We discuss the global stabilization procedure which renders a general class of feedback nonlinear systems exponential convergent, Our stabilizer consists of a nested saturation function, which is a nonlinear combinati...We discuss the global stabilization procedure which renders a general class of feedback nonlinear systems exponential convergent, Our stabilizer consists of a nested saturation function, which is a nonlinear combination of saturation functions. Here we prove the exponential convergence of the stabilizer for the first time and give numerical examples to illustrate the efficiency of the result given above,展开更多
In this paper, we demonstrate that the finite-dimensional approximations to the solutions of a linear bond-based peridynamic boundary value problem converge to the exact solution exponentially with the analyticity ass...In this paper, we demonstrate that the finite-dimensional approximations to the solutions of a linear bond-based peridynamic boundary value problem converge to the exact solution exponentially with the analyticity assumption of the forcing term, therefore greatly improve the convergence rate derived in literature.展开更多
In this paper, the dynamic behaviors of fuzzy cellular neural networks (FCNNs) with time-varying coefficients and delays are considered. Some criteria are established to ensure the exponential convergence or exponen...In this paper, the dynamic behaviors of fuzzy cellular neural networks (FCNNs) with time-varying coefficients and delays are considered. Some criteria are established to ensure the exponential convergence or exponential stability of such neural networks. The effectiveness of obtained results is illustrated by a numerical example.展开更多
In this paper, we propose a Sample Average Approximation (SAA) method for a class of Stochastic Mathematical Programs with Complementarity Constraints (SMPCC) recently considered by Birbil, G/irkan and Listes [3]....In this paper, we propose a Sample Average Approximation (SAA) method for a class of Stochastic Mathematical Programs with Complementarity Constraints (SMPCC) recently considered by Birbil, G/irkan and Listes [3]. We study the statistical properties of obtained SAA estimators. In particular we show that under moderate conditions a sequence of weak stationary points of SAA programs converge to a weak stationary point of the true problem with probability approaching one at exponential rate as the sample size tends to infinity. To implement the SAA method more efficiently, we incorporate the method with some techniques such as Scholtes' regularization method and the well known smoothing NCP method. Some preliminary numerical results are reported.展开更多
The models of competitive neural network(CNN)was in recent past proposed to describe the dynamics of cortical cognitive maps with unsupervised synaptic modifications,where there are two types of memories:Long-term mem...The models of competitive neural network(CNN)was in recent past proposed to describe the dynamics of cortical cognitive maps with unsupervised synaptic modifications,where there are two types of memories:Long-term memories(LTM)and short-term memories(STM),LTM presents unsupervised and slow synaptic modifications and STM characterize the fast neural activity.This paper is concerned with a class of neutral type CNN’s with mixed delay and D operator.By employing the appropriate differential inequality theory,some sufficient conditions are given to ensure that all solutions of the model converge exponentially to zero vector.Finally,an illustrative example is also given at the end of this paper to show the effectiveness of the proposed results.展开更多
In this paper, we introduce the definition of Γ-Fisher entropy. For the stochastic differential equation in Rd and the unique invariant probability measure μ, we obtain the exponential convergence of Γ-Fisher entro...In this paper, we introduce the definition of Γ-Fisher entropy. For the stochastic differential equation in Rd and the unique invariant probability measure μ, we obtain the exponential convergence of Γ-Fisher entropy with respect to μ under some condition about Γ and the drift b. Moreover, we revisit the exponential convergence of the usual Fisher entropy under the Bakry Emery condition.展开更多
Let (Xt)t≥0 be a Lévy process taking values in R^d with absolutely continuous marginal distributions. Given a real measurable function f on R^d in Kato's class, we show that the empirical mean 1/t ∫ f(Xs)d...Let (Xt)t≥0 be a Lévy process taking values in R^d with absolutely continuous marginal distributions. Given a real measurable function f on R^d in Kato's class, we show that the empirical mean 1/t ∫ f(Xs)ds converges to a constant z in probability with an exponential rate if and only if f has a uniform mean z. This result improves a classical result of Kahane et al. and generalizes a similar result of L. Wu from the Brownian Motion to general Lévy processes.展开更多
