寻找最优控制的具有约束算子的梯度算法

§1.具有约束算子的梯度算法 考虑连续的受控系统,其运动轨道及控制满足常微分方程组 x=f(x,u,t),x(t_0)=x_0, (1.1)其中x=(x_1,x_2,…,x_n)~T?E~n是系统的状态变量,u=(u_1,u_2,…,u_r)~T?E~r是系统的控制变量;f(·,·,·)=(f_1(·,·,·),…,f_n(·,·,·))~T是由E~n×E~r×E~1到E~n中的向量值函数;t_0是运动的起始时刻;x_0是运动的初始状态;t_f是运动的终结时刻.为简单起见,下面假定t_0,x_0,t_f均已给定。 我们把定义在区间[t_0,t_f]上的每一个在E~r中取值的分段连续函数u(t)=(u_1(t),u_2(t),…,u_r(t))~T称作系统(1.1)的一个控制。所有这样的控制的集合记作H,给定系     

一类非自治非线性系统的渐近稳定性

考虑方程组(E) (dx)/(dt)=f(t,x),其中 x=(x_1,x_2,…,x_n)~T,f(t,x)=(f_1(t,x),f_2(t,x),…,f_n(t,x))~T 在区域 D:t≥t_0≥0,‖x‖≤H,H>0;上连续可微,且 f(t,0)≡0.用 x=x(t;t_0,x_0)表示(E)的具有初值 x(t_0;t_0,x_0)=x_0的解.对于方程组(E),我们有下面的引理:引理 对于方程组(E),如果存在一个正定的函数 V(t,x)满足微分不等式(dV)/(dt)≤ω(t,V) (1)且比较方程     

带约束的最小平方和

<正> §1.引言设 x=(x_1,…,x_n)~T,f(x)=(f_1(x),…,f_m(x))~T,g(x)=(g_1(x),…,g_l(x))~T,其中 f_i(x)(i=1,…,m)与 g_j(x)(j=1,…,l)可以是非线性函数.令     

双曲型守恒律组的一类差分格式及其熵条件

其中u(x,t)=(u_1(x,t),…,u_m(x,t))~T,f(u(x,t))=(f_1(u(x,t)),…,f_m(u(x,t)))~T,f的Jacobian记为 A(u)=?f(u)/?u,具有m个实特征值λ_1(u)≤λ_2(u)≤… ≤λ_m(u)以及完备的古特征向量系{γ_k(u)}_k~m=1.对区域R~+={(x,t)|x∈(-∞,+∞),t∈     

关于广义Schrodinger型方程组第三边值问题的差分方法

本文讨论下述定解问题的差分解法 u_t(x,t)=Au_(xx)(x,t) f(u),(x,t)∈Q_T=(0,L)×(0,T) u_x(0,t)—σ_1u(0,t)=0,σ_1>0,t∈[0,T]; u_x(L,t) σ_2u(L,t)=0,σ_2>0,t∈[0,T]; u(x,0)=■(x),x∈[0,L].其中u(x,t)=(u_1(x,t),…,u_m(x,t)),f(u)=f(f_1(u),…,f_m(u)),■(x)=(■_1(x),…■_m(x))满足适定性条件,且假定     

一类非线性抛物方程的反问题

本文讨论了下述反问题 u_1-△u=β(t)f(u) γ(x,t),x∈Ω,0相似文献   

EXISTENCE OF PERIODIC SOLUTIONS OF SYSTEM OF DIFFERENCE EQUATIONS

This paper studies the nonautonomous nonlinear system of difference equationsΔx(n)=A(n)x(n)+f(n,x(n)),n∈Z,(*) where x(n)∈R~N,A(n)=(a_(ij)(n))N×N is an N×N matrix,with a-(ij)∈C(R,R) for i,j= 1,2,3,...,N,and f=(f_1,f_2,...,f_N)~T∈C(R×R~N,R~N),satisfying A(t+ω)=A(t),f(t+ω,z)=f(t,z) for any t∈R,(t,z)∈R×R~N andωis a positive integer.Sufficient conditions for the existence ofω-periodic solutions to equations (*) are obtained.     

