Constructive observability inequalities for weak generalized solutions of the wave equation with elastic restraint

Problems with one-sided boundary controls and a homogeneous Robin boundary condition set on the uncontrolled end are considered in the class of strong generalized solutions of the variable coefficient wave equation. In the adjoint class of weak generalized solutions of the dual problems with one-sided observations, new constructive observability inequalities are obtained that differ from previously known ones by an optimal threshold time. It is shown that, in the considered functional classes, the estimated constants degenerate as the time interval length approaches the threshold. Numerical illustrations are given showing that the stability of approximate solutions to control problems can be substantially enhanced by taking into account a priori information contained in the resulting observability inequalities.     

Structure of extremals of autonomous convex variational problems

In this paper we study the structure of approximate solutions of autonomous variational problems with convex integrands. We are interested in a turnpike property of the extremals which is independent of the length of the interval, for all sufficiently large intervals. To have this property means, roughly speaking, that the approximate solutions of the variational problems are determined mainly by the integrand, and are essentially independent of the choice of interval and endpoint conditions.     

Continuous invertibility of a boundary control operator for the wave equation on subcritical intervals in classes of weak generalized solutions

For the wave equation, problems with two-sided boundary controls of three basic types are considered. Constructive estimates for the bounded invertibility of the control operator with known estimated constants are obtained in classes of weak generalized solutions on intervals of subcritical length. With the use of these estimates, stable numerical solutions of the considered problems on time intervals of strictly subcritical length can be found by applying a variational method. The main difference of the present estimated constants from those considered previously in the case of strong generalized solutions is that they degenerate as the time interval length approaches its critical value.     

TIME-PERIODIC ISENTROPIC SUPERSONIC EULER FLOWS IN ONE-DIMENSIONAL DUCTS DRIVING BY PERIODIC BOUNDARY CONDITIONS

We show existence of time-periodic supersonic solutions in a finite interval, after certain start-up time depending on the length of the interval, to the one space-dimensional isentropic compressible Euler equations, subjected to periodic boundary conditions. Both classical solutions and weak entropy solutions, as well as high-frequency limiting behavior are considered. The proofs depend on the theory of Cauchy problems of genuinely nonlinear hyperbolic systems of conservation laws.     

Solutions of a dissipative diffusion equation in a Hilbert space

Two types of dissipation of a diffusion process on an abstract space are discussed. The main results are the bound, critical length, and extension of solutions of a dissipative Riccati equation on a Hilbert space. The bound result enables the local solution to be extended to a global solution. The analysis of critical length provides computational and theoretical information about the maximal interval of existence of solutions. It is shown that no bounded solution can be extended beyond the critical length.     

Ein Beitrag zur Suche der Extremstelle einer strikt unimodalen Funktion

Simultaneous interval elimination is considered in order to shorten the length of the interval containing the location of the maximum value of a strictly unimodal function on the unit interval. As well known, the optimal choice of the points of evaluation is not unique, if the maximum value of spacings is used as an objective function. It is shown in this note that the mean value gives unique optimal solutions.     

Turnpike Results for a Class of Discrete-Time Optimal Control Problems Arising in Economic Dynamics

In this paper we establish turnpike results for a class of discrete-time optimal control problems. These control problems arise in economic dynamics and describe a nonstationary model proposed by Robinson, Solow and Srinivasan. We study the structure of approximate solutions which is independent of the length of the interval, for all sufficiently large intervals.     

Quenching of solutions to a class of semilinear parabolic equations with boundary degeneracy

This paper concerns the quenching phenomenon of solutions to a class of semilinear parabolic equations with boundary degeneracy. In the case that the degeneracy is not strong, it is shown that there exists a critical length, which is positive, such that the solution exists globally in time if the length of the spatial interval is less than it, while quenches in a finite time if the length of the spatial interval is greater than it. Whereas in the case that the degeneracy is strong enough, the solution must be quenching in a finite time no matter how long the spatial interval is. Furthermore, for each quenching solution, the set of quenching points is determined and it is proved that its derivative with respect to the time must blow up at the quenching time.     

The non-relativistic limit of Euler–Maxwell equations for two-fluid plasma

This paper is concerned with two-fluid time-dependent non-isentropic Euler–Maxwell equations in a torus for plasmas or semiconductors. By using the method of formal asymptotic expansions, we analyze the non-relativistic limit for periodic problems with the prepared initial data. It is shown that the small parameter problems have unique solutions existing in the finite time interval where the corresponding limit problems (compressible Euler–Poisson equations) have smooth solutions. Moreover, the formal limit is rigorously justified by an iterative scheme and an analysis of asymptotic expansions up to any order.     

On a class of second order variational problems with constraints

We study the structure of optimal solutions for a class of constrained, second order variational problems on bounded intervals. We show that, for intervals of length greater than some positive constant, the optimal solutions are bounded inC ¹ by a bound independent of the length of the interval. Furthermore, for sufficiently large intervals, the ‘mass’ and ‘energy’ of optimal solutions are almost uniformly distributed.     

Tracking nonperiodic trajectories with the overtaking criterion

A tracking problem on an infinite time interval is studied, where the plant is linear with quadratic cost, and the tracked trajectory is not necessarily period. Optimal solutions with respect to the overtaking criterion are studied. Existence and uniqueness of such optimal solutions are proved and they are shown to be given by a linear feedback law which is the same as in the periodic case. A close relation between the solutions of tracking problems where the tracked trajectories are different only for very large times is established.     