For a collective system,the connectedness of the adjacency matrix plays a key role in making the system achieve its emergent feature,such as flocking or multi-clustering.In this paper,we study a nonsymmetric multi-par...For a collective system,the connectedness of the adjacency matrix plays a key role in making the system achieve its emergent feature,such as flocking or multi-clustering.In this paper,we study a nonsymmetric multi-particle system with a constant and local cut-off weight.A distributed communication delay is also introduced into both the velocity adjoint term and the cut-off weight.As a new observation,we show that the desired multi-particle system undergoes both flocking and clustering behaviors when the eigenvalue 1 of the adjacency matrix is semi-simple.In this case,the adjacency matrix may lose the connectedness.In particular,the number of clusters is discussed by using subspace analysis.In terms of results,for both the non-critical and general neighbourhood situations,some criteria of flocking and clustering emergence with an exponential convergent rate are established by the standard matrix analysis for when the delay is free.As a distributed delay is involved,the corresponding criteria are also found,and these small time lags do not change the emergent properties qualitatively,but alter the final value in a nonlinear way.Consequently,some previous works[14]are extended.展开更多
In this paper, an exponential inequality for the maximal partial sums of negatively superadditive-dependent (NSD, in short) random variables is established. By uSing the exponen- tial inequality, we present some gen...In this paper, an exponential inequality for the maximal partial sums of negatively superadditive-dependent (NSD, in short) random variables is established. By uSing the exponen- tial inequality, we present some general results on the complete convergence for arrays of rowwise NSD random variables, which improve or generalize the corresponding ones of Wang et al. [28] and Chen et al. [2]. In addition, some sufficient conditions to prove the complete convergence are provided. As an application of the complete convergence that we established, we further investigate the complete consistency and convergence rate of the estimator in a nonparametric regression model based on NSD errors.展开更多
It is shown that there is an unique ω-period solution x(t, φ^*) for a delayed cellular network and its each solution x(t, φ) converges exponentially to x(t, φ^*) if its each output function is bounded and ...It is shown that there is an unique ω-period solution x(t, φ^*) for a delayed cellular network and its each solution x(t, φ) converges exponentially to x(t, φ^*) if its each output function is bounded and satisfies Lipschitz condition when all input signals are at-periodic functions.展开更多
We propose a novel numerical approach for delay differential equations with vanishing proportional delays based on spectral methods. A Legendre-collocation method is employed to obtain highly accurate numerical approx...We propose a novel numerical approach for delay differential equations with vanishing proportional delays based on spectral methods. A Legendre-collocation method is employed to obtain highly accurate numerical approximations to the exact solution. It is proved theoretically and demonstrated numerically that the proposed method converges exponentially provided that the data in the are smooth. given pantograph delay differential equation展开更多
Distributed Nash equilibrium seeking of aggregative games is investigated and a continuous-time algorithm is proposed.The algorithm is designed by virtue of projected gradient play dynamics and aggregation tracking dy...Distributed Nash equilibrium seeking of aggregative games is investigated and a continuous-time algorithm is proposed.The algorithm is designed by virtue of projected gradient play dynamics and aggregation tracking dynamics,and is applicable to games with constrained strategy sets and weight-balanced communication graphs.The key feature of our method is that the proposed projected dynamics achieves exponential convergence,whereas such convergence results are only obtained for non-projected dynamics in existing works on distributed optimization and equilibrium seeking.Numerical examples illustrate the effectiveness of our methods.展开更多
