are uniformly bounded and uniformly ultimately bounded.
Using the energy estimate and Gagliardo-Nirenberg-type inequalities,the existence and uniform boundedness of the global solutions to a strongly coupled reaction-diffusion system are proved. This system is a generaliza...Using the energy estimate and Gagliardo-Nirenberg-type inequalities,the existence and uniform boundedness of the global solutions to a strongly coupled reaction-diffusion system are proved. This system is a generalization of the two-species Lotka-Volterra predator-prey model with self and cross-diffusion. Suffcient condition for the global asymptotic stability of the positive equilibrium point of the model is given by constructing Lyapunov function.展开更多
It is shown that the famous Banach-Steinhaus theorem can be generalized to some families of nonlinear functionals defined on some topological groups and topological vector space, e.g. the F-spaced lβ(0 <β < 1)...It is shown that the famous Banach-Steinhaus theorem can be generalized to some families of nonlinear functionals defined on some topological groups and topological vector space, e.g. the F-spaced lβ(0 <β < 1) and S[a, b].展开更多
The uniformly ultimate boundedness of discontinuous systems with time-delay in the sense of Filippov solutions is discussed.Based on the Lyapunov-Krasovskii functional,the Lyapunov theorem for the globally strongly un...The uniformly ultimate boundedness of discontinuous systems with time-delay in the sense of Filippov solutions is discussed.Based on the Lyapunov-Krasovskii functional,the Lyapunov theorem for the globally strongly uniformly ultimate boundedness of retarded discontinuous systems is presented.Furthermore,the result is applied to a class of mechanical systems with a retarded discontinuous friction item.展开更多
This note deals with the functional differential equations with infinite delay x’=f（t, x,）, （1）where x∈R<sup>n</sup>, f: [0,∞) ×C<sub>9</sub>→R<sup>n</sup>, C<sub>...This note deals with the functional differential equations with infinite delay x’=f（t, x,）, （1）where x∈R<sup>n</sup>, f: [0,∞) ×C<sub>9</sub>→R<sup>n</sup>, C<sub>g</sub> is the phase space of （1）, which is defined as follows:C=∮(（-∞, 0], R<sup>n</sup>） is the set of all the continuous vector-functions mapping (-∞, 0] toR<sup>n</sup>. The function g:（-∞, 0]→ [1, ∞） is continuous and nonincreasing with g （0）=1 andg（-∞）=∞.展开更多
We prove the uniform Lipschitz bound of solutions for a nonlinear elliptic system modeling the steady state of populations that compete in a heterogeneous environment. This extends known quasi-optimal regularity resul...We prove the uniform Lipschitz bound of solutions for a nonlinear elliptic system modeling the steady state of populations that compete in a heterogeneous environment. This extends known quasi-optimal regularity results and covers the optimal case for this problem. The proof relies upon the blow-up technique and the almost monotonicity formula by Caffarelli, Jerison and Kenig.展开更多
We stress a basic criterion that shows in a simple way how a sequence of real-valued functions can converge uniformly when it is more or less evident that the sequence converges uniformly away from a finite number of ...We stress a basic criterion that shows in a simple way how a sequence of real-valued functions can converge uniformly when it is more or less evident that the sequence converges uniformly away from a finite number of points of the closure of its domain. For functions of a real variable, unlike in most classical textbooks our criterion avoids the search of extrema (by differential calculus) of their general term.展开更多
The authors consider the uniformly most powerful invariant test of the testing problems (Ⅰ) H 0: μ′Σ -1 μ≥CH 1: μ′Σ -1 μ<C and (Ⅱ) H 00 : β′X′Xβσ 2≥CH 11 : β′X′Xβσ 2<C u...The authors consider the uniformly most powerful invariant test of the testing problems (Ⅰ) H 0: μ′Σ -1 μ≥CH 1: μ′Σ -1 μ<C and (Ⅱ) H 00 : β′X′Xβσ 2≥CH 11 : β′X′Xβσ 2<C under m dimensional normal population N m(μ, Σ) and normal linear model (Y, Xβ, σ 2) respectively. Furthermore, an application of the uniformly most powerful invariant test is given.展开更多
