The leaderless and leader-following finite-time consensus problems for multiagent systems(MASs)described by first-order linear hyperbolic partial differential equations(PDEs)are studied.The Lyapunov theorem and the un...The leaderless and leader-following finite-time consensus problems for multiagent systems(MASs)described by first-order linear hyperbolic partial differential equations(PDEs)are studied.The Lyapunov theorem and the unique solvability result for the first-order linear hyperbolic PDE are used to obtain some sufficient conditions for ensuring the finite-time consensus of the leaderless and leader-following MASs driven by first-order linear hyperbolic PDEs.Finally,two numerical examples are provided to verify the effectiveness of the proposed methods.展开更多
In this paper,by making use of the calculous technique and some results of the impulsive differential inequality,oscillatory properties of the solutions of certain nonlinear impulsive delay hyperbolic partial differen...In this paper,by making use of the calculous technique and some results of the impulsive differential inequality,oscillatory properties of the solutions of certain nonlinear impulsive delay hyperbolic partial differential equations with nonlinear diffusion coefficient are investigated.Sufficient conditions for oscillations of such equations are obtained.展开更多
We extend LeVeque's wave propagation algorithm,a widely used finite volume method for hyperbolic partial differential equations,to a third-order accurate method.The resulting scheme shares main properties with the...We extend LeVeque's wave propagation algorithm,a widely used finite volume method for hyperbolic partial differential equations,to a third-order accurate method.The resulting scheme shares main properties with the original method,i.e.,it is based on a wave decomposition at grid cell interfaces,it can be used to approximate hyperbolic problems in divergence form as well as in quasilinear form and limiting is introduced in the form of a wave limiter.展开更多
Dear Editor,This letter focuses on the distributed cooperative regulation problem for a class of networked re-entrant manufacturing systems(RMSs).The networked system is structured with a three-tier architecture:the p...Dear Editor,This letter focuses on the distributed cooperative regulation problem for a class of networked re-entrant manufacturing systems(RMSs).The networked system is structured with a three-tier architecture:the production line,the manufacturing layer and the workshop layer.The dynamics of re-entrant production lines are governed by hyperbolic partial differential equations(PDEs)based on the law of mass conservation.展开更多
In this paper,we study the pressure of C^(1)-smooth partially hyperbolic diffeomorphisms with sup-additive potentials.We give several definitions of the so called unstable(measure theoretic)pressure in terms of Bowen...In this paper,we study the pressure of C^(1)-smooth partially hyperbolic diffeomorphisms with sup-additive potentials.We give several definitions of the so called unstable(measure theoretic)pressure in terms of Bowen’s picture and the capacity picture.We show that all such unstable metric pressures of a given ergodic measure equals the corresponding unstable measure theoretic entropy plus the Lyapunov exponent of the potentials with respect to the ergodic measure.展开更多
In petroleum engineering, the transport phenomenon of proppants in a fracture caused by hydraulic fracturing is captured by hyperbolic partial differential equations(PDEs). The solution of this kind of PDEs may encoun...In petroleum engineering, the transport phenomenon of proppants in a fracture caused by hydraulic fracturing is captured by hyperbolic partial differential equations(PDEs). The solution of this kind of PDEs may encounter smooth transitions, or there can be large gradients of the field variables. The numerical challenge posed in a shock situation is that high-order finite difference schemes lead to significant oscillations in the vicinity of shocks despite that such schemes result in higher accuracy in smooth regions. On the other hand, first-order methods provide monotonic solution convergences near the shocks,while giving poorer accuracy in the smooth regions.Accurate numerical simulation of such systems is a challenging task using conventional numerical methods. In this paper, we investigate several shock-capturing schemes.The competency of each scheme was tested against onedimensional benchmark problems as well as published numerical experiments. The numerical results have shown good performance of high-resolution finite volume methods in capturing shocks by resolving discontinuities while maintaining accuracy in the smooth regions. Thesemethods along with Godunov splitting are applied to model proppant transport in fractures. It is concluded that the proposed scheme produces non-oscillatory and accurate results in obtaining a solution for proppant transport problems.展开更多