A GI/G/1 queue with vacations is considered in this paper. We develop an approximating technique on max function of independent and identically distributed (i.i.d.) random variables, that is max{ηi, 1 ≤ i ≤ n}. T...A GI/G/1 queue with vacations is considered in this paper. We develop an approximating technique on max function of independent and identically distributed (i.i.d.) random variables, that is max{ηi, 1 ≤ i ≤ n}. The approximating technique is used to obtain the fluid approximation for the queue length, workload and busy time processes. Furthermore, under uniform topology, if the scaled arrival process and the scaled service process converge to the corresponding fluid processes with an exponential rate, we prove by the approximating technique that the scaled processes characterizing the queue converge to the corresponding fluid limits with the exponential rate only for large N. Here the scaled processes include the queue length process, workload process and busy time process.展开更多
We study an explicit exponential scheme for the time discretisation of stochastic SchrS- dinger Equations Driven by additive or Multiplicative It6 Noise. The numerical scheme is shown to converge with strong order 1 i...We study an explicit exponential scheme for the time discretisation of stochastic SchrS- dinger Equations Driven by additive or Multiplicative It6 Noise. The numerical scheme is shown to converge with strong order 1 if the noise is additive and with strong order 1/2 for multiplicative noise. In addition, if the noise is additive, we show that the exact solutions of the linear stochastic Sehr6dinger equations satisfy trace formulas for the expected mass, energy, and momentum (i. e., linear drifts in these quantities). Furthermore, we inspect the behaviour of the numerical solutions with respect to these trace formulas. Several numerical simulations are presented and confirm our theoretical results.展开更多
In this paper, we consider the distributed optimization problem, where the goal is to minimize the global objective function formed by a sum of agents' local smooth and strongly convex objective functions, over un...In this paper, we consider the distributed optimization problem, where the goal is to minimize the global objective function formed by a sum of agents' local smooth and strongly convex objective functions, over undirected connected graphs. Several distributed accelerated algorithms have been proposed for solving such a problem in the existing literature. In this paper, we provide insights for understanding these existing distributed algorithms from an ordinary differential equation(ODE) point of view. More specifically, we first derive an equivalent second-order ODE, which is the exact limit of these existing algorithms by taking the small step-size. Moreover, focusing on the quadratic objective functions, we show that the solution of the resulting ODE exponentially converges to the unique global optimal solution. The theoretical results are validated and illustrated by numerical simulations.展开更多
In this paper,we propose a hybrid spectral method for a type of nonlocal problems,nonlinear Volterra integral equations(VIEs)of the second kind.The main idea is to use the shifted generalized Log orthogonal functions(...In this paper,we propose a hybrid spectral method for a type of nonlocal problems,nonlinear Volterra integral equations(VIEs)of the second kind.The main idea is to use the shifted generalized Log orthogonal functions(GLOFs)as the basis for the first interval and employ the classical shifted Legendre polynomials for other subintervals.This method is robust for VIEs with weakly singular kernel due to the GLOFs can efficiently approximate one-point singular functions as well as smooth functions.The well-posedness and the related error estimates will be provided.Abundant numerical experiments will verify the theoretical results and show the high-efficiency of the new hybrid spectral method.展开更多
We perform the analysis of the hp finite element approximation for the solution to singularly perturbed transmission problems, using Spectral Boundary Layer Meshes. In [12] it was shown that this method yields robust ...We perform the analysis of the hp finite element approximation for the solution to singularly perturbed transmission problems, using Spectral Boundary Layer Meshes. In [12] it was shown that this method yields robust exponential convergence, as the degree p of the approximating polynomials is increased, when the error is measured in the energy norm associated with the boundary value problem. In the present article we sharpen the result by showing that the hp-Finite Element Method (FEM) on Spectral Boundary Layer Meshes leads to robust exponential convergence in a stronger, more balanced norm. Several numerical results illustrating and extending the theory are also nresented.展开更多
