变量变换和几类里卡提方程的通解

研究了几类里卡提方程,利用变量变换法将其化为伯努利方程或可分离变量微分方程,从而求出方程的通解,还得到方程的一个特解.并给出了几个例子来验证主要结论.     

一类里卡提方程的两个周期解的存在性和稳定性

研究了一类里卡提方程,得到了方程的两个周期解的存在性和稳定性的充分性条件.     

变量代换法和几类里卡提方程的通解

研究了几类里卡提方程,利用变量代换法得到了方程的通解的几组充分性条件.     

特定条件下广义伯努利方程的求解方法及其应用

将伯努利方程进行推广,给出了广义伯努利方程的一般形式,进一步分别利用全微分法和变量替换法推导了在特定条件下求广义伯努利方程的隐式解公式.同时将该公式用于求解Gompterz模型和Riccati模型.     

具任意次非线性项的Camassa-Holm方程的双孤子新解

给出辅助方程、函数变换与变量分离解相结合的方法,构造了具任意次非线性项的Camassa-Holm方程的双孤子和双周期新解.首先,通过两个辅助方程、函数变换与变量分离解,将具任意次非线性项的Camassa-Holm方程的求解问题转化为非线性代数方程的求解问题.然后,借助符号计算系统Mathematica求出该方程组的解,并用辅助方程的相关结论,构造了双周期解和双孤子新解.     

具无穷时滞非线性生态竞争系统的正周期解

讨论了一类无穷时滞非线性生态竞争系统的正周期解,利用变量变换和不动点定理,得到了该系统的正周期解的存在性和存在唯一性的充分性条件,获得了一些新的结果.     

两个具有相同Hirota双线性方程的(3+1)-维演化方程的拟周期波解

应用Hirota双线性方法,构造了一个用Riemannθ函数表示的双线性方程的拟周期波解.应用到两个(3+1)-维演化方程:一个是与AKNS可积方程族相关的可积模型,另一个是著名的Jimbo-Miwa方程,分别得到了这两个演化方程的拟周期波解.     

Vakhnenko方程的光滑周期波的波长

主要研究Vakhnenko方程的光滑周期行波解的波长.通过变量变换,Vakhnenko方程可以转化为一个平面多项式微分系统.利用动力系统的临界周期分支方法研究这个多项式微分系统,其主要结果是给出了周期函数T(h)或波长函数λ(a)的单调性质.与Kd V方程比较,波长函数λ(a)单调递减到一个有限的数,而不是单调递增到无穷.结果表明,对于固定波速c,Vakhnenko方程不存在任意小或任意大波长的光滑周期行波解.     

推广的（2＋1）维浅水波方程的精确解及其混沌行为

利用Riccati方程映射法和变量分离法,得到了推广的（2＋1）维浅水波系统的变量分离解（包括孤波解、周期波解和有理函数解）.根据得到的孤波解,构造出了方程的单孤子和双孤子结构,研究了孤子的混沌行为.     

一个(3+1)维非线性方程的B(a)cklund变换和精确解

利用双Bell多项式方法构造了一个(3+1)维非线性方程的双线性形式,得到了该方程的双线性Bcklund变换和相应的Lax对.同时利用Riemann theta函数,获得了该方程的周期波解.     

Fourier expansion based recursive algorithms for periodic Riccati and Lyapunov matrix differential equations

Combining Fourier series expansion with recursive matrix formulas, new reliable algorithms to compute the periodic, non-negative, definite stabilizing solutions of the periodic Riccati and Lyapunov matrix differential equations are proposed in this paper. First, periodic coefficients are expanded in terms of Fourier series to solve the time-varying periodic Riccati differential equation, and the state transition matrix of the associated Hamiltonian system is evaluated precisely with sine and cosine series. By introducing the Riccati transformation method, recursive matrix formulas are derived to solve the periodic Riccati differential equation, which is composed of four blocks of the state transition matrix. Second, two numerical sub-methods for solving Lyapunov differential equations with time-varying periodic coefficients are proposed, both based on Fourier series expansion and the recursive matrix formulas. The former algorithm is a dimension expanding method, and the latter one uses the solutions of the homogeneous periodic Riccati differential equations. Finally, the efficiency and reliability of the proposed algorithms are demonstrated by four numerical examples.     

