Domain Decomposition of Optimal Control Problems for Dynamic Networks of Elastic Strings

We consider optimal control problems related to exact- and approximate controllability of dynamic networks of elastic strings. In this note we concentrate on problems with linear dynamics, no state and no control constraints. The emphasis is on approximating target states and velocities in part of the network using a dynamic domain decomposition method (d³m) for the optimality system on the network. The decomposition is established via a Uzawa-type saddle-point iteration associated with an augmented Lagrangian relaxation of the transmission conditions at multiple joints. We consider various cost functions and prove convergence of the infinite dimensional scheme for an exemplaric choice of the cost. We also give numerical evidence in the case of simple exemplaric networks.     

Global Null Controllability of the 1-Dimensional Nonlinear Slow Diffusion Equation

The authors prove the global null controllability for the 1-dimensional nonlinear slow diffusion equation by using both a boundary and an internal control. They assume that the internal control is only time dependent. The proof relies on the return method in combination with some local controllability results for nondegenerate equations and rescaling techniques.     

Exact Null Controllability of a Nonlinear Thermoelastic Contact Problem

We study the controllability properties of a nonlinear parabolic system that models the temperature evolution of a one-dimensional thermoelastic rod that may come into contact with a rigid obstacle. Basically the system dynamics is described by a one-dimensional nonlocal heat equation with a nonlinear and nonlocal boundary condition of Newmann type. We focus on the control problem and treat the case when the control is distributed over the whole space domain. In this case the system is proved to be exactly null controllable provided the parameters of the system are smooth. The proof is based on changing the control variable and using Aubins Compactness Lemma to obtain an invariant set for the linearized controllability map. Then, by proving that the found solution is sufficiently smooth, we get the null controllability for the original system.     

Controllability of the 3D Compressible Euler System

The paper is devoted to the controllability problem for 3D compressible Euler system. The control is a finite-dimensional external force acting only on the velocity equation. We show that the velocity and density of the fluid are simultaneously controllable. In particular, the system is approximately controllable and exactly controllable in projections.     

A comment on the papers: Controllability results for functional semilinear differential inclusions in Fréchet spaces [Nonlinear Anal. 61 (3) (2005) 405–423] and Controllability of impulsive neutral functional differential inclusions with infinite delay [Nonlinear Anal. 60 (8) (2005) 1533–1552]

In this note we make some observations relating two papers recently published in Nonlinear Analysis Theory, Methods & Applications.     

Exact Boundary Observability of Unsteady Supercritical Flows in a Tree-Like Network of Open Canals

The author establishes the exact boundary observability of unsteady supercritical flows in a tree-like network of open canals with general topology. An implicit duality between the exact boundary controllability and the exact boundary observability is also given for unsteady supercritical flows.     

Controllability for Neutral Stochastic Functional Integrodifferential Infinite Delay Systems in Abstract Space

In this paper, sufficient conditions are given for the controllability of a class of neutral stochastic functional integrodifferential equations with infinite delay in abstract space. The Nussbaum fixed point theorem is adapted to obtain the controllability results, which greatly extends the previous results to the stochastic settings with the help of analytic semigroups. An example is provided to illustrate the theory.     

Exact boundary controllability for quasilinear wave equations in a planar tree-like network of strings

In this paper the local exact boundary controllability for quasilinear wave equations on a planar tree-like network of strings is established and the number of boundary controls is equal to the number of simple nodes minus 1.     

Exact Boundary Controllability on a Tree-Like Network of Nonlinear Planar Timoshenko Beams

This paper concerns a system of equations describing the vibrations of a planar network of nonlinear Timoshenko beams.The authors derive the equations and appropriate nodal conditions,determine equilibrium solutions and,using the methods of quasilinear hyperbolic systems,prove that for tree-like networks the natural initial-boundary value problem admits semi-global classical solutions in the sense of Li [Li,T.T.,Controllability and Observability for Quasilinear Hyperbolic Systems,AIMS Ser.Appl.Math.,vol 3,American Institute of Mathematical Sciences and Higher Education Press,2010] existing in a neighborhood of the equilibrium solution.The authors then prove the local exact controllability of such networks near such equilibrium configurations in a certain specified time interval depending on the speed of propagation in the individual beams.     