A novel,highly efficient and accurate adaptive higher-order finite element method(hp-FEM)is used to simulate a multi-frequency resistivity loggingwhile-drilling(LWD)tool response in a borehole environment.Presented in...A novel,highly efficient and accurate adaptive higher-order finite element method(hp-FEM)is used to simulate a multi-frequency resistivity loggingwhile-drilling(LWD)tool response in a borehole environment.Presented in this study are the vector expression of Maxwell’s equations,three kinds of boundary conditions,stability weak formulation of Maxwell’s equations,and automatic hpadaptivity strategy.The new hp-FEM can select optimal refinement and calculation strategies based on the practical formation model and error estimation.Numerical experiments show that the new hp-FEM has an exponential convergence rate in terms of relative error in a user-prescribed quantity of interest against the degrees of freedom,which provides more accurate results than those obtained using the adaptive h-FEM.The numerical results illustrate the high efficiency and accuracy of the method at a given LWD tool structure and parameters in different physical models,which further confirm the accuracy of the results using the Hermes library(http://hpfem.org/hermes)with a multi-frequency resistivity LWD tool response in a borehole environment.展开更多
In this paper,we aim to develop distributed continuous-time algorithms over directed graphs to seek the Nash equilibrium in a noncooperative game.Motivated by the recent consensus-based designs,we present a distribute...In this paper,we aim to develop distributed continuous-time algorithms over directed graphs to seek the Nash equilibrium in a noncooperative game.Motivated by the recent consensus-based designs,we present a distributed algorithm with a proportional gain for weight-balanced directed graphs.By further embedding a distributed estimator of the left eigenvector associated with zero eigenvalue of the graph Laplacian,we extend it to the case with arbitrary strongly connected directed graphs having possible unbalanced weights.In both cases,the Nash equilibrium is proven to be exactly reached with an exponential convergence rate.An example is given to illustrate the validity of the theoretical results.展开更多
This paper is concerned with control and optimization for a sampled-data system with quantization and actuator saturation. Based quantization and actuator saturation a controller is introduced. The corresponding close...This paper is concerned with control and optimization for a sampled-data system with quantization and actuator saturation. Based quantization and actuator saturation a controller is introduced. The corresponding closed loop system is transformed into a system with input saturation and bounded external disturbance. A new Lyapunov functional is constructed to derive a sample-interval dependent condition on the existence of a state feedback controller such that the closed-loop system is exponentially convergent to an ultimate ellipsoid for the initial condition starting from some initial ellipsoid. Based on the condition, the desired controller is designed. Furthermore, optimization problems about the sample-interval, the ultimate ellipsoid and the initial ellipsoid are formulated. An example is given to illustrate the effectiveness of the proposed method.展开更多
文摘We discuss the global stabilization procedure which renders a general class of feedback nonlinear systems exponential convergent, Our stabilizer consists of a nested saturation function, which is a nonlinear combination of saturation functions. Here we prove the exponential convergence of the stabilizer for the first time and give numerical examples to illustrate the efficiency of the result given above,
文摘In this paper, we demonstrate that the finite-dimensional approximations to the solutions of a linear bond-based peridynamic boundary value problem converge to the exact solution exponentially with the analyticity assumption of the forcing term, therefore greatly improve the convergence rate derived in literature.
基金supported by the National Natural Science Foundation of China (No. 50578064)the Foundation of Science and Technology of Guangdong Province in China (No. 2009B011400046)
文摘In this paper, the dynamic behaviors of fuzzy cellular neural networks (FCNNs) with time-varying coefficients and delays are considered. Some criteria are established to ensure the exponential convergence or exponential stability of such neural networks. The effectiveness of obtained results is illustrated by a numerical example.
文摘In this paper, we propose a Sample Average Approximation (SAA) method for a class of Stochastic Mathematical Programs with Complementarity Constraints (SMPCC) recently considered by Birbil, G/irkan and Listes [3]. We study the statistical properties of obtained SAA estimators. In particular we show that under moderate conditions a sequence of weak stationary points of SAA programs converge to a weak stationary point of the true problem with probability approaching one at exponential rate as the sample size tends to infinity. To implement the SAA method more efficiently, we incorporate the method with some techniques such as Scholtes' regularization method and the well known smoothing NCP method. Some preliminary numerical results are reported.
