This paper considers the rational expectations model with multiplicative noise and input delay,where the system dynamics rely on the conditional expectations of future states.The main contribution is to obtain a suffi...This paper considers the rational expectations model with multiplicative noise and input delay,where the system dynamics rely on the conditional expectations of future states.The main contribution is to obtain a sufficient condition for the exact controllability of the rational expectations model.In particular,we derive a sufficient Gramian matrix condition and a rank condition for the delay-free case.The key is the solvability of the backward stochastic difference equations with input delay which is derived from the forward and backward stochastic system.展开更多
We consider first order quasilinear hyperbolic systems with vertical characteristics. It was shown in [4] that such systems can be exactly controllable with the help of internal controls applied to the equations corr...We consider first order quasilinear hyperbolic systems with vertical characteristics. It was shown in [4] that such systems can be exactly controllable with the help of internal controls applied to the equations corresponding to zero eigenvalues. However, it is possible that, for physical or engineering reasons, we can not put any control on the equations corresponding to zero eigenvalues. In this paper, we will establish the exact controllability only by means of physically meaningfnl internal controls applied to the equations corresponding to non-zero eigenvalues. We also show the exact controllability for a very simplified model by means of switching controls.展开更多
In this paper a new time explicit asymptotic method is presented through introducing an auxiliary parameter for the solution of an exact controllability problem of scattering waves. Based on an exact control function...In this paper a new time explicit asymptotic method is presented through introducing an auxiliary parameter for the solution of an exact controllability problem of scattering waves. Based on an exact control function, the asymptotic iteration is controlled by the auxiliary parameter and the optimal auxiliary parameter is updated during the iteration based on the existing or current iterated solutions of the wave equation. The numerical results show that the new method presented has a significant advantage over the purely asymptotic method in the history of convergence and has the ability to solve the scattering by the multi bodies.展开更多
Necessary and sufficient conditions for the exact controllability and exact observability of a descriptor infinite dimensional system are obtained in the sense of distributional solution.These general results are used...Necessary and sufficient conditions for the exact controllability and exact observability of a descriptor infinite dimensional system are obtained in the sense of distributional solution.These general results are used to examine the exact controllability and exact observability of the Dzektser equation in the theory of seepage and the exact controllability of wave equation.展开更多
In this paper,the authors mainly consider the exact controllability for degenerate wave equation,which degenerates at the interior point,and boundary controls acting at only one of the boundary points.The main results...In this paper,the authors mainly consider the exact controllability for degenerate wave equation,which degenerates at the interior point,and boundary controls acting at only one of the boundary points.The main results are that,it is possible to control both the position and the velocity at every point of the body and at a certain time T for the wave equation with interior weakly degeneracy.Moreover,it is shown that the exact controllability fails for the wave equation with interior strongly degeneracy.In order to steer the system to a certain state,one needs controls to act on both boundary points for the wave equation with interior strongly degeneracy.The difficulties are addressed by means of spectral analysis.展开更多
Exact controllability of singular distributed parameter control system is discussed via functional analysis and the theory of generalized operator semi-group in Hilbert space, Necessary and sufficient conditions conce...Exact controllability of singular distributed parameter control system is discussed via functional analysis and the theory of generalized operator semi-group in Hilbert space, Necessary and sufficient conditions concerning the exact controllability are given. Relations between exact controllability and stability of singular distributed parameter system are specified.展开更多
By means of the existence and uniqueness of semi-global C1 solution to the mixed initial-boundary value problem with general nonlinear boundary conditions for first order quasilinear hyperbolic systems with zero eigen...By means of the existence and uniqueness of semi-global C1 solution to the mixed initial-boundary value problem with general nonlinear boundary conditions for first order quasilinear hyperbolic systems with zero eigenvalues,the local exact boundary controllability for higher order quasilinear hyperbolic equations is established.展开更多
In this article, by using theory of linear evolution system and Schauder fixed point theorem, we establish a sufficient result of exact null controllability for a non-autonomous functional evolution system with nonloc...In this article, by using theory of linear evolution system and Schauder fixed point theorem, we establish a sufficient result of exact null controllability for a non-autonomous functional evolution system with nonlocal conditions. In particular, the compactness condition or Lipschitz condition for the function g in the nonlocal conditions appearing in various literatures is not required here. An example is also provided to show an application of the obtained result.展开更多