In this paper, the problem of cubature Kalman fusion filtering(CKFF) is addressed for multi-sensor systems under amplify-and-forward(AaF) relays. For the purpose of facilitating data transmission, AaF relays are utili...In this paper, the problem of cubature Kalman fusion filtering(CKFF) is addressed for multi-sensor systems under amplify-and-forward(AaF) relays. For the purpose of facilitating data transmission, AaF relays are utilized to regulate signal communication between sensors and filters. Here, the randomly varying channel parameters are represented by a set of stochastic variables whose occurring probabilities are permitted to exhibit bounded uncertainty. Employing the spherical-radial cubature principle, a local filter under AaF relays is initially constructed. This construction ensures and minimizes an upper bound of the filtering error covariance by designing an appropriate filter gain. Subsequently, the local filters are fused through the application of the covariance intersection fusion rule. Furthermore, the uniform boundedness of the filtering error covariance's upper bound is investigated through establishing certain sufficient conditions. The effectiveness of the proposed CKFF scheme is ultimately validated via a simulation experiment concentrating on a three-phase induction machine.展开更多
In this paper, a fault-tolerant-based online critic learning algorithm is developed to solve the optimal tracking control issue for nonaffine nonlinear systems with actuator faults.First, a novel augmented plant is co...In this paper, a fault-tolerant-based online critic learning algorithm is developed to solve the optimal tracking control issue for nonaffine nonlinear systems with actuator faults.First, a novel augmented plant is constructed by fusing the system state and the reference trajectory, which aims to transform the optimal fault-tolerant tracking control design with actuator faults into the optimal regulation problem of the conventional nonlinear error system. Subsequently, in order to ensure the normal execution of the online learning algorithm, a stability criterion condition is created to obtain an initial admissible tracking policy. Then, the constructed model neural network(NN) is pretrained to recognize the system dynamics and calculate trajectory control. The critic and action NNs are constructed to output the approximate cost function and approximate tracking control,respectively. The Hamilton-Jacobi-Bellman equation of the error system is solved online through the action-critic framework. In theoretical analysis, it is proved that all concerned signals are uniformly ultimately bounded according to the Lyapunov principle.The tracking control law can approach the optimal tracking control within a finite approximation error. Finally, two experimental examples are conducted to indicate the effectiveness and superiority of the developed fault-tolerant tracking control scheme.展开更多
基金Supported by the National Natural Science Foundation of China (No.60372012) and NSF of Chongqing (No.0831)
文摘Stability, boundedness and persistence are three important aspects for an ecological model. In this paper, a further analysis of a class of anaerobic digestion ecological models is performed. Based on the Liupunov Method, the local stability of all equilibria in the system is got. According to the vector fields described by the system, the proof of the boundedness of the solution on the anaerobic digestion processes is completed in three steps. The method proposed in the discussion on the boundedness can be generalized to the similar problems. Results in this paper give information on how to run the ecological system well by adjusting the system parameters.
文摘We consider a system of neutral equations with unbounded delay, and derive conditions on Liapunov functionals to ensure that the solutions are uniformly bounded and uniformly ultimately bounded.
基金Partially supported by the National Natural Science Foundation of China (10671158)the NSFof Gansu Province (3ZS061-A25-015)+1 种基金the Scientific Research Fund of Gansu Provincial Educatio nDepartment (0601-21)NWNU-KJCXGC-03-18, 39 Foundations
文摘Using the energy estimate and Gagliardo-Nirenberg-type inequalities,the existence and uniform boundedness of the global solutions to a strongly coupled reaction-diffusion system are proved. This system is a generalization of the two-species Lotka-Volterra predator-prey model with self and cross-diffusion. Suffcient condition for the global asymptotic stability of the positive equilibrium point of the model is given by constructing Lyapunov function.
文摘It is shown that the famous Banach-Steinhaus theorem can be generalized to some families of nonlinear functionals defined on some topological groups and topological vector space, e.g. the F-spaced lβ(0 <β < 1) and S[a, b].
基金supported by the National Natural Science Foundation of China (No. 60874006)
文摘The uniformly ultimate boundedness of discontinuous systems with time-delay in the sense of Filippov solutions is discussed.Based on the Lyapunov-Krasovskii functional,the Lyapunov theorem for the globally strongly uniformly ultimate boundedness of retarded discontinuous systems is presented.Furthermore,the result is applied to a class of mechanical systems with a retarded discontinuous friction item.