文摘The models of competitive neural network(CNN)was in recent past proposed to describe the dynamics of cortical cognitive maps with unsupervised synaptic modifications,where there are two types of memories:Long-term memories(LTM)and short-term memories(STM),LTM presents unsupervised and slow synaptic modifications and STM characterize the fast neural activity.This paper is concerned with a class of neutral type CNN’s with mixed delay and D operator.By employing the appropriate differential inequality theory,some sufficient conditions are given to ensure that all solutions of the model converge exponentially to zero vector.Finally,an illustrative example is also given at the end of this paper to show the effectiveness of the proposed results.
基金Supported by the National Natural Science Foundation of China (11001208)the Fundamental Research Funds for the Central Universities
文摘In this paper, we introduce the definition of Γ-Fisher entropy. For the stochastic differential equation in Rd and the unique invariant probability measure μ, we obtain the exponential convergence of Γ-Fisher entropy with respect to μ under some condition about Γ and the drift b. Moreover, we revisit the exponential convergence of the usual Fisher entropy under the Bakry Emery condition.
文摘Let (Xt)t≥0 be a Lévy process taking values in R^d with absolutely continuous marginal distributions. Given a real measurable function f on R^d in Kato's class, we show that the empirical mean 1/t ∫ f(Xs)ds converges to a constant z in probability with an exponential rate if and only if f has a uniform mean z. This result improves a classical result of Kahane et al. and generalizes a similar result of L. Wu from the Brownian Motion to general Lévy processes.
基金supported by the National Natural Science Foundation of China(11671011).
文摘For a collective system,the connectedness of the adjacency matrix plays a key role in making the system achieve its emergent feature,such as flocking or multi-clustering.In this paper,we study a nonsymmetric multi-particle system with a constant and local cut-off weight.A distributed communication delay is also introduced into both the velocity adjoint term and the cut-off weight.As a new observation,we show that the desired multi-particle system undergoes both flocking and clustering behaviors when the eigenvalue 1 of the adjacency matrix is semi-simple.In this case,the adjacency matrix may lose the connectedness.In particular,the number of clusters is discussed by using subspace analysis.In terms of results,for both the non-critical and general neighbourhood situations,some criteria of flocking and clustering emergence with an exponential convergent rate are established by the standard matrix analysis for when the delay is free.As a distributed delay is involved,the corresponding criteria are also found,and these small time lags do not change the emergent properties qualitatively,but alter the final value in a nonlinear way.Consequently,some previous works[14]are extended.
基金Supported by the National Natural Science Foundation of China(11501004,11501005,11526033,11671012)the Natural Science Foundation of Anhui Province(1508085J06,1608085QA02)+1 种基金the Key Projects for Academic Talent of Anhui Province(gxbj ZD2016005)the Research Teaching Model Curriculum of Anhui University(xjyjkc1407)
文摘In this paper, an exponential inequality for the maximal partial sums of negatively superadditive-dependent (NSD, in short) random variables is established. By uSing the exponen- tial inequality, we present some general results on the complete convergence for arrays of rowwise NSD random variables, which improve or generalize the corresponding ones of Wang et al. [28] and Chen et al. [2]. In addition, some sufficient conditions to prove the complete convergence are provided. As an application of the complete convergence that we established, we further investigate the complete consistency and convergence rate of the estimator in a nonparametric regression model based on NSD errors.
基金the Important Research Fund for the National committee of China (No.20040816012)
文摘It is shown that there is an unique ω-period solution x(t, φ^*) for a delayed cellular network and its each solution x(t, φ) converges exponentially to x(t, φ^*) if its each output function is bounded and satisfies Lipschitz condition when all input signals are at-periodic functions.
