Normal form theory is a very effective method when we study degenerate bifurcations of nonlinear dynamical systems. In this paper by using adjoint operator method, normal forms of order 3 and 4 for nonlinear dynamical...Normal form theory is a very effective method when we study degenerate bifurcations of nonlinear dynamical systems. In this paper by using adjoint operator method, normal forms of order 3 and 4 for nonlinear dynamical system with nilpotent linear part and Z(2)-asymmetry are computed. According to normal forms obtained, universal unfoldings for some degenerate bifurcation cases of codimension 3 and simple global characterizations, are studied.展开更多
In this article we are interested in the numerical computation of spectra of non-self adjoint quadratic operators,in two and three spatial dimensions.Indeed,in the multidimensional case very few results are known on t...In this article we are interested in the numerical computation of spectra of non-self adjoint quadratic operators,in two and three spatial dimensions.Indeed,in the multidimensional case very few results are known on the location of the eigenvalues.This leads to solve nonlinear eigenvalue problems.In introduction we begin with a review of theoretical results and numerical results obtained for the one dimensional case.Then we present the numerical methods developed to compute the spectra(finite difference discretization)for the two and three dimensional cases.The numerical results obtained are presented and analyzed.One difficulty here is that we have to compute eigenvalues of strongly non-self-adjoint operators which are unstable.This work is in continuity of a previous work in one spatial dimension[3].展开更多
The ultimate solution to anthropogenic air pollution depends on an adjustment and upgrade of industrial and energy structures. Before this process can be completed, reducing the anthropogenic pollutant emissions is an...The ultimate solution to anthropogenic air pollution depends on an adjustment and upgrade of industrial and energy structures. Before this process can be completed, reducing the anthropogenic pollutant emissions is an effective measure. This is a problem belonging to "Natural Cybernetics", i.e., the problem of air pollution control should be solved together with the weather prediction; however, this is very complicated. Considering that heavy air pollution usually occurs in stable weather conditions and that the feedbacks between air pollutants and meteorological changes are insufficient, we propose a simplified natural cybernetics method. Here, an off-line air pollution evolution equation is first solved with data from a given anthropogenic emission inventory under the predicted weather conditions, and then, a related "incomplete adjoint problem" is solved to obtain the optimal reduction of anthropogenic emissions. Usually, such solution is sufficient for satisfying the air quality and economical/social requirements. However, a better solution can be obtained by iteration after updating the emission inventory with the reduced anthropogenic emissions. Then, this paper discusses the retrieval of the pollutant emission source with a known spatio-temporal distribution of the pollutant concentrations, and a feasible mathematical method to achieve this is proposed. The retrieval of emission source would also help control air pollution.展开更多
The classical Volterra operator V and its adjoint operator V^(*)play important roles in the complex space L^(2)[0, 1]. As to the properties of linear combination of V and V^(*), we present the equivalent condition ens...The classical Volterra operator V and its adjoint operator V^(*)play important roles in the complex space L^(2)[0, 1]. As to the properties of linear combination of V and V^(*), we present the equivalent condition ensuring z_(1)V +z_(2)V^(*)(z_(1), z_(2) ∈ C) satisfies the accretive property. Then an accurate representation of the numerical range of(u+iv)I +mV +n V^(*)(u, v, m, n ∈ R, m+n ≥ 0)is described in this paper.展开更多
An absolute value equation is established for linear combinations of two operators.When the parameters take special values, the parallelogram law of operator type is given. In addition, the operator equation in litera...An absolute value equation is established for linear combinations of two operators.When the parameters take special values, the parallelogram law of operator type is given. In addition, the operator equation in literature [3] and its equivalent deformation are obtained.Based on the equivalent deformation of the operator equation and using the properties of conjugate number as well as the operator, an absolute value identity of multiple operators is given by means of mathematical induction. As Corollaries, Bohr inequalities are extended to multiple operators and some related inequalities are reduced to, such as inequalities in [2]and [3].展开更多
