John E. Lagnese

An example on the effect of time delays in boundary feedback stabilization of wave equations

This note is concerned with the erect of time delays in boundary feedback stabilization schemes for wave equations. The question to be addressed is whether such delays can destabilize a system which is uniformly asymptotically stable in the absence of delays.

Decay of solutions of wave equations in a bounded region with boundary dissipation

Abstract An energy decay rate is obtained for solutions of wave type equations in a bounded region in Rn whose boundary consists partly of a nontrapping reflecting surface and partly of an energy absorbing surface.

Modeling, Analysis and Control of Dynamic Elastic Multi-Link Structures

Part 1: general overview on the contents of the book. Part 2 Modelling of networks of elastic strings: modelling of nonlinear elastic strings networks of nonlinear elastic strings linearization wellposedness of the network equations controllability of networks of elastic strings - exact controllability of tree networks, lack of controllability for networks with closed circuits stabilizability of string networks string networks with masses at the nodes. Part 3 Networks of thermoelastic beams modelling of a thin thermoelastic curved beam the equations of motion - some remarks on warping and torsion rotating beams - dynamic stiffening nonlinear nonshearable 3-d beams - approximation-generalizations linear shearable 3-d beams nonlinear shearable 2-d beams - approximation-generalizations a list of beam models - damping networks of beams - geometric joint conditions, rigid joints pinned joints, dynamic joint conditions, rigid joints, pinned joints rotating two-link nonlinear shearable beams. Part 4 A general hyperbolic model for networks: the general model some special cases - string networks, networks of planar Timoshenko beams, networks of linear shearable beams, networks of initially curved bresse beams, beams and strings existence and regularity of solutions energy estimates for hyperbolic systems exact controllability of the network model stabilizability of the network model. Part 5 Spectral analysis and numerical simulation: eigenvalue problems for networks - notation networks of strings setworks of Timoshenko beams Euler-Bernoulli beams the eigenvalue problem for mechanical networks - the string case, homogeneous network of strings, examples, the homogenous Timoshenko network numerical simulations of controlled networks, introductory remarks, absorbing controls, cirecting controls finite element approximations of networks - dry friction at joints. Part 6 Interconnected membranes: modelling of dynamic nonlinear elastic membranes - equations of motion, edge conditions, Hamiltons Principle systems of interconnected elastic membranes - geometric junction conditions, dynamic conditions, linearization, well-posedness of the linear model controllability of linked isotropic membranes - observability estimates for the homogeneous problem, a priori estimates for serially connected membranes, a priori estimates for single jointed membrane systems, the reachable states, serially connected membranes, membrane transmission problems. Part 7 Systems of linked plates: modelling of dynamic nonlinear elastic plates (Part contents)

Uniform stabilization of a nonlinear beam by nonlinear boundary feedback

We consider the planar motion of a uniform prismatic beam of length L. We want to derive a model that reflects the effect of stretching on bending, which necessarily leads to nonlinear partial differential equations for the motion of the beam. We will, however, assume that the constitutive equations for bending are linear. This is in agreement with existing engineering literature (see, for example [S] and the bibliography therein). It should be remarked that the effect of stretching on bending becomes significant if, in particular, a rigid rotation is superimposed on the motion. We do not consider such a rotation here, even though it could be handled within the present framework. We assume that the beam, in its reference state, occupies the region described in rectangular coordinates by

Time-domain decomposition of optimal control problems for the wave equation

We consider the problem of boundary optimal control of a wave equation with boundary dissipation by the way of time-domain decomposition of the corresponding optimality system. We develop an iterative algorithm which shows that the decomposed optimality system corresponds to local-in-time optimal control problems which can be treated in parallel. We show convergence of the algorithm. Finally, we provide a time discretization which is reminiscent of an instantaneous control scheme. We thereby also contribute to the problem of convergence of such schemes.

Control of planar networks of Timoshenko beams

The present study is concerned with the questions of controllability and stabilizability of planar networks of vibrating beams consisting of several Timoshenko beams connected to each other by rigid joints at all interior nodes of the system. Some of the exterior nodes are either clamped or free; controls may be applied at the remaining exterior nodes and/or at interior joints in the form of forces and/or bending moments. For a given configuration, is it at all possible to drive all vibrations to the rest configuration in a given finite time interval by means of controls acting at some or all of the available (nonclamped) nodes of the network and, if so, where should such controls be placed? Alternatively, a control objective is to construct energy absorbing boundary-feedback controls that will guarantee uniform energy decay. It is demonstrated that if such a network does not contain closed loops and if at most one of the exterior nodes is clamped, exact controllability and uniform stabilizability of the ...

Dynamic Domain Decomposition in Approximate and Exact Boundary Control in Problems of Transmission for Wave Equations

This paper is concerned with dynamic domain decomposition for optimal boundary control and for approximate and exact boundary controllability of wave propagation in heterogeneous media. We consider a cost functional which penalizes the deviation of the final state of the solution of the global problem from a specified target state. For any fixed value of the penalty parameter, optimality conditions are derived for both the global optimal control problem and for local optimal control problems obtained by a domain decomposition and a saddle-point--type iteration. Convergence of the iterations to the solution of the global optimality system is established. We then pass to the limit in the iterations as the penalty parameter increases without bound and show that the limiting local iterations converge to the solution of the optimality system associated with the problem of finding the minimum norm control that drives the solution of the global problem to a specified target state.

Modelling and stabilization of nonlinear plates

Some nonlinear models for the transient deformations of a thin, elastic plate subject to surface tractions and edge loadings are derived. Energy dissipation is then introduced into the system by way of nonlinear state feedbacks in forces and moments along a part of the edge of the plate. The feedback functions are assumed monotone as graphs and to satisfy certain growth assumptions at zero and at infinity. The main purpose of the paper is to derive asymptotic energy decay rates for the closed-loop system. The decay rates obtained are exponential when the feedback functions behave linearly at zero, otherwise they are algebraic, with the specific rates of energy decay depending only on the growth of the feedback functions at zero.

Boundary stabilization of thin elastic plates

Abstract : In this paper we shall consider the question of uniform stabilization of thin, elastic plates through the action of forces and moments on the edge of the plate (or on a part of the edge of the plate). Two particular plate models will be considered: The classical fourth order Kirchoff model, but incorporating rotational inertia, and the sixth order Mindlin-Timoshenko model. The difference in the two models, from a physical point of view, is that the M-T model incorporates transverse shear effects while the Kirchhoff model does not. Actually, the M-T model is a hyperbolic system three coupled second order partial differential equations in two dependent variables. The unknowns, denoted by w, psi, phi are the vertical component w of displacement and angles which are measures of the amount of transverse shear. The three equations are coupled through terms which are multiples of a factor K called the coefficient of elasticity in shear.

Uniform boundary stabilization of von Karman plates

The author considers the determination of feedback controls which act along the edge of a thin plate and whose purpose is to stabilize the motion of the plate uniformly in situations where the deviation of the plate from equilibrium is not necessarily small, and therefore, is not adequately modeled by linear plate theory. The analysis deals mainly with the classical von Karman plate model. It is shown that the physical assumptions usually associated with the von Karman model lead to a coupled system of three nonlinear second-order partial differential equations for the three components of the displacement vector. The von Karman model is then obtained by imposing an additional hypothesis which does not seem to be altogether reasonable from a physical standpoint. The uniform asymptotic stability of the von Karman model is then studied.<<ETX>>