High-order accurate weighted essentially non-oscillatory(WENO)schemes are a class of broadly applied numerical methods for solving hyperbolic partial differential equations(PDEs).Due to highly nonlinear property of th...High-order accurate weighted essentially non-oscillatory(WENO)schemes are a class of broadly applied numerical methods for solving hyperbolic partial differential equations(PDEs).Due to highly nonlinear property of the WENO algorithm,large amount of computational costs are required for solving multidimensional problems.In our previous work(Lu et al.in Pure Appl Math Q 14:57–86,2018;Zhu and Zhang in J Sci Comput 87:44,2021),sparse-grid techniques were applied to the classical finite difference WENO schemes in solving multidimensional hyperbolic equations,and it was shown that significant CPU times were saved,while both accuracy and stability of the classical WENO schemes were maintained for computations on sparse grids.In this technical note,we apply the approach to recently developed finite difference multi-resolution WENO scheme specifically the fifth-order scheme,which has very interesting properties such as its simplicity in linear weights’construction over a classical WENO scheme.Numerical experiments on solving high dimensional hyperbolic equations including Vlasov based kinetic problems are performed to demonstrate that the sparse-grid computations achieve large savings of CPU times,and at the same time preserve comparable accuracy and resolution with those on corresponding regular single grids.展开更多
We first define a kind of new function space, called the space of twice conormal distributions. With some estimates on these function spaces, we can prove that if there exist two characteristic hypersurfaces bearing s...We first define a kind of new function space, called the space of twice conormal distributions. With some estimates on these function spaces, we can prove that if there exist two characteristic hypersurfaces bearing strong and weak singularities respectively intersect transversally, then some new singularities will take place anti their strength will be no more than the original weak one.展开更多
By using fluid dynamics theory with the effects of adsorption and reaction, the chromatography model with a reaction A →B was established as a system of two hyperbolic partial differential equations (PDE’s)....By using fluid dynamics theory with the effects of adsorption and reaction, the chromatography model with a reaction A →B was established as a system of two hyperbolic partial differential equations (PDE’s). In some practical situations, the reaction chromatography model was simplified a semi-coupled system of two linear hyperbolic PDE’s. In which, the reactant concentration wave model was the initial-boundary value problem of a self-closed hyperbolic PDE, while the resultant concentration wave model was the initial-boundary value problem of hyperbolic PDE coupling reactant concentration. The general explicit expressions for the concentration wave of the reactants and resultants were derived by Laplace transform. The δ-pulse and wide pulse injections were taken as the examples to discuss detailedly, and then the stability analysis between the resultant solutions of the two modes of pulse injection was further discussed. It was significant for further analysis of chromatography, optimizing chromatographic separation, determining the physical and chemical characters.展开更多
In this paper,we study transitive partially hyperbolic di eomorphisms with one-dimensional topologically neutral center,meaning that the length of the iterate of small center segments remains small.Such systems are dy...In this paper,we study transitive partially hyperbolic di eomorphisms with one-dimensional topologically neutral center,meaning that the length of the iterate of small center segments remains small.Such systems are dynamically coherent.We show that there exists a continuous metric along the center foliation which is invariant under the dynamics.As an application,we classify the transitive partially hyperbolic di eomorphisms on 3-manifolds with topologically neutral center.展开更多
