Duality for a Class of Multiobjective Control Problems

In this paper, a class of multiobjective control problems is considered, where the objective and constraint functions involved are f(t, x(t), ?(t), y(t), z(t)) with x(t) ∈ Rⁿ, y(t) ∈ Rⁿ, and z(t) ∈ R^m, where x(t) and z(t) are the control variables and y(t) is the state variable. Under the assumption of invexity and its generalization, duality theorems are proved through a parametric approach to related properly efficient solutions of the primal and dual problems.     

Boundary blow-up for differential equations with indefinite weight

We obtain results of existence and multiplicity of solutions for the second-order equation x″+q(t)g(x)=0, with x(t) defined for all t∈]0,1[ and such that x(t)→+∞ as t→0⁺ and t→1⁻. We assume g having superlinear growth at infinity and q(t) possibly changing sign on [0,1].     

Boundary regularity for viscosity solutions to degenerate elliptic PDEs

Sufficient conditions are given for the solutions to the (fully nonlinear, degenerate) elliptic equation F(x,u,Du,D²u)=0 in Ω to satisfy |u(x)−u(y)|?Cα|x−y| for some α∈(0,1) when x∈Ω and y∈∂Ω.     

A finite difference solution to a two-dimensional parabolic inverse problem

In this paper, we consider the problem of finding u = u(x, y, t) and p = p(t) which satisfy u_t = u_xx + u_yy + p(t)u + ? in R × [0, T], u(x, y, 0) = f(x, y), (x, y) ∈ R = [0, 1] × [0, 1], u is known on the boundary of R and u(x^∗, y^∗, t) = E(t), 0 < t ? T, where E(t) is known and (x^∗, y^∗) is a given point of R. Through a function transformation, the nonlinear two-dimensional diffusion problem is transformed into a linear problem, and a backward Euler scheme is constructed. It is proved by the maximum principle that the scheme is uniquely solvable, unconditionally stable and convergent in L_∞ norm. The convergence orders of u and p are of O(τ + h²). The impact of initial data errors on the numerical solution is also considered. Numerical experiments are presented to illustrate the validity of the theoretical results.     

Minimum augmentation of local edge-connectivity between vertices and vertex subsets in undirected graphs

Given an undirected multigraph G=(V,E), a family W of sets W⊆V of vertices (areas), and a requirement function r:W→Z⁺ (where Z⁺ is the set of nonnegative integers), we consider the problem of augmenting G by the smallest number of new edges so that the resulting graph has at least r(W) edge-disjoint paths between v and W for every pair of a vertex v∈V and an area W∈W. So far this problem was shown to be NP-hard in the uniform case of r(W)=1 for each W∈W, and polynomially solvable in the uniform case of r(W)=r?2 for each W∈W. In this paper, we show that the problem can be solved in time, even if r(W)?2 holds for each W∈W, where n=|V|, m=|{{u,v}|(u,v)∈E}|, p=|W|, and r^*=max{r(W)∣W∈W}.     

An inverse problem for computing a leading coefficient in the Sturm-Liouville operator by using the boundary data

We consider an inverse problem for identifying a leading coefficient α(x) in −(α(x)y′(x))′ + q(x)y(x) = H(x), which is known as an inverse coefficient problem for the Sturm-Liouville operator. We transform y(x) to u(x, t) =  (1 + t)y(x) and derive a parabolic type PDE in a fictitious time domain of t. Then we develop a Lie-group adaptive method (LGAM) to find the coefficient function α(x). When α(x) is a continuous function of x, we can identify it very well, by giving boundary data of y, y′ and α. The efficiency of LGAM is confirmed by comparing the numerical results with exact solutions. Although the data used in the identification are limited, we can provide a rather accurate solution of α(x).     

Automatic Controls for Nonlinear Dynamical Systems with Lipschitzian Trajectories

We would like to investigate on the solution to the automatic control problem given by the differential equation y′(t) = f(t, y(t), w(t)) for a given initial function x in the initial domain D(x, ω, Y) for almost all t in the interval I, with controls given by w(t) = g(t, y(t), T(y)(t)), where T is a nonanticipating and Lipschitzian operator. The result will be generalized for a dynamical system y′(t) = f(t, y(t), T(y), u(t)).     