Quadratic optimal control of stable well-posed linear systems

We consider the infinite horizon quadratic cost minimization problem for a stable time-invariant well-posed linear system in the sense of Salamon and Weiss, and show that it can be reduced to a spectral factorization problem in the control space. More precisely, we show that the optimal solution of the quadratic cost minimization problem is of static state feedback type if and only if a certain spectral factorization problem has a solution. If both the system and the spectral factor are regular, then the feedback operator can be expressed in terms of the Riccati operator, and the Riccati operator is a positive self-adjoint solution of an algebraic Riccati equation. This Riccati equation is similar to the usual algebraic Riccati equation, but one of its coefficients varies depending on the subspace in which the equation is posed. Similar results are true for unstable systems, as we have proved elsewhere.

     

On Periodic Solutions of the Periodic

By treating the periodic Riccati equation ${\rm\dot{z}=a(t)z^2+b(t)z+c(t)}$ as a dynamical system on the sphere S, the number and stability of its periodic solutions are determined. Using properties of Moebius transformations, an exact algebraic relation is obtained between any periodic solution and any complex-valued periodic solution. This leads to a new method for constructing the periodic solutions.     

On stability of zero solution of an essentially nonlinear second-order differential equation

Small periodic (with respect to time) perturbations of an essentially nonlinear differential equation of the second order are studied. It is supposed that the restoring force of the unperturbed equation contains both a conservative and a dissipative part. It is also supposed that all solutions of the unperturbed equation are periodic. Thus, the unperturbed equation is an oscillator. The peculiarity of the considered problem is that the frequency of oscillations is an infinitely small function of the amplitude. The stability problem for the zero solution is considered. Lyapunov investigated the case of autonomous perturbations. He showed that the asymptotic stability or the instability depends on the sign of a certain constant and presented a method to compute it. Liapunov’s approach cannot be applied to nonautonomous perturbations (in particular, to periodic ones), because it is based on the possibility to exclude the time variable from the system. Modifying Lyapunov’s method, the following results were obtained. “Action–angle” variables are introduced. A polynomial transformation of the action variable, providing a possibility to compute Lyapunov’s constant, is presented. In the general case, the structure of the polynomial transformation is studied. It turns out that the “length” of the polynomial is a periodic function of the exponent of the conservative part of the restoring force in the unperturbed equation. The least period is equal to four.     

一类广义Riccati方程的三个可积判据

考虑一类广义Riccati方程,通过函数变换,在所给条件下,将这类方程等价地化为变量分离方程,从而得到了该方程可积的三个充分性判据,并给出方程通解的参数表达形式,扩大了Riccati方程的可解性范围.     

关于离散代数Riccati方程的扰动分析

1．引言若不特别说明,以下记号都是常规的,可参见山．在最优控制中占有核心地位的代数Xiccati方程（ARE）有两种基本形式：连续型的ARE（CARE）：离散型的＊＊E（＊＊RE）：其中只兄见NE贮””,GIE皿””m,GZE＊m”m,in＜2,＊T二K＞风＊T二N＞几＊2二＊2＞凡从应用角度看,主要关心＊＊E的对称半正定解．对于＊＊RE,已有大量的研究工作．特别,陈春晖门、徐树方问、Ghwimi－Laub同等研究了扰动理论．对于DARE,它与＊＊＊E的一个明显区别是：＊＊M是二次矩阵方程而＊＊＊E具有高度非线性．这种区别使得DARE远为复杂…     

具有阻尼项的二阶半线性偏微分方程解的振动性

研究了一类具有阻尼项的二阶半线性偏微分方程div(A(x)‖▽u(x)‖p-2▽u(x))+〈■(x),‖▽u(x)‖p-2▽u(x)〉+C(x)u(x)p-2u(x)=0,p>1运用偏Riccati变换和H函数方法,获得了该方程解的振动性的若干充分条件.     

时标上的三阶非线性中立型变时滞动力方程的振动性与渐进性

考虑了时标上一类三阶非线性中立型变时滞动力方程的振动性和渐进性.利用Riccati变换技巧得到了方程所有解振动或渐进趋于零的充分条件.特别地,所得新结果发展了一些已有结论.并给出了例子加以说明本文的主要结果.