Abstract In this paper we investigate the quasi-shadowing property for C1 random dynamical sys-terns on their random partially hyperbolic sets. It is shown that for any pseudo orbit {xk}＋∞ -∞ on a random partially ...Abstract In this paper we investigate the quasi-shadowing property for C1 random dynamical sys-terns on their random partially hyperbolic sets. It is shown that for any pseudo orbit {xk}＋∞ -∞ on a random partially hyperbolic set there exists a ＂center＂ pseudo orbit {Yk}＋∞ -∞ shadowing it in the sense that yk＋l is obtained from the image of yk by a motion along the center direction. Moreover, when the random partially hyperbolic set has a local product structure, the above ＂center＂ pseudo orbit{yk}＋∞ -∞ can be chosen such that yk＋1 and the image of yk lie in their common center leaf.展开更多
The author considers the Cauchy problem for quasilinear inhomogeneous hyperbolic systems.Under the assumption that the system is weakly dissipative,Hanouzet and Natalini established the global existence of smooth solu...The author considers the Cauchy problem for quasilinear inhomogeneous hyperbolic systems.Under the assumption that the system is weakly dissipative,Hanouzet and Natalini established the global existence of smooth solutions for small initial data(in Arch.Rational Mech.Anal.,Vol.169,2003,pp.89-117).The aim of this paper is to give a completely different proof of this result with slightly different assumptions.展开更多
In this paper, the robustness of the orbit structure is investigated for a partially hyperbolic endomorphism f on a compact manifold M. It is first proved that the dynamical structure of its orbit space （the inverse ...In this paper, the robustness of the orbit structure is investigated for a partially hyperbolic endomorphism f on a compact manifold M. It is first proved that the dynamical structure of its orbit space （the inverse limit space） M^f of f is topologically quasi-stable under C^0-small perturbations in the following sense： For any covering endomorphism g C^0-close to f, there is a continuous map φ from M^9 to Π-∞^∞ M such that for any {yi}i∈z∈φ（M^9）, yi＋1 and f（yi） differ only by a motion along the center direction. It is then proved that f has quasi-shadowing property in the following sense： For any pseudo-orbit {xi}i∈z, there is a sequence of points {yi}i∈z tracing it, in which yi＋1 is obtained from f（yi） by a motion along the center direction.展开更多
Katok’s entropy formula is an important formula in entropy theory.It plays significant roles in large deviation theories,multifractal analysis,quantitative recurrence and so on.This paper is devoted to establishing K...Katok’s entropy formula is an important formula in entropy theory.It plays significant roles in large deviation theories,multifractal analysis,quantitative recurrence and so on.This paper is devoted to establishing Katok’s entropy formula of unstable metric entropy which is the entropy caused by the unstable part of partially hyperbolic systems.We also construct a similar formula which can be used to study the quantitative recurrence in the unstable manifold for partially hyperbolic diffeomorphisms.展开更多
Let f ： M → M be a partially hyperbolic diffeomorphism on a closed Riemannian manifold with uniformly compact center foliation. We show that if the center foliation of f is of dimension one then the topological entr...Let f ： M → M be a partially hyperbolic diffeomorphism on a closed Riemannian manifold with uniformly compact center foliation. We show that if the center foliation of f is of dimension one then the topological entropy is constant on a small C1 neighborhood of f.展开更多
In this paper,local unstable metric entropy,local unstable topological entropy and local unstable pressure for partially hyperbolic endomorphisms are introduced and investigated.Specially,two variational principles co...In this paper,local unstable metric entropy,local unstable topological entropy and local unstable pressure for partially hyperbolic endomorphisms are introduced and investigated.Specially,two variational principles concerning relationships among the above mentioned numbers are formulated.展开更多
A construction of multiple knot B-spline wavelets has been given in[C.K.Chui and E.Quak,Wavelet on a bounded interval,In:D.Braess and L.L.Schumaker,editors.Numerical methods of approximation theory.Basel:Birkhauser Ve...A construction of multiple knot B-spline wavelets has been given in[C.K.Chui and E.Quak,Wavelet on a bounded interval,In:D.Braess and L.L.Schumaker,editors.Numerical methods of approximation theory.Basel:Birkhauser Verlag;(1992),pp.57–76].In this work,we first modify these wavelets to solve the elliptic(partially)Dirichlet boundary value problems by Galerkin and Petrov Galerkin methods.We generalize this construction to two dimensional case by Tensor product space.In addition,the solution of the system discretized by Galerkin method with modified multiple knot B-spline wavelets is discussed.We also consider a nonlinear partial differential equation for unsteady flows in an open channel called Saint-Venant.Since the solving of this problem by some methods such as finite difference and finite element produce unsuitable approximations specially in the ends of channel,it is solved by multiple knot B-spline wavelet method that yields a very well approximation.Finally,some numerical examples are given to support our theoretical results.展开更多