In this paper, the boundary control problem of a distributed parameter system described by the Schrodinger equation posed on finite interval α≤x≤β:{ iyt +yzz+|y|^2y = 0, y(α, t) = h1 (t), y(β, t)=h2...In this paper, the boundary control problem of a distributed parameter system described by the Schrodinger equation posed on finite interval α≤x≤β:{ iyt +yzz+|y|^2y = 0, y(α, t) = h1 (t), y(β, t)=h2(t) for t〉0 (S)is considered. It is shown that by choosing appropriate control inputs (hi), (j = 1, 2) one can always guide the system (S) from a given initial state φ∈H^S(α,β), (s ∈ R) to a terminal state φ∈ H^s(α,β), in the time period [0, T]. The exact boundary controllability is obtained by considering a related initial value control problem of SchrSdinger equation posed on the whole line R. The discovered smoothing properties of Schrodinger equation have played important roles in our approach; this may be the first step to prove the results on boundary controllability of (semi-linear) nonlinear Schrodinger equation.展开更多
This paper deals with the problem of approximate controllability of infinite dimensional linear systems in nonreflexive state spaces. A necessary and sufficient condition for approximate controllability via L^p([0, T...This paper deals with the problem of approximate controllability of infinite dimensional linear systems in nonreflexive state spaces. A necessary and sufficient condition for approximate controllability via L^p([0, T], U), 1≤p〈∞ is obtained,where L^p( [0, T], U) is the control function space.展开更多
In this paper the exact internal controllability for a coupled system of wave equations with arbitrarily given coupling matrix is established.Based on this result,the exact internal synchronization and the exact inter...In this paper the exact internal controllability for a coupled system of wave equations with arbitrarily given coupling matrix is established.Based on this result,the exact internal synchronization and the exact internal synchronization by p-groups are successfully considered.展开更多
In this paper,we propose a second-order quasilinear hyperbolic system.By means of the theory on semi-global C^(1)solution to first-order quasilinear hyperbolic systems,we establish the existence and uniqueness of semi...In this paper,we propose a second-order quasilinear hyperbolic system.By means of the theory on semi-global C^(1)solution to first-order quasilinear hyperbolic systems,we establish the existence and uniqueness of semi-global C^(2)solution to this second-order quasilinear hyperbolic system.After then,the general constructive framework is utilized to obtain the local exact boundary controllability for this second-order system.展开更多
We analyze the exponential decay property of solutions of the semilinear wave equation in bounded domain Ω of R^N with a damping term which is effective on the exterior of a ball and boundary conditions of the Cauchy...We analyze the exponential decay property of solutions of the semilinear wave equation in bounded domain Ω of R^N with a damping term which is effective on the exterior of a ball and boundary conditions of the Cauchy-Ventcel type. Under suitable and natural assumptions on the nonlinearity, we prove that the exponential decay holds locally uniformly for finite energy solutions provided the nonlinearity is subcritical at infinity. Subcriticality means, roughly speaking, that the nonlinearity grows at infinity at most as a power p 〈 5. The results obtained in R^3 and RN by B. Dehman, G. Lebeau and E. Zuazua on the inequalities of the classical energy (which estimate the total energy of solutions in terms of the energy localized in the exterior of a ball) and on Strichartz's estimates, allow us to give an application to the stabilization controllability of the semilinear wave equation in a bounded domain of R^N with a subcritical nonlinearity on the domain and its boundary, and conditions on the boundary of Cauchy-Ventcel type.展开更多
In this paper,we show by Hilbert Uniqueness Method that the boundary value problem of fifth-order KdV equation{y_(t)-y_(5x)=0,(x,t)∈(0,2π)×(0,T),y(t,2π)-y(t,0)=h_(0)(t),y_(x)(t,2π)-y_(x)(t,0)=h_(1)(t),y_(2x)(...In this paper,we show by Hilbert Uniqueness Method that the boundary value problem of fifth-order KdV equation{y_(t)-y_(5x)=0,(x,t)∈(0,2π)×(0,T),y(t,2π)-y(t,0)=h_(0)(t),y_(x)(t,2π)-y_(x)(t,0)=h_(1)(t),y_(2x)(t,2π)-y_(2x)(t,0)=h_(2)(t),y_(3x)(t,2π)-y_(3x)(t,0)=h_(3)(t),y_(4x)(t,2π)-y_(4x)(t,0)=h_(4)(t),(with boundary data as control inputs)is exact controllability.展开更多
The internal control problem is considered, based on the linear displacement equations of shallow shell. It is shown, with some checkable geometric conditions on control region, that the undergoing shallow shell is ex...The internal control problem is considered, based on the linear displacement equations of shallow shell. It is shown, with some checkable geometric conditions on control region, that the undergoing shallow shell is exactly controllable by using Hilbert uniqueness method (HUM), piecewise multiplier method and Riemannian geometry method. Then some examples are given to show the assumed geometric conditions.展开更多