多目标规划的真有效解

考虑问题(P) (?)其中 f(x)=(f_1(x),…,f_m(x))~T,g(x)=(g_1(x),…,g_l(x))~T,一切 f_i(x),g_j(x)为定义在 n 维欧氏空间 E_n 中某开域上的实值函数(为简单起见,不妨认为定义域就是 E_n);D为 E_l 中的凸锥.记约束集为 R={x|g(x)∈D}.设(?)∈R;Λ为 E_m 中包含原点0的闭凸锥.称(?)为有效解,若不存在 x∈R 使     

惩罚函数法

引言非线性规划问题大致可分为两类:一类是无约束最优化问题:极小化f(x),x=(x_1,…,x_n)~T∈E~n;(0.1)另一类是约束最优化问题:极小化f(x),x=(x_1,…,x_n)~T∈E~n;约束g_j(x)≤0,j=1,…,m;(0.2)h_k(x)=0,k=1,…,l。     

INVARIANT SETS AND THE HUKUHARA-KNESER PROPERTY FOR PARABOLIC SYSTEMS

In this paper we are concerned with the nonlinear boundary value problem forparabolic system(Lu=f(x,t,u,▽u),x∈Ω,0相似文献   

非光滑多目标规划非控解和真有效解

考虑问题(P) (?)其中 f(x)=(f_1(x),…,f_m(x))~T,g(x)=(g_1(x),…,g_l(x))~T,C 是 n 维欧氏空间 E_n中的闭集,f_i(x)(i=1,…,m)和 g_j(x)(j=1,…,l)为在 C 的某个邻域中的 Lipschitz函数,D 为 E_l 中的闭凸锥。记R={x|g(x)∈D,x∈C}。设 A 为 E_m 中的非零凸锥。(?)∈R 称为 f(x)(对 A)的非控解,若不存在 x∈R 使     

On the Partial Asymptotic Stability for Linear Time-Varying System

Consider system where y=(x_1,…,x_m)~T, z=(x_(m 1),…,x_n)~T, A(t),B(t), C(t) and D(t) are the corresponding continuous matrices, (suppose that p=n-m>0). Let K_p(t)=(B_1~T(t),…,B_p~T(t)) (B_1~T(t)=B~T(t),B_i~T(t)=D~T(t)B_(i-1)~T(t) B_(i-1)~T(t), i=2,…, p. Suppose that rank K_p(t)=h,let L_i(t) is a transposed matrix from the columns at which the nonzero subeterminant of order h in K_p(t) was located, L_2 satisfies that L_1(t)L_2~T=0 and det L_0(t)≠0. Let U=L(t)x, we can reduce system (1) to (2) as follows.     

一类函数奇偶性问题的简捷解法

例1 已知,f(x)为奇函数,当x>0时,f(x)=x~2-x-2,求x<0时f(x)的解析式。解∵ g_1(x)=-x与 g_2(x)=-x~2 2 (x<0) x~2-2 (x>0)都是定义在(-∞,0) ∪(0, ∞)上的奇函数,故g_1(x) g_2(x)也是定义在上述定义域的奇函数,由已知条件及符合条件的函数是唯一的,得x<0时,f(x)的解析式是-x~2-x 2。一般地,容易证明下列结论: 命题 f_1(x)与f_2(x)分别是定义在D＇∪D上的奇函数与偶函数(其中上D与D＇关于原点对称),当x∈D时,f(x)=f_1(x) f_2(x),则当x∈D＇时,     

GLOBAL EXISTENCE AND BLOW-UP OF SOLUTIONS FOR GENERALIZED POCHHAMMER-CHREE EQUATIONS

In this article,we study the initial boundary value problem of generalized Pochhammer-Chree equation u_(tt)-u_(xx)-u_(xxt)-u_(xxtt)=f(u) xx,x ∈Ω,t 0,u(x,0) = u0(x),u t(x,0)=u1(x),x ∈Ω,u(0,t) = u(1,t) = 0,t≥0,where Ω=(0,1).First,we obtain the existence of local W k,p solutions.Then,we prove that,if f(s) ∈ΩC k+1(R) is nondecreasing,f(0) = 0 and |f(u)|≤C1|u| u 0 f(s)ds+C2,u 0(x),u 1(x) ∈ΩW k,p(Ω) ∩ W 1,p 0(Ω),k ≥ 1,1 p ≤∞,then for any T 0 the problem admits a unique solution u(x,t) ∈ W 2,∞(0,T;W k,p(Ω) ∩ W 1,p 0(Ω)).Finally,the finite time blow-up of solutions and global W k,p solution of generalized IMBq equations are discussed.     

