Boundary Controllability of a Dynamical System Governed by the Wave Equation on a Class of Graphs (Trees)

The boundary control problem for the wave equation on a planar graph consisting of strings of variable densities with Dirichlet control at boundary vertices is considered. The exact controllability in L₂-classes of controls and states is established in the case where the graph is a tree; a sharp estimate of the time of controllability is given. Bibliography: 7 titles. Dedicated to J. E. Lagnese on the occasion of his 65th birthday __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 308, 2004, pp. 23–47.     

Exact Boundary Controllability for a Kind of Second-Order Quasilinear Hyperbolic Systems

Based on the theory of semi-global C ¹ solution and the local exact boundary controllability for first-order quasilinear hyperbolic systems, the local exact boundary controllability for a kind of second-order quasilinear hyperbolic systems is obtained by a constructive method.     

Optimal boundary displacement control of string vibrations with a nonlocal evenness condition of the second kind

We study the behavior of a string with the nonlocal boundary condition u _x (l, t) = u _x ($ x^\circ $ x^\circ , t). A displacement control u(0, t) = μ(t) bringing the string from an arbitrarily given initial state to an arbitrarily given terminal state is applied at the left endpoint of the string. For the initial and terminal functions, we find necessary and sufficient conditions for the controllability of the string. Under these conditions, we carry out optimization; i.e., of all admissible controls, we choose a control minimizing the boundary energy integral.     

Exact boundary controllability of nodal profile for 1-D quasilinear wave equations

Based on the theory of semi-global C ² solution for 1-D quasilinear wave equations, the local exact boundary controllability of nodal profile for 1-D quasilinear wave equations is obtained by a constructive method, and the corresponding global exact boundary controllability of nodal profile is also obtained under certain additional hypotheses.     

Global Exact Boundary Controllability for Cubic Semi-linear Wave Equations and Klein-Gordon Equations

The authors prove the global exact boundary controllability for the cubic semi-linear wave equation in three space dimensions, subject to Dirichlet, Neumann, or any other kind of boundary controls which result in the well-posedness of the corresponding initial-boundary value problem. The exponential decay of energy is first established for the cubic semi-linear wave equation with some boundary condition by the multiplier method, which reduces the global exact boundary controllability problem to a local one. The proof is carried out in line with [2, 15]. Then a constructive method that has been developed in [13] is used to study the local problem. Especially when the region is star-complemented, it is obtained that the control function only need to be applied on a relatively open subset of the boundary. For the cubic Klein-Gordon equation, similar results of the global exact boundary controllability are proved by such an idea.     

Localized null controllability and corresponding minimal norm control blowup rates of thermoelastic systems

In this paper, we consider the problem of null controllability for an elastic operator under square root damping. Such partial differential equation models can be described by analytic semigroups on the basic space of finite energy. Thus by inherent smoothing coming from the parabolic-like behavior of the dynamics, the problem of null controllability is appropriate for consideration. In particular, we will show that the solution variables can be steered to the zero state by means of iterations of locally supported steering controls acting on appropriate finite dimensional systems. The hinged boundary conditions considered here admit of a diagonalization of the spatial operator. The control strategy implemented in [A. Benabdallah, M. Naso, Null controllability of a thermoelastic plate, Abstr. Appl. Anal. 7 (2002) 585-599] is used to construct a suboptimal control for the problem, but here we expand upon their results by providing a bound for the energy function Emin(T), T>0. Our results are valid for localized mechanical and thermal control. The strategy relies heavily on the availability of a Carleman's estimate for finite linear combinations of eigenfunctions of the Dirichlet Laplacian.     