Exact Boundary Controllability and Exact Boundary Observability for a Coupled System of Quasilinear Wave Equations

Based on the theory of semi-global classical solutions to quasilinear hyperbolic systems, the authors apply a unified constructive method to establish the local exact boundary(null) controllability and the local boundary(weak) observability for a coupled system of 1-D quasilinear wave equations with various types of boundary conditions.     

双曲系统的节点状态的精确边界能控性

在此综述性文章中,我们将回顾关于节点状态的精确边界能控性的已有结果,并对此主题之进一步研究给出若干建议     

Exact Controllability for a Class of Nonlinear Evolution Control Systems

In this paper, we study the exact controllability of the nonlinear control systems. The controllability results by using the monotone operator theory are es-tablished. No compactness assumptions are im...     

含分布滞量的非线性双曲偏微分方程组解的振动性

研究了一类具有连续分布时滞变量的非线性双曲型偏微分方程组解的振动性.获得了该方程组在Robin边值条件和Dircichlet边值条件下解振动的充分条件.     

Exact boundary observability for a kind of second order quasilinear hyperbolic systems and its applications

In this paper, by means of a constructive method based on the existence and uniqueness of the semi-global C² solution, we establish the local exact boundary observability for a kind of second order quasilinear hyperbolic systems. As an application, we obtain the one-sided local exact boundary observability for a kind of first order quasilinear hyperbolic systems of diagonal form with boundary conditions in which the diagonal variables corresponding to the positive eigenvalues and those corresponding to the negative eigenvalues are decoupled.     

Local Exact Boundary Synchronization for a Kind of First Order Quasilinear Hyperbolic Systems

In this paper, the synchronization for a kind of first order quasilinear hyperbolic system is taken into account. In this system, all the equations share the same positive wave speed. To realize the synchronization, a uniform constructive method is adopted, rather than an iteration process usually used in dealing with nonlinear systems. Furthermore, similar results on the exact boundary synchronization by groups can be obtained for a kind of first order quasilinear hyperbolic system of equations with different positive wave speeds by groups.     

河渠非定常流的精确边界能控性

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。     

On Controllability of an Elastic Ring

We study the exact controllability problem for a ring under stretching tension that varies in time. We are looking for a couple of forces, which drive the state solution to rest. We show that applying two forces is necessary for controllability and the ring is controllable in the time interval greater than the optical length of the string. We also explain why one force would not be enough to control the ring. We use the method of moments to reduce the controllability problem to a moment problem for the controlling forces. The solution of that problem is based on an auxiliary basis property result. Both method of moments and proof of the basis property are developed for the model with time-dependent parameters. S.A.’s research was supported in part by the NSF, grant ARC–0724860. B.B.’s research was supported in part by University of Tennessee at Chattanooga Faculty Research Grant. S.I.’s research was supported in part by the Russian Foundation for Basic Research, grants 08-01-00595a and 08-01-00676a.     

On the Decay of Solutions for Some Nonlinear Dissipative Hyperbolic Equations

In this paper we show the decay of solutions to the initial-boundary value problem for somenonlinear hyperbolic equation with a nonlinear dissipative term,by using a difference inequality.     

具有Hille-Yosida算子的非线性随机脉冲泛函微分包含的可控性

本文研究具有Hille-Yosida算子的非线性随机脉冲泛函微分包含的可控性.假设多值非线性和脉冲函数满足由非紧性测度表示的正则性条件,利用非紧性测度理论和多值凝聚不动点定理,得到这类微分包含的可控性的充分条件.