Scattering of Sound by Large Finite Geometries

Diffraction problems with finite geometries do not usually haveexact solutions and so asymptotic methods, which require a largeor small parameter, are employed to obtain approximate results.This paper presents a formal method for approximating the acousticpotential when the finite length in the geometry is large comparedto an acoustic wavelength. The method presented is exactly analogousto other approaches, including the modified Wiener-Hopf technique,but is advantageous because of its relative ease of use andapplicability to many problems. This is shown by using the methodon problems with resonances and complicated geometries.     

Initial Boundary Value Problems of the Camassa–Holm Equation

In this paper we study initial boundary value problems of the Camassa–Holm equation on the half line and on a compact interval. Using rigorously the conservation of symmetry, it is possible to convert these boundary value problems into Cauchy problems for the Camassa–Holm equation on the line and on the circle, respectively. Applying thus known results for the latter equations we first obtain the local well-posedness of the initial boundary value problems under consideration. Then we present some blow-up and global existence results for strong solutions. Finally we investigate global and local weak solutions for the equation on the half line and on a compact interval, respectively. An interesting result of our analysis shows that the Camassa–Holm equation on a compact interval possesses no nontrivial global classical solutions.     

New solitary wave solutions and periodic wave solutions for the compound KdV equation

In this paper, for compound KdV equation, four new solitary wave solutions in the form of hyperbolic secant function and six periodic wave solutions in the form of cosine function are obtained by using undetermined coefficient method. On three different layers, the velocity interval which ensures that bell-shaped solitary wave solutions and periodic wave solutions exist synchronously is obtained, respectively. The length of the interval is related to coefficients of the two nonlinear terms.     

A Generalized Time–Cost Trade-Off Transportation Problem

In the classical transportation problem if the unit costs and transportation durations are considered, the time-cost trade-off solutions can be determined by the well-known threshold approach assuming that all the transportations are permitted to be simultaneous in time. If all the unit costs are linear functions of time over a specified interval of time, a parametric technique can be applied for identifying all the time-cost trade-off solutions pertaining to this interval. In this paper, the unit costs considered are piecewise linear non-increasing functions of time and transportations are allowed to be simultaneous. It is shown that a parametric method involving a finite sequence of parametric transportation problems reveals all the time-cost trade-off solutions of this generalized trade-off problem. Computational experience is included. If the transportation problem has considerable degeneracy, the parametric approach may pose some computational difficulty. This difficulty can be reduced by using an alternative method involving the bicriteria optimization approach of Aneja and Nair. Also, a direct method is outlined for the case where a finite set of discrete alternatives of unit cost-time pairs is available.     

Uniqueness for second order nonlinear boundary value problems with applications to almost periodic solutions

Summary The problem of uniqueness of bounded solutions for various nonlinear boundary value problems is considered. Results for problems with an unbounded interval are obtained. The results obtained for the uniqueness of bounded solutions on the interval (−∞, ∞) are applied to the problem of proving the existence of almost periodic solutions. Entrata in Redazione il 4 aprile 1968.     

Interval Branch and Bound with Local Sampling for Constrained Global Optimization

In this article, we introduce a global optimization algorithm that integrates the basic idea of interval branch and bound, and new local sampling strategies along with an efficient data structure. Also included in the algorithm are procedures that handle constraints. The algorithm is shown to be able to find all the global optimal solutions under mild conditions. It can be used to solve various optimization problems. The local sampling (even if done stochastically) is used only to speed up the convergence and does not affect the fact that a complete search is done. Results on several examples of various dimensions ranging from 1 to 100 are also presented to illustrate numerical performance of the algorithm along with comparison with another interval method without the new local sampling and several noninterval methods. The new algorithm is seen as the best performer among those tested for solving multi-dimensional problems.     

An example of non-decreasing solution for the KdV equation posed on a bounded interval

An initial-boundary value problem for the KdV equation posed on a bounded interval is considered. The theory of Jacobi elliptic functions is used to obtain a new kind of stationary waves which are spatially periodic with a period equal to an interval length. The properties of those solutions are studied.     

随机补货间隔且需求依赖库存水平的库存控制模型

定期补货库存模型在实践中被广泛使用,尤其是在单一供应商中购买多种不同产品的库存系统中更为常见.然而,大多数定期补货库存模型都假设补货的时间间隔是恒定不变的.但在实践中,补货的时间间隔也可能是一个随机的时间长度.提出了一个随机补货时间间隔和需求依赖于当前展示库存水平的库存控制模型,且补货间隔服从指数分布和均匀分布,同时允许短缺发生并且短缺量部分延期供给,并研究了模型最优解的存在性与唯一性.最后,给出了数值算例来说明模型在实际中的应用.     

一类带扰动的随机脉冲泛函微分方程解的渐近性北大核心CSCD

该文讨论了一类带扰动的随机脉冲泛函微分方程解的渐近性.通过比较扰动方程的解和原方程的解,得到了两者逼近的充分条件.首先,两者在有限的时间区间上相互逼近;其次,当扰动趋于零时,区间长度趋于无穷大,在这个区间上两个解仍然是相互逼近的.最后,举例说明了结果的有效性.