This paper focuses on linear-quadratic(LQ)optimal control for a class of systems governed by first-order hyperbolic partial differential equations(PDEs).Different from most of the previous works,an approach of discret...This paper focuses on linear-quadratic(LQ)optimal control for a class of systems governed by first-order hyperbolic partial differential equations(PDEs).Different from most of the previous works,an approach of discretization-then-continuousization is proposed in this paper to cope with the infinite-dimensional nature of PDE systems.The contributions of this paper consist of the following aspects:(1)The differential Riccati equations and the solvability condition of the LQ optimal control problems are obtained via the discretization-then-continuousization method.(2)A numerical calculation way of the differential Riccati equations and a practical design way of the optimal controller are proposed.Meanwhile,the relationship between the optimal costate and the optimal state is established by solving a set of forward and backward partial difference equations(FBPDEs).(3)The correctness of the method used in this paper is verified by a complementary continuous method and the comparative analysis with the existing operator results is presented.It is shown that the proposed results not only contain the classic results of the standard LQ control problem of systems governed by ordinary differential equations as a special case,but also support the existing operator results and give a more convenient form of computation.展开更多
Combining difference method and boundary integral equation method,we propose a new numerical method for solving initial-boundary value problem of second order hyperbolic partial differential equations defined on a bou...Combining difference method and boundary integral equation method,we propose a new numerical method for solving initial-boundary value problem of second order hyperbolic partial differential equations defined on a bounded or unbounded domain in R~3 and obtain the error estimates of the approximate solution in energy norm and local maximum norm.展开更多
In this paper, we consider a bound on a general version of the integralinequalities for functions and also study the qualitative behavior of the solutions of certainclasses of the hyperbolic partial delay differential...In this paper, we consider a bound on a general version of the integralinequalities for functions and also study the qualitative behavior of the solutions of certainclasses of the hyperbolic partial delay differential equations under the integral inequalities.展开更多
基金the National Natural Science Foundation of China(Nos.11671282 and 12171339)。
文摘The leaderless and leader-following finite-time consensus problems for multiagent systems(MASs)described by first-order linear hyperbolic partial differential equations(PDEs)are studied.The Lyapunov theorem and the unique solvability result for the first-order linear hyperbolic PDE are used to obtain some sufficient conditions for ensuring the finite-time consensus of the leaderless and leader-following MASs driven by first-order linear hyperbolic PDEs.Finally,two numerical examples are provided to verify the effectiveness of the proposed methods.
基金Supported by the Natural Science Foundation of China(10471086)Supported by the Science Research Foundation of Department of Education of Hunan Province(07C164)
文摘In this paper,by making use of the calculous technique and some results of the impulsive differential inequality,oscillatory properties of the solutions of certain nonlinear impulsive delay hyperbolic partial differential equations with nonlinear diffusion coefficient are investigated.Sufficient conditions for oscillations of such equations are obtained.
基金This work was supported by the DFG through HE 4858/4-1
文摘We extend LeVeque's wave propagation algorithm,a widely used finite volume method for hyperbolic partial differential equations,to a third-order accurate method.The resulting scheme shares main properties with the original method,i.e.,it is based on a wave decomposition at grid cell interfaces,it can be used to approximate hyperbolic problems in divergence form as well as in quasilinear form and limiting is introduced in the form of a wave limiter.