基金Project partially supported by the National Natural Science Foundation of China.
文摘This note deals with the functional differential equations with infinite delay x’=f（t, x,）, （1）where x∈R<sup>n</sup>, f: [0,∞) ×C<sub>9</sub>→R<sup>n</sup>, C<sub>g</sub> is the phase space of （1）, which is defined as follows:C=∮(（-∞, 0], R<sup>n</sup>） is the set of all the continuous vector-functions mapping (-∞, 0] toR<sup>n</sup>. The function g:（-∞, 0]→ [1, ∞） is continuous and nonincreasing with g （0）=1 andg（-∞）=∞.
文摘We prove the uniform Lipschitz bound of solutions for a nonlinear elliptic system modeling the steady state of populations that compete in a heterogeneous environment. This extends known quasi-optimal regularity results and covers the optimal case for this problem. The proof relies upon the blow-up technique and the almost monotonicity formula by Caffarelli, Jerison and Kenig.
文摘We stress a basic criterion that shows in a simple way how a sequence of real-valued functions can converge uniformly when it is more or less evident that the sequence converges uniformly away from a finite number of points of the closure of its domain. For functions of a real variable, unlike in most classical textbooks our criterion avoids the search of extrema (by differential calculus) of their general term.
文摘The authors consider the uniformly most powerful invariant test of the testing problems (Ⅰ) H 0: μ′Σ -1 μ≥CH 1: μ′Σ -1 μ<C and (Ⅱ) H 00 : β′X′Xβσ 2≥CH 11 : β′X′Xβσ 2<C under m dimensional normal population N m(μ, Σ) and normal linear model (Y, Xβ, σ 2) respectively. Furthermore, an application of the uniformly most powerful invariant test is given.
基金supported in part by the National Natural Science Foundation of China(12171124,61933007)the Natural Science Foundation of Heilongjiang Province of China(ZD2022F003)+2 种基金the National High-End Foreign Experts Recruitment Plan of China(G2023012004L)the Royal Society of UKthe Alexander von Humboldt Foundation of Germany
文摘In this paper, the problem of cubature Kalman fusion filtering(CKFF) is addressed for multi-sensor systems under amplify-and-forward(AaF) relays. For the purpose of facilitating data transmission, AaF relays are utilized to regulate signal communication between sensors and filters. Here, the randomly varying channel parameters are represented by a set of stochastic variables whose occurring probabilities are permitted to exhibit bounded uncertainty. Employing the spherical-radial cubature principle, a local filter under AaF relays is initially constructed. This construction ensures and minimizes an upper bound of the filtering error covariance by designing an appropriate filter gain. Subsequently, the local filters are fused through the application of the covariance intersection fusion rule. Furthermore, the uniform boundedness of the filtering error covariance's upper bound is investigated through establishing certain sufficient conditions. The effectiveness of the proposed CKFF scheme is ultimately validated via a simulation experiment concentrating on a three-phase induction machine.
基金supported in part by the National Natural Science Foundation of China(62222301,62373012,62473012,62021003)the National Science and Technology Major Project(2021ZD0112302,2021ZD0112301)the Beijing Natural Science Foundation(JQ19013)
文摘In this paper, a fault-tolerant-based online critic learning algorithm is developed to solve the optimal tracking control issue for nonaffine nonlinear systems with actuator faults.First, a novel augmented plant is constructed by fusing the system state and the reference trajectory, which aims to transform the optimal fault-tolerant tracking control design with actuator faults into the optimal regulation problem of the conventional nonlinear error system. Subsequently, in order to ensure the normal execution of the online learning algorithm, a stability criterion condition is created to obtain an initial admissible tracking policy. Then, the constructed model neural network(NN) is pretrained to recognize the system dynamics and calculate trajectory control. The critic and action NNs are constructed to output the approximate cost function and approximate tracking control,respectively. The Hamilton-Jacobi-Bellman equation of the error system is solved online through the action-critic framework. In theoretical analysis, it is proved that all concerned signals are uniformly ultimately bounded according to the Lyapunov principle.The tracking control law can approach the optimal tracking control within a finite approximation error. Finally, two experimental examples are conducted to indicate the effectiveness and superiority of the developed fault-tolerant tracking control scheme.
