In this paper the existence of solutions of the singularly perturbed boundary value problems on infinite interval for the second order nonlinear equation containing a small parameterε>0,εy'=f(x,y,y'),y...In this paper the existence of solutions of the singularly perturbed boundary value problems on infinite interval for the second order nonlinear equation containing a small parameterε>0,εy'=f(x,y,y'),y'(0)=a,y(∞)=βis examined,where are constants,and i=0,1.Moreover,asymptotic estimates of the solutions for the above problems are given.展开更多
Various kinds of Riemann boundary value problems (BVPs) for analytic functions on closed curves or on open arc, doubly periodic Riemann BVPs, doubly quasi-periodic Riemann BVPs, and BVPs for polyanalytic functions hav...Various kinds of Riemann boundary value problems (BVPs) for analytic functions on closed curves or on open arc, doubly periodic Riemann BVPs, doubly quasi-periodic Riemann BVPs, and BVPs for polyanalytic functions have been widely investigated in [1-8]. The main ap- proach is to use the decomposition of polyanalytic functions and their generalization to transform the boundary value problems to their corresponding boundary value problems for analytic functions. Recently, inverse Riemann BVPs for generalized analytic functions or bianalytic functions have been investigated in [9-12]. In this paper, we consider a kind of Riemann BVP of non-normal type on the infinite straight line and discuss the solvable conditions and the general solution for it.展开更多
We consider a Hilbert boundary value problem with an unknown parametric function on arbitrary infinite straight line passing through the origin. We propose to transform the Hilbert boundary value problem to Riemann bo...We consider a Hilbert boundary value problem with an unknown parametric function on arbitrary infinite straight line passing through the origin. We propose to transform the Hilbert boundary value problem to Riemann boundary value problem, and address it by defining symmetric extension for holomorphic functions about an arbitrary straight line passing through the origin. Finally, we develop the general solution and the solvable conditions for the Hilbert boundary value problem.展开更多
This article discusses the general boundary value problem for the nonlinear uniformly elliptic equation of second order in D (0.1) and the boundary condition,(0.2) in a multiply connected infinite domain D with the bo...This article discusses the general boundary value problem for the nonlinear uniformly elliptic equation of second order in D (0.1) and the boundary condition,(0.2) in a multiply connected infinite domain D with the boundary T. The above boundary value problem is called Problem G. Problem G extends the work [8] in which the equation (0.1) includes a nonlinear lower term and the boundary condition (0.2) is more general. If the complex equation (0.1) and the boundary condition (0.2) meet certain assumptions, some solvability results for Problem G can be obtained. By using reduction to absurdity, we first discuss a priori estimates of solutions and solvability for a modified problem. Then we present results on solvability of Problem G.展开更多
We use the Fokas method to analyze the derivative nonlinear Schrodinger （DNLS） equation iqt （x, t） = -qxx （x, t）＋（rq^2）x on the interval [0, L]. Assuming that the solution q（x, t） exists, we show that it ca...We use the Fokas method to analyze the derivative nonlinear Schrodinger （DNLS） equation iqt （x, t） = -qxx （x, t）＋（rq^2）x on the interval [0, L]. Assuming that the solution q（x, t） exists, we show that it can be represented in terms of the solution of a matrix Riemann- Hilbert problem formulated in the plane of the complex spectral parameter ξ. This problem has explicit （x, t） dependence, and it has jumps across {ξ∈C｜Imξ^4 = 0}. The relevant jump matrices are explicitely given in terms of the spectral functions {a（ξ）, b（ξ）}, {A（ξ）, B（ξ）}, and {A（ξ）, B（ξ）}, which in turn are defined in terms of the initial data q0（x） = q（x, 0）, the bound- ary data g0（t）= q（0, t）, g1（t） = qx（0, t）, and another boundary values f0（t） = q（L, t）, f1（t） = qx（L, t）. The spectral functions are not independent, but related by a compatibility condition, the so-called global relation.展开更多
Itis proved that for ε≥0 and δ≥0 the two -point boundary value problemhas a unique solution (y(t,ε,δ),z(t,ε,δ)) under certain hypotheses with the aid of the appropriate Green's function integral operator....Itis proved that for ε≥0 and δ≥0 the two -point boundary value problemhas a unique solution (y(t,ε,δ),z(t,ε,δ)) under certain hypotheses with the aid of the appropriate Green's function integral operator.The unique solution (ξ,η,v.(s)) of the free boundary problem is constructed utilizing the solution (y(t,ε.0),z(t,ε,0)).The fine boundary problem is shown to be a singular perturbation problem when the function k(t) possesses intervals of degeneracy展开更多