文摘Dear Editor,This letter focuses on the distributed cooperative regulation problem for a class of networked re-entrant manufacturing systems(RMSs).The networked system is structured with a three-tier architecture:the production line,the manufacturing layer and the workshop layer.The dynamics of re-entrant production lines are governed by hyperbolic partial differential equations(PDEs)based on the law of mass conservation.
基金supported by a NSFC grant(11501066)the Department of Education in Chongqing City(KJQN201900724 and KJQN202100722)in Chongqing Jiaotong Universitysupported by the Chongqing Key Laboratory of Analytic Mathematics and Applications。
文摘In this paper,we study the pressure of C^(1)-smooth partially hyperbolic diffeomorphisms with sup-additive potentials.We give several definitions of the so called unstable(measure theoretic)pressure in terms of Bowen’s picture and the capacity picture.We show that all such unstable metric pressures of a given ergodic measure equals the corresponding unstable measure theoretic entropy plus the Lyapunov exponent of the potentials with respect to the ergodic measure.
基金the research funding for this study provided by NSERC through CRDPJ 387606-09
文摘In petroleum engineering, the transport phenomenon of proppants in a fracture caused by hydraulic fracturing is captured by hyperbolic partial differential equations(PDEs). The solution of this kind of PDEs may encounter smooth transitions, or there can be large gradients of the field variables. The numerical challenge posed in a shock situation is that high-order finite difference schemes lead to significant oscillations in the vicinity of shocks despite that such schemes result in higher accuracy in smooth regions. On the other hand, first-order methods provide monotonic solution convergences near the shocks,while giving poorer accuracy in the smooth regions.Accurate numerical simulation of such systems is a challenging task using conventional numerical methods. In this paper, we investigate several shock-capturing schemes.The competency of each scheme was tested against onedimensional benchmark problems as well as published numerical experiments. The numerical results have shown good performance of high-resolution finite volume methods in capturing shocks by resolving discontinuities while maintaining accuracy in the smooth regions. Thesemethods along with Godunov splitting are applied to model proppant transport in fractures. It is concluded that the proposed scheme produces non-oscillatory and accurate results in obtaining a solution for proppant transport problems.
文摘High-order accurate weighted essentially non-oscillatory(WENO)schemes are a class of broadly applied numerical methods for solving hyperbolic partial differential equations(PDEs).Due to highly nonlinear property of the WENO algorithm,large amount of computational costs are required for solving multidimensional problems.In our previous work(Lu et al.in Pure Appl Math Q 14:57–86,2018;Zhu and Zhang in J Sci Comput 87:44,2021),sparse-grid techniques were applied to the classical finite difference WENO schemes in solving multidimensional hyperbolic equations,and it was shown that significant CPU times were saved,while both accuracy and stability of the classical WENO schemes were maintained for computations on sparse grids.In this technical note,we apply the approach to recently developed finite difference multi-resolution WENO scheme specifically the fifth-order scheme,which has very interesting properties such as its simplicity in linear weights’construction over a classical WENO scheme.Numerical experiments on solving high dimensional hyperbolic equations including Vlasov based kinetic problems are performed to demonstrate that the sparse-grid computations achieve large savings of CPU times,and at the same time preserve comparable accuracy and resolution with those on corresponding regular single grids.
文摘We first define a kind of new function space, called the space of twice conormal distributions. With some estimates on these function spaces, we can prove that if there exist two characteristic hypersurfaces bearing strong and weak singularities respectively intersect transversally, then some new singularities will take place anti their strength will be no more than the original weak one.