Numerical simulation of the boundary exact control for the system of linear elasticity

The problem of computing numerically the boundary exact control for the system of linear elasticity in two dimensions is addressed. A numerical method which has been recently proposed in [P. Pedregal, F. Periago, J. Villena, A numerical method of local energy decay for the boundary controllability of time-reversible distributed parameter systems. Stud. Appl. Math. 121 (1) (2008) 27–47] is implemented. Two cases are considered: first, a rectangular domain with Dirichlet controls acting on two adjacent edges, and secondly, a circular domain with Neumann controls distributed along the whole boundary.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> and <Emphasis Type="Italic">L</Emphasis><Superscript>∞</Superscript> norm estimates of the cost of the controllability for heat equations

This paper is concerned with the bound of the cost of approximate controllability and null controllability of heat equations, i.e., the minimal Lp norm and L∞ norm of a control needed to control the system approximately or a control needed to steer the state of the system to zero. The methods we use combine observability inequalities, energy estimates for heat equations and the dual theory.     

Exact boundary controllability 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 controllability for a kind of second‐order quasilinear hyperbolic systems. As an application, we obtain the one‐sided local exact boundary controllability for the 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. Copyright © 2009 John Wiley & Sons, Ltd.     

On exact boundary controllability for linearly coupled wave equations

In this paper we study exact boundary controllability for a system of two linear wave equations coupled by lower order terms. We obtain square integrable control of Neuman type for initial state with finite energy, in nonsmooth domains of the plane.     

Exact Boundary Controllability of Weak Solutions for a Kind of First Order Hyperbolic System — the HUM Method*

The exact boundary controllability and the exact boundary observability for the 1-D first order linear hyperbolic system were studied by the constructive method in the framework of weak solutions in the work [Lu, X. and Li, T. T., Exact boundary controllability of weak solutions for a kind of first order hyperbolic system — the constructive method, Chin. Ann. Math. Ser. B, 42(5), 2021, 643–676]. In this paper, in order to study these problems from the viewpoint of duality, the authors establish ...     

Exact controllability of semilinear abstract systems with application to waves and plates boundary control problems

This paper studies (global) exact controllability of abstract semilinear equations. Applications include boundary control problems for wave and plate equations on the explicitly identified spaces of exact controllability of the corresponding linear systems.Contents. 1. Motivating examples, corresponding results, literature. 1.1. Motivating examples and corresponding results. 1.2. Literature. 2. Abstract formulation. Statement of main result. Proof. 2.1. Abstract formulation. Exact controllability problem. 2.2. Assumptions and statement of main result. 2.3. Proof of Theorem 2.1. 3. Application: a semilinear wave equation with Dirichlet boundary control. Problem (1.1). 3.1. The case = 1 in Theorem 1.1 for problem (1.1). 3.2. The case = 0 in Theorem 1.1 for problem (1.1). 4. Application: a semilinear Euler—Bernoulli equation with boundary controls. Problem (1.14). 4.1. Verification of assumption (C.1): exact controllability of the linear system. 4.2. Abstract setting for problem (1.14). 4.3. Verification of assumptions (A.1)–(A.5). 4.4. Verification of assumption (C.2). 5. Proof of Theorem 1.2 and of Remark 1.2. Appendix A: Proof of Theorem 3.1. Appendix B: Proof of (4.9) and of (4.10b). References.Research partially supported by the National Science Foundation under Grant DMS-8902811 and by the Air Force Office of Scientific Research under Grant AFOSR-87-0321. The main results of this paper are announced in: Proceedings of the 28th Conference on Decision and Control, Tampa, Florida, December 1989, pp 2291–2294.     

Boundary Controllability of a Stationary Stokes System with Linear Convection Observed on an Interior Curve

We study the approximate controllability of a stationary Stokes system with linearized convection in a bounded domain of ^N. The control acts on a part of the boundary and the velocity field is observed on an interior curve (N=2) or surface (N=3). We establish the L ²-approximate controllability under certain compatibility conditions and suitable geometrical assumptions on the curve or surface. We build controls of minimal L ²-norm by duality. To compute the control, we propose a numerical method, based on duality techniques, consisting in the minimization of a nonquadratic functional coupled to a Stokes system. It is tested in several situations leading to interesting numerical results.     