Homogeneous tri-additive forms and derivations

Gleason [A.M. Gleason, The definition of a quadratic form, Amer. Math. Monthly 73 (1966) 1049-1066] determined all functionals Q on K-vector spaces satisfying the parallelogram law Q(x+y)+Q(x-y)=2Q(x)+2Q(y) and the homogeneity Q(λx)=λ²Q(x). Associated with Q is a unique symmetric bi-additive form S such that Q(x)=S(x,x) and 4S(x,y)=Q(x+y)-Q(x-y). Homogeneity of Q corresponds to that of S: S(λx,λy)=λ²S(x,y). The associated S is not necessarily bi-linear.Let V be a vector space over a field K, char(K)≠2,3. A tri-additive form T on V is a map of V³ into K that is additive in each of its three variables. T is homogeneous of degree 3 if T(λx,λy,λz)=λ³T(x,y,z) for all .We determine the structure of tri-additive forms that are homogeneous of degree 3. One of the keys to this investigation is to find the general solution of the functional equation

F(t)+t³G(1/t)=0,     

On solvability of two-dimensional Lp-Minkowski problem

Given p≠0 and a positive continuous function g, with g(x+T)=g(x), for some 0<T<1 and all real x, it is shown that for suitable choice of a constant C>0 the functional has a minimizer in the class of positive functions u∈C¹(R) for which u(x+T)=u(x) for all x∈R. This minimizer is used to prove the existence of a positive periodic solution y∈C²(R) of two-dimensional L_p-Minkowski problem y^1−p(x)(y″(x)+y(x))=g(x), where p∉{0,2}.     

Multiple solutions of some boundary value problems with parameters

In this paper, we study the existence and multiplicity of nontrivial solutions for the following second-order Dirichlet nonlinear boundary value problem with odd order derivative: −u^″(t)+au^′(t)+bu(t)=f(t,u(t)) for all t∈[0,1] with u(0)=u(1)=0, where a,b∈R¹, f∈C¹([0,1]×R¹,R¹). By using the Morse theory, we impose certain conditions on f which are able to guarantee that the problem has at least one nontrivial solution, two nontrivial solutions and infinitely many solutions, separately.     

Nonlinear dissipative wave equations with space-time dependent potential

We study the long time behavior of solutions for damped wave equations with absorption. These equations are generally accepted as models of wave propagation in heterogeneous media with space-time dependent friction a(t,x)u_t and nonlinear absorption |u|^p−1u (Ikawa (2000) [17]). We consider 1<p<(n+2)/(n−2) and separable a(t,x)=λ(x)η(t) with λ(x)∼(1+|x|)^−α and η(t)∼(1+t)^−β satisfying conditions (A1) or (A2) which are given. The main results are precise decay estimates for the energy, L² and L^p+1 norms of solutions. We also observe the following behavior: if α∈[0,1), β∈(−1,1) and 0<α+β<1, there are three different regions for the decay of solutions depending on p; if α∈(−∞,0) and β∈(−1,1), there are only two different regions for the decay of the solutions depending on p.     

Semilinear differential inclusions via weak topologies

The paper deals with the multivalued initial value problem x^′∈A(t,x)x+F(t,x) for a.a. t∈[a,b], x(a)=x₀ in a separable, reflexive Banach space E. The nonlinearity F is weakly upper semicontinuous in x and the investigation includes the case when both A and F have a superlinear growth in x. We prove the existence of local and global classical solutions in the Sobolev space W1,p([a,b],E) with 1<p<∞. Introducing a suitably defined Lyapunov-like function, we are able to investigate the topological structure of the solution set. Our main tool is a continuation principle in Frechét spaces and we prove the required pushing condition in two different ways. Some examples complete the discussion.     