In this paper, we study a fractional differential equation cD0^a＋u（t）＋f（t,u（t））=0,t∈（0＋∞）satisfying the boundary conditions： ……where 1〈a≤2,cD0^a＋ is the standard Caputo fractional derivative of orde...In this paper, we study a fractional differential equation cD0^a＋u（t）＋f（t,u（t））=0,t∈（0＋∞）satisfying the boundary conditions： ……where 1〈a≤2,cD0^a＋ is the standard Caputo fractional derivative of order a. The main tools used in the paper is a contraction principle in the Banach space and the fixed point theorem due to D. O＇Regan. Under a compactness criterion, the existence of solutions are established.展开更多
In this paper, the author discusses the multiple positive solutions for an infinite boundary value problem of first order impulsive superlinear integro-differential equations on the half line by means of the fixed poi...In this paper, the author discusses the multiple positive solutions for an infinite boundary value problem of first order impulsive superlinear integro-differential equations on the half line by means of the fixed point theorem of cone expansion and compression with norm type.展开更多
In this paper, we consider a two-point boundary value problem for a system of second order ordinary differential equations. Under some conditions, we show the existence of positive solution to the system of second ord...In this paper, we consider a two-point boundary value problem for a system of second order ordinary differential equations. Under some conditions, we show the existence of positive solution to the system of second order ordinary differential equa-tions.展开更多
An infinite horizon linear quadratic optimal control problem for analytic semigroup with unbounded control in Hilbert space is considered.The state weight operator is allowed to be inddefinite while the control weight...An infinite horizon linear quadratic optimal control problem for analytic semigroup with unbounded control in Hilbert space is considered.The state weight operator is allowed to be inddefinite while the control weight operator is coercive.Under the exponential stabilization condition,it is proved that any optimal control and its optimal trajectory are continuous.The positive real lemma as a necessary and sufficient condition for the unique solvability of this problem is established.The closed-loop synthesis of optimal control is given via the solution to the algebraic Riccati equation.展开更多
This paper presents a new approach for solving a class of infinite horizon nonlinear optimal control problems (OCPs).In this approach,a nonlinear two-point boundary value problem (TPBVP),derived from Pontryagin's ...This paper presents a new approach for solving a class of infinite horizon nonlinear optimal control problems (OCPs).In this approach,a nonlinear two-point boundary value problem (TPBVP),derived from Pontryagin's maximum principle,is transformed into a sequence of linear time-invariant TPBVPs.Solving the latter problems in a recursive manner provides the optimal control law and the optimal trajectory in the form of uniformly convergent series.Hence,to obtain the optimal solution,only the techniques for solving linear ordinary differential equations are employed.An efficient algorithm is also presented,which has low computational complexity and a fast convergence rate.Just a few iterations are required to find an accurate enough suboptimal trajectory-control pair for the nonlinear OCP.The results not only demonstrate the efficiency,simplicity,and high accuracy of the suggested approach,but also indicate its effectiveness in practical use.展开更多
文摘In this paper the existence of solutions of the singularly perturbed boundary value problems on infinite interval for the second order nonlinear equation containing a small parameterε>0,εy'=f(x,y,y'),y'(0)=a,y(∞)=βis examined,where are constants,and i=0,1.Moreover,asymptotic estimates of the solutions for the above problems are given.
文摘Various kinds of Riemann boundary value problems (BVPs) for analytic functions on closed curves or on open arc, doubly periodic Riemann BVPs, doubly quasi-periodic Riemann BVPs, and BVPs for polyanalytic functions have been widely investigated in [1-8]. The main ap- proach is to use the decomposition of polyanalytic functions and their generalization to transform the boundary value problems to their corresponding boundary value problems for analytic functions. Recently, inverse Riemann BVPs for generalized analytic functions or bianalytic functions have been investigated in [9-12]. In this paper, we consider a kind of Riemann BVP of non-normal type on the infinite straight line and discuss the solvable conditions and the general solution for it.
文摘We consider a Hilbert boundary value problem with an unknown parametric function on arbitrary infinite straight line passing through the origin. We propose to transform the Hilbert boundary value problem to Riemann boundary value problem, and address it by defining symmetric extension for holomorphic functions about an arbitrary straight line passing through the origin. Finally, we develop the general solution and the solvable conditions for the Hilbert boundary value problem.
