Boundary controllability of a viscoelastic beam

It is shown thai, a viscoelastic beam with long memory is exactly controllable by L₂ (0 ,T)-boundary controls in any time T > 0. Furthermore, a weak bang-hang principle is derived In the elastic case     

Approximate boundary controllability for the system of linear elasticity

The approximate controllability for variable coefficients, isotropic, evolution elasticity system is considered. The appropriate unique continuation theorem for solutions of the system is stated. Copyright © 2004 John Wiley & Sons, Ltd.     

A linear wave equation with a nonlinear boundary condition of viscoelastic type

The paper is devoted to the study of a linear wave equation with a nonlinear boundary condition of viscoelastic type. The existence of a weak solution is proved by using the Faedo–Galerkin method. The uniqueness, regularity and asymptotic expansion of the solution are also discussed.     

Time optimal rendezvous of three linear systems

The minimum-time rendezvous problem of three linear controlled systems is studied, and the existence and necessary conditions for a solution are derived.This research was supported by NSF Grant No. GK-31628.     

On boundary controllability of a vibrating plate

A vibrating plate is here taken to satisfy the model equation:u _tt + ²u = 0 (where ²u:= (u); = Laplacian) with boundary conditions of the form:u _v = 0 and(u) _v = = control. Thus, the state is the pair [u, u _t] and controllability means existence of on := (0,T)× transfering any[u, u _t]₀ to any[u, u _t]_T. The formulation is given by eigenfunction expansion and duality. The substantive results apply to a rectangular plate. For largeT one has such controllability with = O(T ^–1/2). More surprising is that (based on a harmonic analysis estimate [11]) one has controllability for arbitrarily short times (in contrast to the wave equation:u _tt = u) with log = O(T ^–1) asT0. Some related results on minimum time control are also included.This research was partially supported under the grant AFOSR-82-0271.     

Small-time local controllability and continuity of the optimal time function for linear systems

For a finite-dimensional linear system, in which the control is restricted to belong to a completely arbitrary setJ, we give a simple necessary and sufficient condition for small-time local controllability from a pointp. The condition is equivalent to a characterization of the property that the Bellman function for the corresponding minimum-time optimal control problem is continuous atp.This work was partially supported by the National Science Foundation, Grant No. MCS-78-02442.     

Exponential decay of non‐linear wave equation with a viscoelastic boundary condition

We study in this paper the global existence and exponential decay of solutions of the non‐linear unidimensional wave equation with a viscoelastic boundary condition. We prove that the dissipation induced by the memory effect is strong enough to secure global estimates, which allow us to show existence of global smooth solution for small initial data. We also prove that the solution decays exponentially provided the resolvent kernel of the relaxation function, k decays exponentially. When k decays polynomially, the solution decays polynomially and with the same rate. Copyright © 2000 John Wiley & Sons, Ltd.     

Time optimal boundary controls for the heat equation

The fact that the time optimal controls for parabolic equations have the bang–bang property has been recently proved for controls distributed inside the considered domain (interior control). The main result in this article asserts that the boundary controls for the heat equation have the same property, at least in rectangular domains. This result is proved by combining methods from traditionally distinct fields: the Lebeau–Robbiano strategy for null controllability and estimates of the controllability cost in small time for parabolic systems, on one side, and a Remez-type inequality for Müntz spaces and a generalization of Turán?s inequality, on the other side.     

Exact boundary controllability of a shallow intrinsic shell model

We consider a variant of a Koiter shell model based on the intrinsic geometry methods of Michael Delfour and Jean-Paul Zolésio. This model, derived in [J. Cagnol, I. Lasiecka, C. Lebiedzik, J.-P. Zolésio, Uniform stability in structural acoustic models with flexible curved walls, J. Differential Equations 186 (1) (2003) 88-121], relies heavily on the oriented distance function which describes the geometry. Here, we establish continuous observability estimates in the Dirichlet case with an explicit observability time, under an additional shallowness assumption and a checkable geometric condition. This yields (by duality) exact controllability for this class of intrinsically modelled shells.     