一阶微分方程的渐近平稳和周期边值问题

假定函数 f∈C[R_+×R,R],我们考虑非线性问题u'=f(t,u),u(t_0)=u_0,t_0≥0.(A)[1]附录的定理 A.1.2就(A)的渐近平稳(Asymptotic Equilibrium)给出如下的定理 A。假定 g(t,u)∈C[R_+×R_+,|R_+]对于每个 t 关于 u 单调非减,且使得|f(t,u)|≤g(t,|u|),(t,u)∈R_+×R.如果问题u′=g(t,u),u(t_0)=u_0≥0的所有解 u(t)在[t_0,∞)上有界,那么问题(A)渐近平稳.利用这个定理,[1]在假定,f(t,u)满足单边的 Lipschitz 条件     

ASYMPTOTICALLY OPTIMAL EMPIRICAL BAYES ESTIMATION FOR PARAMETERS OF TWO-SIDED TRUNCATION DISTRIBUTION FAMILIES

Consider the two-sided truncation distrbution families written in the formf(x,θ)dx=w(θ_1, θ_2)h(x)I_([θ_1,θ_2])(x)dx, where θ=(θ_1,θ_2).T(x)=(t_1(x), t_2(x))=(min(x_1,…,x_m), max(x_1, …,x_m))is a sufficient statistic and its marginal density is denoted by f(t)dμ~T. The prior distribution of θ belongs to the familyF={G:∫‖θ‖~2dG(θ)<∞}.In this paper, the author constructs the empirical Bayes estimator (EBE) of θ, φ_n (t), by using the kernel estimation of f(t). Under a quite general assumption imposed upon f(t) and h(x), it is shown that φ_n(t) is an asymptotically optimal EBE of θ.     

非线性高阶广义 Schrdinger型方程组的周期边界问题

本文考虑系数矩阵为非负定与非奇异的高阶抛物型方程组周期边界问题:=(-1)~(m 1)α_(Ij)(t) f\-1(u\-1,…u\-1),×∈R,t∈R,(Ⅰ)u\-1(x,t)|_(t=0)=_1(x),u\-1(x 1,t)=u\-1(x,t),x∈R,t∈R ,l=1,2…,J;整体解的存在与唯一问题,其中中 m1为整数,_1(x)是以1为周期的函数。J×t 阶矩阵 A(t)=(α_(xj)(t))是非负定的,即α_(lj)(t)ξ_lξ_j≥0,ξ_j∈R,i∈R_。     

INITIAL VALUE PROBLEM FOR A CLASS OF NONLINEAR PARABOLIC SYSTEM OF FOURTH-ORDER

In this paper, the periodic boundary problem and the initial value problem for the nonlinear system of parabolic type u_1=-A(x, t)u_(x4)+B(x, t)u_(x2)+(g(u))_(x2)+(grad h(u))_x+f(u)are studied, where u(x, t)=(u_1(x, t).…, u_J(x, t) is a J-dimensional unknown vector valued function, f(u) and g(u) are the J-dimensional vector valued function of u(x, t), h(u) is a scalar function of u, A(x, t) and B(x, t) are J×J matrices of functions. The existent, uniqueness and regularities of the generalized global solution and classical global solution of the problems are proved. When J=1, h(u)=0, g(u)=au~3, A=a_1, B=a_2, where a_1, a_2 a are constants, the system is a generalized diffusion model equation in population problem.     

A Remark on the Existence of Entire Large and Bounded Solutions to a(k_1, k_2)-Hessian System with Gradient Term

In this paper, we study the existence of positive entire large and bounded radial positive solutions for the following nonlinear system{S_k_1(λ(D_(u1)~2)) + a_1(|x|) |▽_(u_1) |~(k_1)= p_1(|x|) f_1(u_2) for x ∈ R~N,S_k_2(λ(D_(u_2)~2)) + a_2(|x|) |▽_(u2) |~(k_2)= p_2(|x|) f_2(u_1) for x ∈ R~N.Here S_k_i(λ(D_(u_i)~2) is the k_i-Hessian operator, a_1, p_1, f_1, a_2, p_2 and f_2 are continuous functions.     

非线性梁振动方程的周期解

一、引言考虑下述问题Ku″ A~2u M(‖A~1/2u‖~2)Au Au′=f(x,t),t>0,x∈Ω,(1.1)u|_t=0~=u_0(x),x∈Ω,(1.2)Ku′|_(t=0)=u_1(x),x∈Ω,(1.3)u=0,x∈(?)Ω,t≥0 (1.4)的ω-周期解的存在性.其中 Ω(?)R~n 为一有界光滑区域,u′=((?)u)/((?)t),u_″=((?)u)/((?)t)~2,K 为有界线性对称算子且满足(Ku,u)≥0,M∈C~1[0,∞),M(ξ)≥-β,ξ≥0.此模型最初由Woinowsky 和 Krieger 提出,方程形式为