In this paper the authors first present the definition and some properties of weak solutions to 1-D first order linear hyperbolic systems. Then they show that the constructive method with modular structure originally ...In this paper the authors first present the definition and some properties of weak solutions to 1-D first order linear hyperbolic systems. Then they show that the constructive method with modular structure originally given in the framework of classical solutions is still very powerful and effective in the framework of weak solutions to prove the exact boundary(null) controllability and the exact boundary observability for first order hyperbolic systems.展开更多
The exact boundary controllability and the exact boundary observability for the 1-D first order linear hyperbolic system were studied by the constructive method in the framework of weak solutions in the work[Lu,X.and ...The exact boundary controllability and the exact boundary observability for the 1-D first order linear hyperbolic system were studied by the constructive method in the framework of weak solutions in the work[Lu,X.and Li,T.T.,Exact boundary con-trollability of weak solutions for a kind of first order hyperbolic system—the constructive method,Chin.Ann.Math.Ser.B,42(5),2021,643-676].In this paper,in order to study these problems from the viewpoint of duality,the authors establish a complete the-ory on the HUM method and give its applications to first order hyperbolic systems.Thus,a deeper relationship between the controllability and the observability can be revealed.Moreover,at the end of the paper,a comparison will be made between these two methods.展开更多
This paper deals with the spatial vibration of an elastic string with masses at the endpoints. The authors derive the corresponding quasilinear wave equation with dynamical boundary conditions, and prove the exact bou...This paper deals with the spatial vibration of an elastic string with masses at the endpoints. The authors derive the corresponding quasilinear wave equation with dynamical boundary conditions, and prove the exact boundary controllability of this system by means of a constructive method with modular structure.展开更多
This paper deals with the exact boundary controllability and the exact boundary synchronization for a 1-D system of wave equations coupled with velocities.These problems can not be solved directly by the usual HUM met...This paper deals with the exact boundary controllability and the exact boundary synchronization for a 1-D system of wave equations coupled with velocities.These problems can not be solved directly by the usual HUM method for wave equations,however,by transforming the system into a first order hyperbolic system,the HUM method for 1-D first order hyperbolic systems,established by Li-Lu(2022)and Lu-Li(2022),can be applied to get the corresponding results.展开更多
In this paper,an abstract multi-term time-fractional differential system is considered and the exact controllability results are investigated.In this theory,we tend to implement the basic tools of fractional calculus ...In this paper,an abstract multi-term time-fractional differential system is considered and the exact controllability results are investigated.In this theory,we tend to implement the basic tools of fractional calculus and measure of noncompactness to come up with a new set of sufficient conditions for the exact controllability by utilisation of Mönch fixed point theorem.Finally,an application is given to illustrate the obtained results.展开更多
基金supported by the National Natural Science Foundation of China under Grants 61821004,62250056,62350710214,U23A20325,62350055the Natural Science Foundation of Shandong Province,China(ZR2021ZD14,ZR2021JQ24)+2 种基金High-level Talent Team Project of Qingdao West Coast New Area,China(RCTD-JC-2019-05)Key Research and Development Program of Shandong Province,China(2020CXGC01208)Science and Technology Project of Qingdao West Coast New Area,China(2019-32,2020-20,2020-1-4).
文摘This paper considers the rational expectations model with multiplicative noise and input delay,where the system dynamics rely on the conditional expectations of future states.The main contribution is to obtain a sufficient condition for the exact controllability of the rational expectations model.In particular,we derive a sufficient Gramian matrix condition and a rank condition for the delay-free case.The key is the solvability of the backward stochastic difference equations with input delay which is derived from the forward and backward stochastic system.
文摘We consider first order quasilinear hyperbolic systems with vertical characteristics. It was shown in [4] that such systems can be exactly controllable with the help of internal controls applied to the equations corresponding to zero eigenvalues. However, it is possible that, for physical or engineering reasons, we can not put any control on the equations corresponding to zero eigenvalues. In this paper, we will establish the exact controllability only by means of physically meaningfnl internal controls applied to the equations corresponding to non-zero eigenvalues. We also show the exact controllability for a very simplified model by means of switching controls.
文摘In this paper a new time explicit asymptotic method is presented through introducing an auxiliary parameter for the solution of an exact controllability problem of scattering waves. Based on an exact control function, the asymptotic iteration is controlled by the auxiliary parameter and the optimal auxiliary parameter is updated during the iteration based on the existing or current iterated solutions of the wave equation. The numerical results show that the new method presented has a significant advantage over the purely asymptotic method in the history of convergence and has the ability to solve the scattering by the multi bodies.