When D: E →F is a linear differential operator of order q between the sections of vector bundles over a manifold X of dimension n, it is defined by a bundle map Φ: J<sub>q</sub>(E) &ra...When D: E →F is a linear differential operator of order q between the sections of vector bundles over a manifold X of dimension n, it is defined by a bundle map Φ: J<sub>q</sub>(E) →F=F<sub>0</sub> that may depend, explicitly or implicitly, on constant parameters a, b, c, ... . A “direct problem” is to find the generating compatibility conditions (CC) in the form of an operator D<sub>1</sub>: F<sub>0</sub> →F<sub>1</sub>. When D is involutive, that is when the corresponding system R<sub>q</sub> = ker (Φ) is involutive, this procedure provides successive first order involutive operators D<sub>1</sub>, ..., D<sub>n</sub>. Though D<sub>1</sub> οD = 0 implies ad (D) οad(D<sub>1</sub>) = 0 by taking the respective adjoint operators, then ad (D) may not generate the CC of ad (D<sub>1</sub>) and measuring such “gaps” led to introduce extension modules in differential homological algebra. They may also depend on the parameters and such a situation is well known in ordinary or partial control theory. When R<sub>q</sub> is not involutive, a standard prolongation/projection (PP) procedure allows in general to find integers r, s such that the image of the projection at order q+r of the prolongation is involutive but it may highly depend on the parameters. However, sometimes the resulting system no longer depends on the parameters and the extension modules do not depend on the parameters because it is known that they do not depend on the differential sequence used for their definition. The purpose of this paper is to study the above problems for the Kerr (m, a), Schwarzschild (m, 0) and Minkowski (0, 0) parameters while computing the dimensions of the inclusions for the respective Killing operators. Other striking motivating examples are also presented.展开更多
We extend the(outer)measureγ_(I) associated to an operator ideal I to a measureγ_(I) for bounded bilinear operators.If I is surjective and closed,and J is the class of those bilinear operators such thatγ_(I)(T)=0,w...We extend the(outer)measureγ_(I) associated to an operator ideal I to a measureγ_(I) for bounded bilinear operators.If I is surjective and closed,and J is the class of those bilinear operators such thatγ_(I)(T)=0,we prove that J coincides with the composition bideal I?B.If I satisfies theΣ_(r)-condition,we establish a simple necessary and sufficient condition for an interpolated operator by the real method to belong to J.Furthermore,if in addition I is symmetric,we prove a formula for the measureγ_(I) of an operator interpolated by the real method.In particular,results apply to weakly compact operators.展开更多
In this paper, based on the invariant subspace theory and adjoint operator concept of linear operator, a new matrix representation method is proposed to calculate the normal forms of n order general nonlinear dyna...In this paper, based on the invariant subspace theory and adjoint operator concept of linear operator, a new matrix representation method is proposed to calculate the normal forms of n order general nonlinear dynamic systems. In the method, there is no need to determine the structure of the class of normal forms in advance. Because the subspace is not related to the dimensions of the system and the order of the normal forms directly, it is determined only by a given vector field. So the normal forms with high orders and dimensions can be calculated by the method without difficulties. In this paper, is used the method for selecting the minimal subspace and solving homological equations in the subspace, the examples show that the method is very effective.展开更多
Designing a controller for the docking maneuver in Probe-Drogue Refueling(PDR) is an important but challenging task, due to the complex system model and the high precision requirement.In order to overcome the disadvan...Designing a controller for the docking maneuver in Probe-Drogue Refueling(PDR) is an important but challenging task, due to the complex system model and the high precision requirement.In order to overcome the disadvantage of only feedback control, a feedforward control scheme known as Iterative Learning Control(ILC) is adopted in this paper.First, Additive State Decomposition(ASD) is used to address the tight coupling of input saturation, nonlinearity and the property of Non Minimum Phase(NMP) by separating these features into two subsystems(a primary system and a secondary system).After system decomposition, an adjoint-type ILC is applied to the Linear Time-Invariant(LTI) primary system with NMP to achieve entire output trajectory tracking, whereas state feedback is used to stabilize the secondary system with input saturation.The two controllers designed for the two subsystems can be combined to achieve the original control goal of the PDR system.Furthermore, to compensate for the receiverindependent uncertainties, a correction action is proposed by using the terminal docking error,which can lead to a smaller docking error at the docking moment.Simulation tests have been carried out to demonstrate the performance of the proposed control method, which has some advantages over the traditional derivative-type ILC and adjoint-type ILC in the docking control of PDR.展开更多