文摘By using fluid dynamics theory with the effects of adsorption and reaction, the chromatography model with a reaction A →B was established as a system of two hyperbolic partial differential equations (PDE’s). In some practical situations, the reaction chromatography model was simplified a semi-coupled system of two linear hyperbolic PDE’s. In which, the reactant concentration wave model was the initial-boundary value problem of a self-closed hyperbolic PDE, while the resultant concentration wave model was the initial-boundary value problem of hyperbolic PDE coupling reactant concentration. The general explicit expressions for the concentration wave of the reactants and resultants were derived by Laplace transform. The δ-pulse and wide pulse injections were taken as the examples to discuss detailedly, and then the stability analysis between the resultant solutions of the two modes of pulse injection was further discussed. It was significant for further analysis of chromatography, optimizing chromatographic separation, determining the physical and chemical characters.
基金The second author was supported by the European Research Council(Grant No.692925)the starting grant from Beihang University.
文摘In this paper,we study transitive partially hyperbolic di eomorphisms with one-dimensional topologically neutral center,meaning that the length of the iterate of small center segments remains small.Such systems are dynamically coherent.We show that there exists a continuous metric along the center foliation which is invariant under the dynamics.As an application,we classify the transitive partially hyperbolic di eomorphisms on 3-manifolds with topologically neutral center.
基金supported by NSFC(Grant Nos.11371120 and 11771118)supported by Fundamental Research Funds for the Central University,China(Grant No.20720170004)
文摘Abstract In this paper we investigate the quasi-shadowing property for C1 random dynamical sys-terns on their random partially hyperbolic sets. It is shown that for any pseudo orbit {xk}＋∞ -∞ on a random partially hyperbolic set there exists a ＂center＂ pseudo orbit {Yk}＋∞ -∞ shadowing it in the sense that yk＋l is obtained from the image of yk by a motion along the center direction. Moreover, when the random partially hyperbolic set has a local product structure, the above ＂center＂ pseudo orbit{yk}＋∞ -∞ can be chosen such that yk＋1 and the image of yk lie in their common center leaf.
基金Project supported by the National Natural Science Foundation of China(No.10728101)the Basic Research Program of China(No.2007CB814800)+1 种基金the Doctoral Program Foundation of the Ministry of Education of Chinathe"111"Project(No.B08018)and SGST(No.09DZ2272900)
文摘The author considers the Cauchy problem for quasilinear inhomogeneous hyperbolic systems.Under the assumption that the system is weakly dissipative,Hanouzet and Natalini established the global existence of smooth solutions for small initial data(in Arch.Rational Mech.Anal.,Vol.169,2003,pp.89-117).The aim of this paper is to give a completely different proof of this result with slightly different assumptions.
基金supported by the National Natural Science Foundation of China(No.11371120)the High-level Personnel for Institutions of Higher Learning in Hebei Province(No.GCC2014052)the Natural Science Foundation of Hebei Province(No.A2013205148)
文摘In this paper, the robustness of the orbit structure is investigated for a partially hyperbolic endomorphism f on a compact manifold M. It is first proved that the dynamical structure of its orbit space （the inverse limit space） M^f of f is topologically quasi-stable under C^0-small perturbations in the following sense： For any covering endomorphism g C^0-close to f, there is a continuous map φ from M^9 to Π-∞^∞ M such that for any {yi}i∈z∈φ（M^9）, yi＋1 and f（yi） differ only by a motion along the center direction. It is then proved that f has quasi-shadowing property in the following sense： For any pseudo-orbit {xi}i∈z, there is a sequence of points {yi}i∈z tracing it, in which yi＋1 is obtained from f（yi） by a motion along the center direction.
基金supported by NNSF of China(12101446,11401581,11971236)The second author was supported by NNSF of China(11401581)+1 种基金the third author was supported by NNSF of China(11671208,11431012)At the end,we would like to express our gratitude to Tianyuan Mathematical Center in Southwest China,Sichuan University and Southwest Jiaotong University for their support and hospitality.
文摘Katok’s entropy formula is an important formula in entropy theory.It plays significant roles in large deviation theories,multifractal analysis,quantitative recurrence and so on.This paper is devoted to establishing Katok’s entropy formula of unstable metric entropy which is the entropy caused by the unstable part of partially hyperbolic systems.We also construct a similar formula which can be used to study the quantitative recurrence in the unstable manifold for partially hyperbolic diffeomorphisms.