Initial boundary value problems for linear viscoelastic flows generated by integro-differential equations

An estimate of the velocity field is obtained for the equation of motion of incompressible media. With the help of this estimate, the integro-differential equations that describe the motion of linear viscoelastic fluids in the two-dimensional case are studied. The existence is proved for a weak, global in time, solution of the Cauchy problem and of the initial boundary value problem with periodic boundary conditions. Bibliography: 18 titles.Dedicated to Olga Aleksandrovna Ladyzhenskaya on the occasion of her jubilee__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 295, 2003, pp. 90–98.     

Exact bounded boundary controllability of vibrations of a two-dimensional membrane

The boundary control of vibrations of a plane membrane is considered. A constraint is imposed on the absolute value of the control function. The goal of the control is to drive the membrane to rest. The proof technique used in this paper can be applied to a membrane of any dimension, but the two-dimensional case is considered for simplicity and illustrative purposes.     

Time-effective moduli of a linear viscoelastic body

An approximate method for solving boundary-value problems for isotropic linear viscoelastic bodies is considered, and a technique of determining the time-effective viscoelastic moduli is presented. Approximate estimates for the functionals of potential energy of strains and stresses are also given. Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. 1, pp. 59–70, January–February, 2000.     

A simple linear model for the optimal exploitation of renewable resources

This paper describes a simple linearized model for the optimal control of a natural resource stock. Applications of the model to fisheries and forestry, as well as to mineral exploration and recovery, are discussed.Research supported partially by the National Research Council of Canada, Grant A-3990.     

Instant controllability of linear autonomous systems

In this paper, the complete controllability of linear autonomous control systems in any interval of time and the local controllability in any interval of time are investigated. Necessary and sufficient conditions are obtained for both types of controllability.     

Uniform boundary controllability of a semi-discrete 1-D wave equation

Summary. A numerical scheme for the controlled semi-discrete 1-D wave equation is considered. We analyze the convergence of the boundary controls of the semi-discrete equations to a control of the continuous wave equation when the mesh size tends to zero. We prove that, if the high modes of the discrete initial data have been filtered out, there exists a sequence of uniformly bounded controls and any weak limit of this sequence is a control for the continuous problem. The number of the eliminated frequencies depends on the mesh size and the regularity of the continuous initial data. The case of the HUM controls is also discussed. Received March 3, 2001 / Published online October 17, 2001     

Spectral boundary controllability of networks of strings

In this Note we give a necessary and sufficient condition for the spectral controllability from one simple node of a general network of strings that undergoes transversal vibrations in a sufficiently large time. This condition asserts that no eigenfunction vanishes identically on the string that contains the controlled node. The proof combines the Beurling–Malliavin's theorem and an asymptotic formula for the eigenvalues of the network. The optimal control time may be characterized as twice the sum of the lengths of all the strings of the network. To cite this article: R. Dáger, E. Zuazua, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 545–550.     

On boundary controllability of one-dimensional vibrating systems

In this paper boundary controllability of one-dimensional vibrating system such as the vibrating string or the vibrating beam is studied. In particular we are concerned with the question whether it is possible to transfer a given initial state of vibration into rest within a given time such that the system stays in rest when the control is turned off. This problem is rephrased as a typical trigonometric moment problem which is solved within the framework of an abstract moment problem in a Hilbert space. The results of null-controllability which are obtained are substantially based on classical results of Ingham and Redheffer concerning trigonometric inequalities and incompleteness of certain sequences of trigonometric functions, respectively. The representation of the general statements follows closely the lines of a paper of Russell. Besides a special case is treated where explicit representations of boundary controls can be given that transfer the system to a permanent rest position. This special case includes amplitude boundary control of the vibrating string and the freely supported beam.