A new type of Galerkin finite element for first-order initial-value problems(IVPs)is proposed.Both the trial and test functions employ the same m-degreed polynomials.The adjoint equation is used to eliminate one degre...A new type of Galerkin finite element for first-order initial-value problems(IVPs)is proposed.Both the trial and test functions employ the same m-degreed polynomials.The adjoint equation is used to eliminate one degree of freedom(DOF)from the test function,and then the so-called condensed test function and its consequent condensed Galerkin element are constructed.It is mathematically proved and numerically verified that the condensed element produces the super-convergent nodal solutions of O(h^(2m+2)),which is equivalent to the order of accuracy by the conventional element of degree m+1.Some related properties are addressed,and typical numerical examples of both linear and nonlinear IVPs of both a single equation and a system of equations are presented to show the validity and effectiveness of the proposed element.展开更多
In this paper, the adjoint of a densely defined block operator matrix L=[A B C D] in a Hilbert space X ×X is studied and the sufficient conditions under which the equality L＊=[A＊ B＊ C＊ D＊] holds are obtained...In this paper, the adjoint of a densely defined block operator matrix L=[A B C D] in a Hilbert space X ×X is studied and the sufficient conditions under which the equality L＊=[A＊ B＊ C＊ D＊] holds are obtained through applying Frobenius-Schur factorization.展开更多
We discuss a transfer line consisting of a reliable machine, an unreliable machine and a storage buffer. This transfer line can be described by a group of partial differential equations with integral boundary conditio...We discuss a transfer line consisting of a reliable machine, an unreliable machine and a storage buffer. This transfer line can be described by a group of partial differential equations with integral boundary conditions. First we show that the operator corresponding to these equations generates a positive contraction C0-semigroup T(t), and prove that T(t) is a quasi-compact operator. Next we verify that 0 is an eigenvalue of this operator and its adjoint operator with geometric multiplicity one. Last, by using the above results we obtain that the time-dependent solution of these equations converges strongly to their steady-state solution.展开更多
In this paper, the optical wave propagation in lossy waveguides is described by the Helmholtz equation with the complex refractive-index, and the Chebyshev pseudospectral method is used to discretize the transverse op...In this paper, the optical wave propagation in lossy waveguides is described by the Helmholtz equation with the complex refractive-index, and the Chebyshev pseudospectral method is used to discretize the transverse operator of the equation. Meanwhile, an operator marching method, a one-way re-formulation based on the Dirichlet- to-Neumann （DtN） map, is improved to solve the equation. NumericM examples show that our treatment is more efficient.展开更多
Using the incomplete adjoint operator method in part I of this series of papers,the total emission source S can be retrieved from the pollutant concentrationsρob obtained from the air pollution monitoring network.Thi...Using the incomplete adjoint operator method in part I of this series of papers,the total emission source S can be retrieved from the pollutant concentrationsρob obtained from the air pollution monitoring network.This paper studies the problem of retrieving anthropogenic emission sources from S.Assuming that the natural source Sn is known,and as the internal source Sc due to chemical reactions is a function of pollutant concentrations,if the chemical reaction equations are complete and the parameters are accurate,Sc can be calculated directly fromρob,and then Sa can be obtained from S.However,if the chemical reaction parameters(denoted asγ)are insufficiently accurate,bothγand Sc should be corrected.This article proposes a"double correction iterative method"to retrieve Sc and correctγand proves that this iterative method converges.展开更多
文摘Normal form theory is a very effective method when we study degenerate bifurcations of nonlinear dynamical systems. In this paper by using adjoint operator method, normal forms of order 3 and 4 for nonlinear dynamical system with nilpotent linear part and Z(2)-asymmetry are computed. According to normal forms obtained, universal unfoldings for some degenerate bifurcation cases of codimension 3 and simple global characterizations, are studied.
基金supported by the Fédération de Mathématiques des Pays de la Loire,CNRS FR 2962。
文摘In this article we are interested in the numerical computation of spectra of non-self adjoint quadratic operators,in two and three spatial dimensions.Indeed,in the multidimensional case very few results are known on the location of the eigenvalues.This leads to solve nonlinear eigenvalue problems.In introduction we begin with a review of theoretical results and numerical results obtained for the one dimensional case.Then we present the numerical methods developed to compute the spectra(finite difference discretization)for the two and three dimensional cases.The numerical results obtained are presented and analyzed.One difficulty here is that we have to compute eigenvalues of strongly non-self-adjoint operators which are unstable.This work is in continuity of a previous work in one spatial dimension[3].