基金This work was supported by the National Natural Science Foundation of China(11926402,61973338).
文摘Necessary and sufficient conditions for the exact controllability and exact observability of a descriptor infinite dimensional system are obtained in the sense of distributional solution.These general results are used to examine the exact controllability and exact observability of the Dzektser equation in the theory of seepage and the exact controllability of wave equation.
基金supported by the National Natural Science Foundation of China under Grant No.12271316the National Natural Science Foundation of China for the Youth under Grant No.11801339+1 种基金Shanxi Sciences Project for Selected Overseas Scholars under Grant No.2018–172the Technical Innovation Team of Jinzhong University under Grant No.202111。
文摘In this paper,the authors mainly consider the exact controllability for degenerate wave equation,which degenerates at the interior point,and boundary controls acting at only one of the boundary points.The main results are that,it is possible to control both the position and the velocity at every point of the body and at a certain time T for the wave equation with interior weakly degeneracy.Moreover,it is shown that the exact controllability fails for the wave equation with interior strongly degeneracy.In order to steer the system to a certain state,one needs controls to act on both boundary points for the wave equation with interior strongly degeneracy.The difficulties are addressed by means of spectral analysis.
基金Supported by the National Natural Science Foundation of China (Grant No. 60674018)
文摘Exact controllability of singular distributed parameter control system is discussed via functional analysis and the theory of generalized operator semi-group in Hilbert space, Necessary and sufficient conditions concerning the exact controllability are given. Relations between exact controllability and stability of singular distributed parameter system are specified.
文摘By means of the existence and uniqueness of semi-global C1 solution to the mixed initial-boundary value problem with general nonlinear boundary conditions for first order quasilinear hyperbolic systems with zero eigenvalues,the local exact boundary controllability for higher order quasilinear hyperbolic equations is established.
基金supported by NSF of China (11171110)Shanghai Leading Academic Discipline Project (B407)
文摘In this article, by using theory of linear evolution system and Schauder fixed point theorem, we establish a sufficient result of exact null controllability for a non-autonomous functional evolution system with nonlocal conditions. In particular, the compactness condition or Lipschitz condition for the function g in the nonlocal conditions appearing in various literatures is not required here. An example is also provided to show an application of the obtained result.
文摘In this paper, the boundary control problem of a distributed parameter system described by the Schrodinger equation posed on finite interval α≤x≤β:{ iyt +yzz+|y|^2y = 0, y(α, t) = h1 (t), y(β, t)=h2(t) for t〉0 (S)is considered. It is shown that by choosing appropriate control inputs (hi), (j = 1, 2) one can always guide the system (S) from a given initial state φ∈H^S(α,β), (s ∈ R) to a terminal state φ∈ H^s(α,β), in the time period [0, T]. The exact boundary controllability is obtained by considering a related initial value control problem of SchrSdinger equation posed on the whole line R. The discovered smoothing properties of Schrodinger equation have played important roles in our approach; this may be the first step to prove the results on boundary controllability of (semi-linear) nonlinear Schrodinger equation.
文摘This paper deals with the problem of approximate controllability of infinite dimensional linear systems in nonreflexive state spaces. A necessary and sufficient condition for approximate controllability via L^p([0, T], U), 1≤p〈∞ is obtained,where L^p( [0, T], U) is the control function space.
基金supported by the National Natural Science Foundation of China(No.11831011)。
文摘In this paper the exact internal controllability for a coupled system of wave equations with arbitrarily given coupling matrix is established.Based on this result,the exact internal synchronization and the exact internal synchronization by p-groups are successfully considered.
基金Supported by the Science and Technology Commission of Shanghai Municipality (Grant No.23ZR1402100)the Fundamental Research Funds for the Central Universities (Grant Nos. 2232022G-13 and 2232023G-13)
文摘In this paper,we propose a second-order quasilinear hyperbolic system.By means of the theory on semi-global C^(1)solution to first-order quasilinear hyperbolic systems,we establish the existence and uniqueness of semi-global C^(2)solution to this second-order quasilinear hyperbolic system.After then,the general constructive framework is utilized to obtain the local exact boundary controllability for this second-order system.