基金The research of HB was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada and by the Research Grants Council of Hong KongThe research of TT was supported by Hong Kong Baptist University,the Research Grants Council of Hong Kong and he was supported in part by the Chinese Academy of Sciences while visiting its Institute of Computational Mathematics.
文摘We propose a novel numerical approach for delay differential equations with vanishing proportional delays based on spectral methods. A Legendre-collocation method is employed to obtain highly accurate numerical approximations to the exact solution. It is proved theoretically and demonstrated numerically that the proposed method converges exponentially provided that the data in the are smooth. given pantograph delay differential equation
基金This work was partially supported by the National Natural Science Foundation of China under Grant 61903027,72171171,62003239Shanghai Municipal Science and Technology Major Project under Grant 2021SHZDZX0100Shanghai Sailing Program under Grant 20YF1453000.
文摘Distributed Nash equilibrium seeking of aggregative games is investigated and a continuous-time algorithm is proposed.The algorithm is designed by virtue of projected gradient play dynamics and aggregation tracking dynamics,and is applicable to games with constrained strategy sets and weight-balanced communication graphs.The key feature of our method is that the proposed projected dynamics achieves exponential convergence,whereas such convergence results are only obtained for non-projected dynamics in existing works on distributed optimization and equilibrium seeking.Numerical examples illustrate the effectiveness of our methods.
基金Supported by the National Natural Science Foundation of China (No. 10826047 and No.10901023)by the Fundamental Research Funds for the Central Universities under Contract BUPT2009RC0707
文摘A GI/G/1 queue with vacations is considered in this paper. We develop an approximating technique on max function of independent and identically distributed (i.i.d.) random variables, that is max{ηi, 1 ≤ i ≤ n}. The approximating technique is used to obtain the fluid approximation for the queue length, workload and busy time processes. Furthermore, under uniform topology, if the scaled arrival process and the scaled service process converge to the corresponding fluid processes with an exponential rate, we prove by the approximating technique that the scaled processes characterizing the queue converge to the corresponding fluid limits with the exponential rate only for large N. Here the scaled processes include the queue length process, workload process and busy time process.
文摘We study an explicit exponential scheme for the time discretisation of stochastic SchrS- dinger Equations Driven by additive or Multiplicative It6 Noise. The numerical scheme is shown to converge with strong order 1 if the noise is additive and with strong order 1/2 for multiplicative noise. In addition, if the noise is additive, we show that the exact solutions of the linear stochastic Sehr6dinger equations satisfy trace formulas for the expected mass, energy, and momentum (i. e., linear drifts in these quantities). Furthermore, we inspect the behaviour of the numerical solutions with respect to these trace formulas. Several numerical simulations are presented and confirm our theoretical results.
基金supported by the National Natural Science Foundation of China (Grant Nos. 91748112,61991403,61991404,and 61991400)。
文摘In this paper, we consider the distributed optimization problem, where the goal is to minimize the global objective function formed by a sum of agents' local smooth and strongly convex objective functions, over undirected connected graphs. Several distributed accelerated algorithms have been proposed for solving such a problem in the existing literature. In this paper, we provide insights for understanding these existing distributed algorithms from an ordinary differential equation(ODE) point of view. More specifically, we first derive an equivalent second-order ODE, which is the exact limit of these existing algorithms by taking the small step-size. Moreover, focusing on the quadratic objective functions, we show that the solution of the resulting ODE exponentially converges to the unique global optimal solution. The theoretical results are validated and illustrated by numerical simulations.
基金The research of C.Zhang is partially supported by NSFC(Grant Nos.11971207,12071172)the Natural Science Foundation of the Jiangsu Higher Education Institutions of China(Grant No.20KJA11002)The research of S.Chen is partially supported by NSFC(Grant No.11801235).
