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1.
We consider the problem of boundary control by a force applied to one end of a string in the case of a given force mode at
the other end. The problem is studied in the sense of the generalized solution of the corresponding mixed initial-boundary
value problem in the Sobolev space. We also solve the problem of choosing an optimal boundary control in the set of all admissible
controls. The generalized solution of the mixed initial-boundary value problem is constructed in closed form, and its uniqueness
is proved. 相似文献
2.
E. I. Moiseev A. A. Kholomeeva 《Proceedings of the Steklov Institute of Mathematics》2012,276(1):153-160
The problem of optimal boundary control by displacement at one end of a string under a specified force mode at the other end is studied in the sense of a generalized solution of the corresponding mixed initial-boundary value problem from a Sobolev space. The problem of choosing an optimal boundary control from an infinite number of admissible controls is solved. A generalized solution of the mixed initial-boundary value problem is constructed explicitly and the uniqueness of the solution is proved. 相似文献
3.
We study a problem of optimal boundary control of vibrations of a one-dimensional elastic string, the objective being to bring the string from an arbitrary initial state into an arbitrary terminal state. The control is by the displacement at one end of the string, and a homogeneous boundary condition containing the time derivative is posed at the other end. We study the corresponding initial-boundary value problem in the sense of a generalized solution in the Sobolev space and prove existence and uniqueness theorems for the solution. An optimal boundary control in the sense of minimization of the boundary energy is constructed in closed analytic form. 相似文献
4.
The article considers uniqueness of the finite-energy generalized solution of the mixed boundary-initial value problem for
a wave equation with elastic clamping. Excitation and relaxation problems for the wave process are solved analytically. The
results are applied to solve the problem of boundary optimal control of the wave process. 相似文献
5.
A. A. Kholomeeva 《Differential Equations》2008,44(5):717-721
We consider the optimization problem for a nonlocal boundary control of vibrations of an elastic string with fixed right end for an arbitrary sufficiently large time interval that is a multiple of twice the string length. We prove the existence and uniqueness of a generalized solution of the first boundary value problem and indicate an explicit expression for the solution. The optimal control is found in closed form. 相似文献
6.
A. A. Nikitin 《Differential Equations》2013,49(5):645-653
We prove the uniqueness of a generalized solution of an initial-boundary value problem for the wave equation with boundary conditions of the third and second kind. In addition, we find a closed-form expression for the analytic solution of that problem with zero initial data. The result plays an important role in the investigation of the boundary control problem. We show how to use the obtained solution for the investigation of the boundary control problem in the case of subcritical time intervals for which the solution of the boundary control problem, if it exists at all, is unique. We obtain necessary and sufficient conditions for the existence of a unique solution in a class admitting the existence of finite energy. 相似文献
7.
8.
M. N. Sadikhov K. I. Khudaverdiev 《Computational Mathematics and Mathematical Physics》2011,51(3):404-422
As is well known, many problems of mathematical physics are reduced to one- and multidimensional initial and initial-boundary
value problems for, generally speaking, strongly nonlinear pseudoparabolic equations. The existence (local and global) and
uniqueness of a classical solution to a one-dimensional mixed problem with homogeneous Riquier-type boundary conditions are
analyzed for a class of fifth-order semilinear pseudoparabolic equations of the Korteweg-de Vries-Burgers type. For the classical
solution of the mixed problem, a uniqueness theorem is proved using the Gronwall-Bellman inequality, a local existence theorem
is proved by combining the generalized contraction mapping principle with the Schauder fixed point principle, and a global
existence theorem is proved by applying the method of a priori estimates. 相似文献
9.
The present work deals with the formulation of the boundary integral equations for the solution of equations under linear theory of generalized thermoelastic diffusion in a three-dimensional Euclidean space. A mixed initial-boundary value problem is considered in the present context and the fundamental solutions of the corresponding coupled differential equations are obtained in the Laplace transform domain by employing the treatment of scalar and vector potential theory. A reciprocal relation of Betti type is established. Then we formulate the boundary integral equations for generalized thermoelastic diffusion on the basis of these fundamental solutions and the reciprocal relation. 相似文献
10.
We consider a boundary value problem for the wave equation with given initial conditions and with boundary conditions of the second kind at one end of the string and boundary conditions of the first kind at the other end of the string. We assume the boundary conditions to ensure that the solution of the problem (in the class of generalized functions) satisfying the initial conditions at the initial time t = 0 satisfies given terminal conditions at the terminal time t = T. We clarify the relationship between the functions µ(t) and ν(t) in the boundary conditions and the given functions specifying the initial and terminal states. We obtain closed-form analytic expressions for the functions µ(t) and ν(t) minimizing the boundary energy functional. 相似文献
11.