基金supported by the National Natural Science Foundation of China (Grant No. 41630530the National Key Research and Development Program of China (Grant No. 2016YFC0209000)
文摘The ultimate solution to anthropogenic air pollution depends on an adjustment and upgrade of industrial and energy structures. Before this process can be completed, reducing the anthropogenic pollutant emissions is an effective measure. This is a problem belonging to "Natural Cybernetics", i.e., the problem of air pollution control should be solved together with the weather prediction; however, this is very complicated. Considering that heavy air pollution usually occurs in stable weather conditions and that the feedbacks between air pollutants and meteorological changes are insufficient, we propose a simplified natural cybernetics method. Here, an off-line air pollution evolution equation is first solved with data from a given anthropogenic emission inventory under the predicted weather conditions, and then, a related "incomplete adjoint problem" is solved to obtain the optimal reduction of anthropogenic emissions. Usually, such solution is sufficient for satisfying the air quality and economical/social requirements. However, a better solution can be obtained by iteration after updating the emission inventory with the reduced anthropogenic emissions. Then, this paper discusses the retrieval of the pollutant emission source with a known spatio-temporal distribution of the pollutant concentrations, and a feasible mathematical method to achieve this is proposed. The retrieval of emission source would also help control air pollution.
基金Supported by the Tianjin Normal University Research Innovation Project for Postgraduate Students (Grant No. 2022KYCX110Y)。
文摘The classical Volterra operator V and its adjoint operator V^(*)play important roles in the complex space L^(2)[0, 1]. As to the properties of linear combination of V and V^(*), we present the equivalent condition ensuring z_(1)V +z_(2)V^(*)(z_(1), z_(2) ∈ C) satisfies the accretive property. Then an accurate representation of the numerical range of(u+iv)I +mV +n V^(*)(u, v, m, n ∈ R, m+n ≥ 0)is described in this paper.
基金Supported by the Key Scientific and Technological Innovation Team Project in Shaanxi Province(2014KCT-15)
文摘An absolute value equation is established for linear combinations of two operators.When the parameters take special values, the parallelogram law of operator type is given. In addition, the operator equation in literature [3] and its equivalent deformation are obtained.Based on the equivalent deformation of the operator equation and using the properties of conjugate number as well as the operator, an absolute value identity of multiple operators is given by means of mathematical induction. As Corollaries, Bohr inequalities are extended to multiple operators and some related inequalities are reduced to, such as inequalities in [2]and [3].
文摘When D: E →F is a linear differential operator of order q between the sections of vector bundles over a manifold X of dimension n, it is defined by a bundle map Φ: J<sub>q</sub>(E) →F=F<sub>0</sub> that may depend, explicitly or implicitly, on constant parameters a, b, c, ... . A “direct problem” is to find the generating compatibility conditions (CC) in the form of an operator D<sub>1</sub>: F<sub>0</sub> →F<sub>1</sub>. When D is involutive, that is when the corresponding system R<sub>q</sub> = ker (Φ) is involutive, this procedure provides successive first order involutive operators D<sub>1</sub>, ..., D<sub>n</sub>. Though D<sub>1</sub> οD = 0 implies ad (D) οad(D<sub>1</sub>) = 0 by taking the respective adjoint operators, then ad (D) may not generate the CC of ad (D<sub>1</sub>) and measuring such “gaps” led to introduce extension modules in differential homological algebra. They may also depend on the parameters and such a situation is well known in ordinary or partial control theory. When R<sub>q</sub> is not involutive, a standard prolongation/projection (PP) procedure allows in general to find integers r, s such that the image of the projection at order q+r of the prolongation is involutive but it may highly depend on the parameters. However, sometimes the resulting system no longer depends on the parameters and the extension modules do not depend on the parameters because it is known that they do not depend on the differential sequence used for their definition. The purpose of this paper is to study the above problems for the Kerr (m, a), Schwarzschild (m, 0) and Minkowski (0, 0) parameters while computing the dimensions of the inclusions for the respective Killing operators. Other striking motivating examples are also presented.
基金supported in part by UCM(Grant No.PR3/23-30811)。
文摘We extend the(outer)measureγ_(I) associated to an operator ideal I to a measureγ_(I) for bounded bilinear operators.If I is surjective and closed,and J is the class of those bilinear operators such thatγ_(I)(T)=0,we prove that J coincides with the composition bideal I?B.If I satisfies theΣ_(r)-condition,we establish a simple necessary and sufficient condition for an interpolated operator by the real method to belong to J.Furthermore,if in addition I is symmetric,we prove a formula for the measureγ_(I) of an operator interpolated by the real method.In particular,results apply to weakly compact operators.
文摘In this paper, based on the invariant subspace theory and adjoint operator concept of linear operator, a new matrix representation method is proposed to calculate the normal forms of n order general nonlinear dynamic systems. In the method, there is no need to determine the structure of the class of normal forms in advance. Because the subspace is not related to the dimensions of the system and the order of the normal forms directly, it is determined only by a given vector field. So the normal forms with high orders and dimensions can be calculated by the method without difficulties. In this paper, is used the method for selecting the minimal subspace and solving homological equations in the subspace, the examples show that the method is very effective.