文摘In this paper,we propose a hybrid spectral method for a type of nonlocal problems,nonlinear Volterra integral equations(VIEs)of the second kind.The main idea is to use the shifted generalized Log orthogonal functions(GLOFs)as the basis for the first interval and employ the classical shifted Legendre polynomials for other subintervals.This method is robust for VIEs with weakly singular kernel due to the GLOFs can efficiently approximate one-point singular functions as well as smooth functions.The well-posedness and the related error estimates will be provided.Abundant numerical experiments will verify the theoretical results and show the high-efficiency of the new hybrid spectral method.
文摘We perform the analysis of the hp finite element approximation for the solution to singularly perturbed transmission problems, using Spectral Boundary Layer Meshes. In [12] it was shown that this method yields robust exponential convergence, as the degree p of the approximating polynomials is increased, when the error is measured in the energy norm associated with the boundary value problem. In the present article we sharpen the result by showing that the hp-Finite Element Method (FEM) on Spectral Boundary Layer Meshes leads to robust exponential convergence in a stronger, more balanced norm. Several numerical results illustrating and extending the theory are also nresented.
基金The work for this paper was supported by the National Natural Science Foundation of China under Projects No.41074099。
文摘A novel,highly efficient and accurate adaptive higher-order finite element method(hp-FEM)is used to simulate a multi-frequency resistivity loggingwhile-drilling(LWD)tool response in a borehole environment.Presented in this study are the vector expression of Maxwell’s equations,three kinds of boundary conditions,stability weak formulation of Maxwell’s equations,and automatic hpadaptivity strategy.The new hp-FEM can select optimal refinement and calculation strategies based on the practical formation model and error estimation.Numerical experiments show that the new hp-FEM has an exponential convergence rate in terms of relative error in a user-prescribed quantity of interest against the degrees of freedom,which provides more accurate results than those obtained using the adaptive h-FEM.The numerical results illustrate the high efficiency and accuracy of the method at a given LWD tool structure and parameters in different physical models,which further confirm the accuracy of the results using the Hermes library(http://hpfem.org/hermes)with a multi-frequency resistivity LWD tool response in a borehole environment.
基金This work was partially supported by the National Natural Science Foundation of China under Grants 61973043,62003239,and 61703368Shanghai Sailing Program under Grant 20YF1453000+1 种基金Shanghai Municipal Science and Technology Major Project No.2021SHZDZX0100Shanghai Municipal Commission of Science and Technology Project No.19511132101.
文摘In this paper,we aim to develop distributed continuous-time algorithms over directed graphs to seek the Nash equilibrium in a noncooperative game.Motivated by the recent consensus-based designs,we present a distributed algorithm with a proportional gain for weight-balanced directed graphs.By further embedding a distributed estimator of the left eigenvector associated with zero eigenvalue of the graph Laplacian,we extend it to the case with arbitrary strongly connected directed graphs having possible unbalanced weights.In both cases,the Nash equilibrium is proven to be exactly reached with an exponential convergence rate.An example is given to illustrate the validity of the theoretical results.
基金supported by the Natural Science Foundation of China under Grant Nos.61374090,and 61473171the Program for Scientific Research Innovation Team in Colleges and Universities of Shandong Provincethe Taishan Scholarship Project of Shandong Province
文摘This paper is concerned with control and optimization for a sampled-data system with quantization and actuator saturation. Based quantization and actuator saturation a controller is introduced. The corresponding closed loop system is transformed into a system with input saturation and bounded external disturbance. A new Lyapunov functional is constructed to derive a sample-interval dependent condition on the existence of a state feedback controller such that the closed-loop system is exponentially convergent to an ultimate ellipsoid for the initial condition starting from some initial ellipsoid. Based on the condition, the desired controller is designed. Furthermore, optimization problems about the sample-interval, the ultimate ellipsoid and the initial ellipsoid are formulated. An example is given to illustrate the effectiveness of the proposed method.