文摘We analyze the exponential decay property of solutions of the semilinear wave equation in bounded domain Ω of R^N with a damping term which is effective on the exterior of a ball and boundary conditions of the Cauchy-Ventcel type. Under suitable and natural assumptions on the nonlinearity, we prove that the exponential decay holds locally uniformly for finite energy solutions provided the nonlinearity is subcritical at infinity. Subcriticality means, roughly speaking, that the nonlinearity grows at infinity at most as a power p 〈 5. The results obtained in R^3 and RN by B. Dehman, G. Lebeau and E. Zuazua on the inequalities of the classical energy (which estimate the total energy of solutions in terms of the energy localized in the exterior of a ball) and on Strichartz's estimates, allow us to give an application to the stabilization controllability of the semilinear wave equation in a bounded domain of R^N with a subcritical nonlinearity on the domain and its boundary, and conditions on the boundary of Cauchy-Ventcel type.
基金supported by the Zhejiang Provincial Natural Science Foundation of China(No.LY18A010024)National Natural Science Foundation of China(No.12075208).
文摘In this paper,we show by Hilbert Uniqueness Method that the boundary value problem of fifth-order KdV equation{y_(t)-y_(5x)=0,(x,t)∈(0,2π)×(0,T),y(t,2π)-y(t,0)=h_(0)(t),y_(x)(t,2π)-y_(x)(t,0)=h_(1)(t),y_(2x)(t,2π)-y_(2x)(t,0)=h_(2)(t),y_(3x)(t,2π)-y_(3x)(t,0)=h_(3)(t),y_(4x)(t,2π)-y_(4x)(t,0)=h_(4)(t),(with boundary data as control inputs)is exact controllability.
基金supported by the National Natural Science Foundation of China(Grant Nos.60334040,60225003,10501044).
文摘The internal control problem is considered, based on the linear displacement equations of shallow shell. It is shown, with some checkable geometric conditions on control region, that the undergoing shallow shell is exactly controllable by using Hilbert uniqueness method (HUM), piecewise multiplier method and Riemannian geometry method. Then some examples are given to show the assumed geometric conditions.
基金supported by the National Natural Science Foundation of China(Nos.11831011,11901082)the Natural Science Foundation of Jiangsu Province(No.BK20190323)the Fundamental Research Funds for the Central Universities of China。
文摘In this paper the authors first present the definition and some properties of weak solutions to 1-D first order linear hyperbolic systems. Then they show that the constructive method with modular structure originally given in the framework of classical solutions is still very powerful and effective in the framework of weak solutions to prove the exact boundary(null) controllability and the exact boundary observability for first order hyperbolic systems.
基金This work was supported by the National Natural Science Foundation of China(Nos.11831011,11901082)the Natural Science Foundation of Jiangsu Province(No.BK20190323)the Fundamental Research Funds for the Central Universities of China.
文摘The exact boundary controllability and the exact boundary observability for the 1-D first order linear hyperbolic system were studied by the constructive method in the framework of weak solutions in the work[Lu,X.and Li,T.T.,Exact boundary con-trollability of weak solutions for a kind of first order hyperbolic system—the constructive method,Chin.Ann.Math.Ser.B,42(5),2021,643-676].In this paper,in order to study these problems from the viewpoint of duality,the authors establish a complete the-ory on the HUM method and give its applications to first order hyperbolic systems.Thus,a deeper relationship between the controllability and the observability can be revealed.Moreover,at the end of the paper,a comparison will be made between these two methods.
基金supported by the National Natural Science Foundation of China(No.11831011).
文摘This paper deals with the spatial vibration of an elastic string with masses at the endpoints. The authors derive the corresponding quasilinear wave equation with dynamical boundary conditions, and prove the exact boundary controllability of this system by means of a constructive method with modular structure.
基金supported by the National Natural Science Foundation of China (Nos. 11831011,11901082)the Natural Science Foundation of Jiangsu Province (No. BK20190323)the Fundamental Research Funds for the Central Universities of China
文摘This paper deals with the exact boundary controllability and the exact boundary synchronization for a 1-D system of wave equations coupled with velocities.These problems can not be solved directly by the usual HUM method for wave equations,however,by transforming the system into a first order hyperbolic system,the HUM method for 1-D first order hyperbolic systems,established by Li-Lu(2022)and Lu-Li(2022),can be applied to get the corresponding results.
基金The work of the first author(Vikram Singh)is supported by the Ministry of Human Resource Development,India under[grant number:MHR-02-23-200-44]’.
文摘In this paper,an abstract multi-term time-fractional differential system is considered and the exact controllability results are investigated.In this theory,we tend to implement the basic tools of fractional calculus and measure of noncompactness to come up with a new set of sufficient conditions for the exact controllability by utilisation of Mönch fixed point theorem.Finally,an application is given to illustrate the obtained results.