基金supported by the National Natural Science Foundation of China(No.61473012)。
文摘Designing a controller for the docking maneuver in Probe-Drogue Refueling(PDR) is an important but challenging task, due to the complex system model and the high precision requirement.In order to overcome the disadvantage of only feedback control, a feedforward control scheme known as Iterative Learning Control(ILC) is adopted in this paper.First, Additive State Decomposition(ASD) is used to address the tight coupling of input saturation, nonlinearity and the property of Non Minimum Phase(NMP) by separating these features into two subsystems(a primary system and a secondary system).After system decomposition, an adjoint-type ILC is applied to the Linear Time-Invariant(LTI) primary system with NMP to achieve entire output trajectory tracking, whereas state feedback is used to stabilize the secondary system with input saturation.The two controllers designed for the two subsystems can be combined to achieve the original control goal of the PDR system.Furthermore, to compensate for the receiverindependent uncertainties, a correction action is proposed by using the terminal docking error,which can lead to a smaller docking error at the docking moment.Simulation tests have been carried out to demonstrate the performance of the proposed control method, which has some advantages over the traditional derivative-type ILC and adjoint-type ILC in the docking control of PDR.
基金Project supported by the National Natural Science Foundation of China(Nos.51878383 and51378293)。
文摘A new type of Galerkin finite element for first-order initial-value problems(IVPs)is proposed.Both the trial and test functions employ the same m-degreed polynomials.The adjoint equation is used to eliminate one degree of freedom(DOF)from the test function,and then the so-called condensed test function and its consequent condensed Galerkin element are constructed.It is mathematically proved and numerically verified that the condensed element produces the super-convergent nodal solutions of O(h^(2m+2)),which is equivalent to the order of accuracy by the conventional element of degree m+1.Some related properties are addressed,and typical numerical examples of both linear and nonlinear IVPs of both a single equation and a system of equations are presented to show the validity and effectiveness of the proposed element.
基金Supported by NSFC(Grant Nos.11101200,11371185,2013ZD01)
文摘In this paper, the adjoint of a densely defined block operator matrix L=[A B C D] in a Hilbert space X ×X is studied and the sufficient conditions under which the equality L＊=[A＊ B＊ C＊ D＊] holds are obtained through applying Frobenius-Schur factorization.
基金This research is supported by Excellent Youth Reward Foundation of the Higher Education Institution of Xinjiang (No: XJEDU2004E05) the Major Project of the Ministry of Education of China(No. 205180).
文摘We discuss a transfer line consisting of a reliable machine, an unreliable machine and a storage buffer. This transfer line can be described by a group of partial differential equations with integral boundary conditions. First we show that the operator corresponding to these equations generates a positive contraction C0-semigroup T(t), and prove that T(t) is a quasi-compact operator. Next we verify that 0 is an eigenvalue of this operator and its adjoint operator with geometric multiplicity one. Last, by using the above results we obtain that the time-dependent solution of these equations converges strongly to their steady-state solution.
基金supported by the National Natural Science Foundation of China(No.11371319)the Zhejiang Provincial Natural Science Foundation of China(No.LY13A010002)
文摘In this paper, the optical wave propagation in lossy waveguides is described by the Helmholtz equation with the complex refractive-index, and the Chebyshev pseudospectral method is used to discretize the transverse operator of the equation. Meanwhile, an operator marching method, a one-way re-formulation based on the Dirichlet- to-Neumann （DtN） map, is improved to solve the equation. NumericM examples show that our treatment is more efficient.
基金supported by the National Natural Science Foundation of China(Grant Nos.41630530&41877316)the Key Research Program of Frontier Sciences,Chinese Academy of Sciences(Grant No.QYZDY-SSW-DQC002)the Youth Innovation Promotion Association,Chinese Academy of Sciences(Grant No.2019079)。
文摘Using the incomplete adjoint operator method in part I of this series of papers,the total emission source S can be retrieved from the pollutant concentrationsρob obtained from the air pollution monitoring network.This paper studies the problem of retrieving anthropogenic emission sources from S.Assuming that the natural source Sn is known,and as the internal source Sc due to chemical reactions is a function of pollutant concentrations,if the chemical reaction equations are complete and the parameters are accurate,Sc can be calculated directly fromρob,and then Sa can be obtained from S.However,if the chemical reaction parameters(denoted asγ)are insufficiently accurate,bothγand Sc should be corrected.This article proposes a"double correction iterative method"to retrieve Sc and correctγand proves that this iterative method converges.