V. V. Tikhomirov 《Computational Mathematics and Modeling》2005,16(1):72-82
The spectral method is applied to solve the mixed initial boundary-value problem for a parabolic equation with nonhomogeneous boundary conditions, one of which is nonlocal. We prove existence and uniqueness of the generalized solution of this problem in the Sobolev class W
2
1,0
and represent it as a biorthogonal series. We also consider optimal control by the right-hand side of the equation, which is constructed as a biorthogonal series in the root functions of the spectral problem.Translated from Nelineinaya Dinamika i Upravlenie, No. 2, pp. 209–220, 2002. 相似文献
12.
A. Yu. Mokin 《Differential Equations》2013,49(1):59-67
We consider an initial-boundary value problem for the heat equation with a nonlocal two-point boundary condition containing a parameter. By separating the variables in an auxiliary function system, we construct a regular solution. We obtain sufficient conditions for the absolute and uniform convergence of the series in the auxiliary system. We prove conditions close to necessary ones for the existence of a regular solution of the initial-boundary value problem. 相似文献
13.
Gunilla Kreiss 《Mathematical Methods in the Applied Sciences》1990,13(6):493-514
In this paper we consider initial-boundary value problems for systems with a small parameter ?. The problems are mixed hyperbolic–parabolic when ? > 0 and hyperbolic when ? = 0. Often the solution can be expanded asymptotically in ? and to first approximation it consists of the solution of the corresponding hyperbolic problem and a boundary layer part. We prove sufficient conditions for the expansion to exist and give estimates of the remainder. We also examine how the boundary conditions should be choosen to avoid O(1) boundary layers. 相似文献
14.
We consider a mixed problem with Dirichlet homogeneous boundary conditions and nonzero initial conditions for a nonlinear coupled evolution system of equations in a domain unbounded in time. The conditions of existence of a generalized solution are obtained. It is shown that no solution of the problem exists at a negative initial value of the energy integral. 相似文献
15.
Xiu-Min Shao & Zhong-Min Yuan 《计算数学(英文版)》1991,9(2):135-148
This paper deals with a coefficient inverse problem of a system of ODEs whose coefficient matrix is the so-called generalized negative definite matrix. To solve the problem, an initial-boundary value problem of a hyperbolic system of PDEs is constructed. The existence and uniqueness of its solution and its asymptotic convergence with respect to one of the variables to the original inverse problem are proved. As a result, the solution of the inverse problem is reduced to the solution of the direct problem. A few numerical examples were solved to show the effectiveness of the method. 相似文献
16.
LOCAL EXACT BOUNDARY CONTROLLABILITY FOR A CLASS OF QUASILINEAR HYPERBOLIC SYSTEMS 总被引:20,自引:9,他引:11
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17.
L. S. Pul’kina 《Differential Equations》2008,44(8):1119-1125
We consider a mixed initial-boundary value problem for a multidimensional (with respect to the space variables) hyperbolic equation with a nonlocal boundary condition containing an integral of the desired solution. We prove the unique solvability of the problem in the space W 2 1 . 相似文献
18.
河渠非定常流的精确边界能控性 总被引:1,自引:1,他引:0
于勇 《高校应用数学学报(A辑)》2004,19(4):379-393
借助于一阶拟线性双曲型方程组混合初边值问题的半整体C^1解理论对单个河道及弦状网络河道中的非定常流动分别讨论了在闸门边界条件下的精确边界能控性问题,并对在泄洪边界条件下的精确边界能控性进行了相应的讨论。 相似文献
19.
We solve a Dirichlet boundary value problem for the Klein–Gordon equation posed in a time‐dependent domain. Our approach is based on a general transform method for solving boundary value problems for linear and integrable nonlinear PDE in two variables. Our results consist of the inversion formula for a generalized Fourier transform, and of the application of this generalized transform to the solution of the boundary value problem. 相似文献
20.
A. N. Bogolyubov Jiexing Gao Yu. V. Mukhartova 《Computational Mathematics and Mathematical Physics》2011,51(9):1614-1621
The problem of excitation of electromagnetic oscillations by a given distribution of charges and currents in a domain with
inhomogeneous chiral filling is examined. The domain in which the problem is considered may either be finite with a perfectly
conducting boundary surface or be the complement of a perfectly conducting bounded body. A special functional space is defined
on which a generalized initial-boundary value problem is formulated. The Galerkin method is used to prove the existence and
uniqueness of a weak solution of this problem. 相似文献