A controllability minimum principle

A controllability minimum principle and two associated transversality conditions are presented, dealing with the controllability of nonlinear systems. The theorems represent necessary conditions for a control function to generate a system path which lies in the boundary of the set of points that are controllable to a target. The theorems presented here are controllability counterparts to Pontryagin's maximum principle, and undoubtedly these results will seem familiar or may have occurred to other researchers in the area of optimal control. The purpose of this paper is to make the distinction explicit and to establish the validity of these controllability theorems on their own merits. The theorems are demonstrated using a simple example and the principal result (a controllability minimum principle) is shown to be equivalent to the Kalman controllability criterion for linear systems.     

Exact Controllability and Boundary Stabilization of Torsional Vibrations of an Internally Damped Flexible Space Structure

In this paper, we study the exact controllability and boundary stabilization of the torsional vibrations of a flexible space structure (such as a solar cell array) modeled by a rectangular panel, incorporating the material damping of the structure. The panel is hoisted at one end by a rigid hub and the other end is totally free. For the attachment of this hub on one side of the panel, the hub dynamics leads to a nonstandard boundary condition. To incorporate internal damping of the material, we assume Voigt-type viscoelasticity of the structure. Exact controllability theory is established using the Hilbert uniqueness method by means of a control torque applied only on the rigid hub of the panel. At the same time, uniform exponential energy decay rate is obtained directly for the solution of this problem.     

Controllability of the nonlinear Schrödinger equation in the vicinity of the ground state

Local exact controllability of the one‐dimensional NLS (subject to zero‐boundary conditions) with distributed control is shown to hold in a H¹‐neighbourhood of the nonlinear ground state. The Hilbert Uniqueness Method (HUM), due to Lions, is applied to the linear control problem that arises by linearization around the ground state. The application of HUM crucially depends on the spectral properties of the linearized NLS operator which are given in detail. Copyright © 2007 John Wiley & Sons, Ltd.     

Insensitizing controls for the parabolic equation with equivalued surface boundary conditions

This paper is devoted to the study of the existence of insensitizing controls for the parabolic equation with equivalued surface boundary conditions. The insensitizing problem consists in finding a control function such that some energy functional of the equation is locally insensitive to a perturbation of the initial data. As usual, this problem can be reduced to a partially null controllability problem for a cascade system of two parabolic equations with equivalued surface boundary conditions. Compared the problems with usual boundary conditions, in the present case we need to derive a new global Carleman estimate, for which, in particular one needs to construct a new weight function to match the equivalued surface boundary conditions.     

Boundary controllability for conservative PDEs

Boundary observability and controllability problems for evolution equations governed by PDEs have been greatly studied in the past years. However, the problems were studied on a case-by-case basis, only for some particular types of boundary controls, and, moreover, several unnatural restrictions concerning lower-order terms were used.Our goal here is to give a general approach for boundary controllability problems, which is valid for all evolution PDEs of hyperbolic or ultrahyperbolic type, all boundary controls for which the corresponding homogeneous problem is well-posed, and all well-posedness spaces for the homogeneous problem. The first example of such equations is the class of hyperbolic equations, but valid examples are also equations such as the Schroedinger equation and various models for the plate equation.This work is essentially based on some apriori estimates of Carleman's type obtained by the author in a previous paper [29].This research was partially supported by the National Science Foundation under Grant NSF-DMS-8903747.     

Approximate boundary controllability of Sobolev-type stochastic differential systems

The objective of this paper is to investigate the approximate boundary controllability of Sobolev-type stochastic differential systems in Hilbert spaces. The control function for this system is suitably constructed by using the infinite dimensional controllability operator. Sufficient conditions for approximate boundary controllability of the proposed problem in Hilbert space is established by using contraction mapping principle and stochastic analysis techniques. The obtained results are extended to stochastic differential systems with Poisson jumps. Finally, an example is provided which illustrates the main results.     

A Unified Boundary Controllability Theory for Hyperbolic and Parabolic Partial Differential Equations

With use of the method of spherical means we are able to show that control processes modelled by the wave equation in a domain Ω ? Rⁿ are exactly controllable in finite time by control forces applied at the boundary of the spatial region Ω. The introduction of certain concepts from harmonic analysis together with use of the Fourier transform enables us to apply this result to obtain finite time exact controllability theorems for processes modelled by the heat equation in the same region Ω with controls of the same type.