文摘This article discusses the general boundary value problem for the nonlinear uniformly elliptic equation of second order in D (0.1) and the boundary condition,(0.2) in a multiply connected infinite domain D with the boundary T. The above boundary value problem is called Problem G. Problem G extends the work [8] in which the equation (0.1) includes a nonlinear lower term and the boundary condition (0.2) is more general. If the complex equation (0.1) and the boundary condition (0.2) meet certain assumptions, some solvability results for Problem G can be obtained. By using reduction to absurdity, we first discuss a priori estimates of solutions and solvability for a modified problem. Then we present results on solvability of Problem G.
基金supported by grants from the National Science Foundation of China (10971031 11271079+2 种基金 11075055)Doctoral Programs Foundation of the Ministry of Education of Chinathe Shanghai Shuguang Tracking Project (08GG01)
文摘We use the Fokas method to analyze the derivative nonlinear Schrodinger （DNLS） equation iqt （x, t） = -qxx （x, t）＋（rq^2）x on the interval [0, L]. Assuming that the solution q（x, t） exists, we show that it can be represented in terms of the solution of a matrix Riemann- Hilbert problem formulated in the plane of the complex spectral parameter ξ. This problem has explicit （x, t） dependence, and it has jumps across {ξ∈C｜Imξ^4 = 0}. The relevant jump matrices are explicitely given in terms of the spectral functions {a（ξ）, b（ξ）}, {A（ξ）, B（ξ）}, and {A（ξ）, B（ξ）}, which in turn are defined in terms of the initial data q0（x） = q（x, 0）, the bound- ary data g0（t）= q（0, t）, g1（t） = qx（0, t）, and another boundary values f0（t） = q（L, t）, f1（t） = qx（L, t）. The spectral functions are not independent, but related by a compatibility condition, the so-called global relation.
文摘Itis proved that for ε≥0 and δ≥0 the two -point boundary value problemhas a unique solution (y(t,ε,δ),z(t,ε,δ)) under certain hypotheses with the aid of the appropriate Green's function integral operator.The unique solution (ξ,η,v.(s)) of the free boundary problem is constructed utilizing the solution (y(t,ε.0),z(t,ε,0)).The fine boundary problem is shown to be a singular perturbation problem when the function k(t) possesses intervals of degeneracy
文摘In this paper, we study a fractional differential equation cD0^a＋u（t）＋f（t,u（t））=0,t∈（0＋∞）satisfying the boundary conditions： ……where 1〈a≤2,cD0^a＋ is the standard Caputo fractional derivative of order a. The main tools used in the paper is a contraction principle in the Banach space and the fixed point theorem due to D. O＇Regan. Under a compactness criterion, the existence of solutions are established.
文摘In this paper, the author discusses the multiple positive solutions for an infinite boundary value problem of first order impulsive superlinear integro-differential equations on the half line by means of the fixed point theorem of cone expansion and compression with norm type.
基金Project supported financially by the National Natural Science Foundation of China (1087111610671167)
文摘In this paper, we consider a two-point boundary value problem for a system of second order ordinary differential equations. Under some conditions, we show the existence of positive solution to the system of second order ordinary differential equa-tions.
基金This work is partially supported by the National Key Project of Chinathe National Nature Science Foundation of China No.19901030NSF of the Chinese State Education Ministry and Lab.of Math.for Nonlinear Sciences at Fudan University
文摘An infinite horizon linear quadratic optimal control problem for analytic semigroup with unbounded control in Hilbert space is considered.The state weight operator is allowed to be inddefinite while the control weight operator is coercive.Under the exponential stabilization condition,it is proved that any optimal control and its optimal trajectory are continuous.The positive real lemma as a necessary and sufficient condition for the unique solvability of this problem is established.The closed-loop synthesis of optimal control is given via the solution to the algebraic Riccati equation.
文摘This paper presents a new approach for solving a class of infinite horizon nonlinear optimal control problems (OCPs).In this approach,a nonlinear two-point boundary value problem (TPBVP),derived from Pontryagin's maximum principle,is transformed into a sequence of linear time-invariant TPBVPs.Solving the latter problems in a recursive manner provides the optimal control law and the optimal trajectory in the form of uniformly convergent series.Hence,to obtain the optimal solution,only the techniques for solving linear ordinary differential equations are employed.An efficient algorithm is also presented,which has low computational complexity and a fast convergence rate.Just a few iterations are required to find an accurate enough suboptimal trajectory-control pair for the nonlinear OCP.The results not only demonstrate the efficiency,simplicity,and high accuracy of the suggested approach,but also indicate its effectiveness in practical use.