Positive and dead core solutions of singular BVPs for third-order differential equations

The paper discusses the existence of positive and dead core solutions of the singular differential equation (?(u^″))^′=λf(t,u,u^′,u^″) satisfying the boundary conditions u(0)=A, u(T)=A, min{u(t):t∈[0,T]}=0. Here λ is a nonnegative parameter, A is a positive constant and the Carathéodory function f(t,x,y,z) is singular at the value 0 of its space variable y.     

On the determination of the basin of attraction of a periodic orbit in two-dimensional systems

We consider the general nonlinear differential equation with x∈R² and develop a method to determine the basin of attraction of a periodic orbit. Borg's criterion provides a method to prove existence, uniqueness and exponential stability of a periodic orbit and to determine a subset of its basin of attraction. In order to use the criterion one has to find a function W∈C¹(R²,R) such that LW(x)=W^′(x)+L(x) is negative for all x∈K, where K is a positively invariant set. Here, L(x) is a given function and W^′(x) denotes the orbital derivative of W. In this paper we prove the existence and smoothness of a function W such that LW(x)=−μ‖f(x)‖. We approximate the function W, which satisfies the linear partial differential equation W^′(x)=〈∇W(x),f(x)〉=−μ‖f(x)‖−L(x), using radial basis functions and obtain an approximation w such that Lw(x)<0. Using radial basis functions again, we determine a positively invariant set K so that we can apply Borg's criterion. As an example we apply the method to the Van-der-Pol equation.     

A non-separable solution of the diffusion equation based on the Galerkin’s method using cubic splines

The two dimensional diffusion equation of the form is considered in this paper. We try a bi-cubic spline function of the form as its solution. The initial coefficients C_i,j(0) are computed simply by applying a collocation method; C_i,j = f(x_i, y_j) where f(x, y) = u(x, y, 0) is the given initial condition. Then the coefficients C_i,j(t) are computed by X(t) = e^tQX(0) where X(t) = (C_0,1, C_0,1, C_0,2, … , C_0,N, C_1,0, … , C_N,N) is a one dimensional array and the square matrix Q is derived from applying the Galerkin’s method to the diffusion equation. Note that this expression provides a solution that is not necessarily separable in space coordinates x, y. The results of sample calculations for a few example problems along with the calculation results of approximation errors for a problem with known analytical solution are included.     

Periodic solutions to one-dimensional wave equation with x-dependent coefficients

In this paper, we study the nonlinear one-dimensional periodic wave equation with x-dependent coefficients u(x)ytt−(ux(x)yx)+g(x,t,y)=f(x,t) on (0,π)×R under the boundary conditions a₁y(0,t)+b₁yx(0,t)=0, a₂y(π,t)+b₂yx(π,t)=0 ( for i=1,2) and the periodic conditions y(x,t+T)=y(x,t), yt(x,t+T)=yt(x,t). Such a model arises from the forced vibrations of a nonhomogeneous string and the propagation of seismic waves in nonisotropic media. A main concept is the notion “weak solution” to be given in Section 2. For T is the rational multiple of π, we prove some important properties of the weak solution operator. Based on these properties, the existence and regularity of weak solutions are obtained.     

Existence of solutions to a class of nonlinear second order two-point boundary value problems

In this paper, the existence and multiplicity results of solutions are obtained for the second order two-point boundary value problem −u^″(t)=f(t,u(t)) for all t∈[0,1] subject to u(0)=u^′(1)=0, where f is continuous. The monotone operator theory and critical point theory are employed to discuss this problem, respectively. In argument, quadratic root operator and its properties play an important role.     

An optimal filtering method for stable analytic continuation

In this paper, we consider the problem of numerical analytic continuation of an analytic function f(z)=f(x+iy) on a strip domain Ω⁺={z=x+iy∈C∣x∈R,0<y<y₀}, where the data is given approximately only on the real axis y=0. This problem is severely ill-posed: the solution does not depend continuously on the given data. A novel method (filtering) is used to solve this problem and an optimal error estimate with Hölder type is proved. Numerical examples show that this method works effectively.