基金supported by NSFC(No:11371120)GCCHB(No:GCC2014052)supported by NSFHB(No:A2014205154)
文摘Let f ： M → M be a partially hyperbolic diffeomorphism on a closed Riemannian manifold with uniformly compact center foliation. We show that if the center foliation of f is of dimension one then the topological entropy is constant on a small C1 neighborhood of f.
基金supported by the National Natural Science Foundation of China (Nos. 11771118,11801336, 12171400)the Innovation Fund Designated for Graduate Students of Hebei Province (No.CXZZBS2018101)+1 种基金China Scholarship Council (CSC for short)China Postdoctoral Science Foundation (No. 2021M691889).
文摘In this paper,local unstable metric entropy,local unstable topological entropy and local unstable pressure for partially hyperbolic endomorphisms are introduced and investigated.Specially,two variational principles concerning relationships among the above mentioned numbers are formulated.
文摘A construction of multiple knot B-spline wavelets has been given in[C.K.Chui and E.Quak,Wavelet on a bounded interval,In:D.Braess and L.L.Schumaker,editors.Numerical methods of approximation theory.Basel:Birkhauser Verlag;(1992),pp.57–76].In this work,we first modify these wavelets to solve the elliptic(partially)Dirichlet boundary value problems by Galerkin and Petrov Galerkin methods.We generalize this construction to two dimensional case by Tensor product space.In addition,the solution of the system discretized by Galerkin method with modified multiple knot B-spline wavelets is discussed.We also consider a nonlinear partial differential equation for unsteady flows in an open channel called Saint-Venant.Since the solving of this problem by some methods such as finite difference and finite element produce unsuitable approximations specially in the ends of channel,it is solved by multiple knot B-spline wavelet method that yields a very well approximation.Finally,some numerical examples are given to support our theoretical results.
基金supported by the National Natural Science Foundation of China under Grant Nos.61821004 and 62250056the Natural Science Foundation of Shandong Province under Grant Nos.ZR2021ZD14 and ZR2021JQ24+1 种基金Science and Technology Project of Qingdao West Coast New Area under Grant Nos.2019-32,2020-20,2020-1-4,High-level Talent Team Project of Qingdao West Coast New Area under Grant No.RCTDJC-2019-05Key Research and Development Program of Shandong Province under Grant No.2020CXGC01208.
文摘This paper focuses on linear-quadratic(LQ)optimal control for a class of systems governed by first-order hyperbolic partial differential equations(PDEs).Different from most of the previous works,an approach of discretization-then-continuousization is proposed in this paper to cope with the infinite-dimensional nature of PDE systems.The contributions of this paper consist of the following aspects:(1)The differential Riccati equations and the solvability condition of the LQ optimal control problems are obtained via the discretization-then-continuousization method.(2)A numerical calculation way of the differential Riccati equations and a practical design way of the optimal controller are proposed.Meanwhile,the relationship between the optimal costate and the optimal state is established by solving a set of forward and backward partial difference equations(FBPDEs).(3)The correctness of the method used in this paper is verified by a complementary continuous method and the comparative analysis with the existing operator results is presented.It is shown that the proposed results not only contain the classic results of the standard LQ control problem of systems governed by ordinary differential equations as a special case,but also support the existing operator results and give a more convenient form of computation.
基金China State Major Key Project for Basic Researches
文摘Combining difference method and boundary integral equation method,we propose a new numerical method for solving initial-boundary value problem of second order hyperbolic partial differential equations defined on a bounded or unbounded domain in R~3 and obtain the error estimates of the approximate solution in energy norm and local maximum norm.
文摘In this paper, we consider a bound on a general version of the integralinequalities for functions and also study the qualitative behavior of the solutions of certainclasses of the hyperbolic partial delay differential equations under the integral inequalities.
