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The articles in this volume focus on control theory of systems governed by nonlinear linear partial differential equations, identification and optimal design of such systems, and modelling of advanced materials. Optimal design of systems governed by PDEs is a relatively new area of study, now particularly relevant because of interest in optimization of fluid flow in domains of variable configuration, advanced and composite materials studies and "smart" materials which include possibilities for built in sensing and control actuation. The book will be of interest to both applied mathematicians and to engineers.



A Shape Optimization Problem in Inverse Acoustics

Given the incident field, a plane wave or a superposition thereof, and the scattered far field, the problem addressed herewith consists of determining the shape of a sound soft obstacle, which is axially symmetric, star shaped with respect to the origin and smooth. The unknowns are the parameters, by which shape is represented as the linear combination of e.g., trigonometric functions. Some methods of solution proposed in the past consist in transforming the problem into the constrained minimization of an objective function, which consisted of the boundary defect and a penalty term. The boundary defect is a squared L 2 norm of the incident plus approximate scattered field at the obstacle surface. The penalty term usually compares the computed and measured scattered far fields. The method presented herewith retains the minimization approach: it however replaces the constraint contained in the penalty term by a functional relationship between the coefficients, which appear in approximate representations of the far and resp., boundary scattered field. Said relationship is expressed by the approximate back propagation (ABP) operator, a product of some matrices, the entries of which depend on the shape parameters because they are inner products of suitable basis functions on the obstacle surface. As a consequence, a boundary defect alone has to be minimized with respect to the shape parameters. Some properties of the ABP operator are presented: they depend on those of the basis functions (spherical wave functions, their real parts and the normal derivatives of both at the obstacle surface) and are related e.g., to the properties of the T matrix method, used in the solution of forward scattering problems. Two classes of numerical problems are considered herewith: i) shape identification from the boundary coefficients of the scattered field and ii) shape identification from the far field coefficients. In either case the performance of the minimization algorithm agrees with an error estimate based on applying the projection theorem to the boundary defect. All examples have been selected to comply with a known uniqueness result, which would apply if the scattered field were exactly known: uniqueness is however lost because of approximation. The role of supplementary information in improving numerical reconstruction is demonstrated. Some preliminary results are also given for shape identification from the approximate scattering amplitude.
Giovanni Crosta

On a Variational Equation for Thin Shells

We present a mathematical construction of a boundary equation for thin shells from classical elasticity by using the tangential differential calculus and the oriented distance function. We specify the appropriate function spaces and give an existence and uniqueness theorem. The assumptions on the displacement field around a mean surface are analogous to the ones found in the work of W.T. Koiter and P.M. Naghdi.
M. C. Delfour, J. P. Zolésio

Oriented Distance Functions in Shape Analysis and Optimization

The object of this paper is twofold. We first present constructions which induce topologies on subsets of a fixed domain or hold-all D in ℝ N by using set parametrized functions in an appropriate function space. Secondly we study the role of the family of oriented distance functions (also known as algebraic or signed distance functions) in the analysis of shape optimization problems. They play an important role in the introduction of topologies which retain the classical geometric properties associated with sets: convexity, exterior normals, mean curvature, C k boundaries, etc.
Michel C. Delfour, Jean-Paul Zolésio

On the Density of the Range of the Semigroup for Semilinear Heat Equations

We consider the semilinear heat equation u t − Δu + f(u) = 0 in a bounded domain Ω ⊂ R n , n ≥ 1, for t > 0 with Dirichlet boundary conditions u = 0 on ∂Ω × (0,∞). For T > 0 fixed we consider the map S(T) : C 0 (Ω) → C 0 (Ω) such that S(T )u 0 = u (x, T) where u is the solution of this heat equation with initial data u(x, 0) = u 0(x) and C 0(Ω) is the space of uniformly continuous functions on Ω that vanish on its boundary. When f is globally Lipschitz and for any T > 0 we prove that the range of S(T)is dense in C 0(Ω). Our method of proof combines backward uniqueness results, a variational approach to the problem of the density of the range of the semigroup for linear heat equations with potentials and a fixed point technique. These methods are similar to those developed by the authors in an earlier paper in the study of the approximate controllability of semilinear heat equations.
Caroline Fabre, Jean-Pierre Puel, Enrique Zuazua

Relaxation in Semilinear Infinite Dimensional Systems Modelling Fluid Flow Control Problems

Fluid flow control problems lead to models described by semilinear abstract differential equations in Hilbert space, where relaxed controls must be incorporated to guarantee existence of optimal controls. We show that trajectories driven by relaxed controls can be uniformly approximated by trajectories driven by ordinary controls.
H. O. Fattorini, S. S. Sritharan

Multidimensional Inverse Scattering Problems in Deterministic and Random Media

Suppose that a medium with slowly changing spatial properties is enclosed in a bounded 3-dimensional domain and it is subjected to a scattering by plane waves of a fixed frequency. Let measurements of the wave scattering field induced by this medium be available in the region outside of this domain. We study how to extract the properties of the medium from the information contained in the measurements. We are concerned with the weak scattering case of the above inverse scattering problem (ISP). That is, the unknown spatial variations of the medium are assumed to be close to a constant. Examples can be found in studies of wave propagation in oceans, in the atmosphere and in some biological media.
In this paper we study the Inverse Scattering Problems in both deterministic and random media. Since the problems are nonlinear, the methods for their linearization (the Born approximation) have been developed. However, such an approach often does not produce good results. In our method, the Born approximation is just the first iteration and further iterations improve the identification by an order of magnitude.
The iterative sequence is defined in the framework of a Quasi-Newton method. Using the measurements of the scattering field from a carefully chosen set of directions we are able to recover (finitely many) Fourier coefficients of the sought parameters of the model. Numerical experiments for the scattering from coaxial circular cylinders as well as for simulated data are presented, which show a high precision of the identification. We also discuss some results based on Carleman’s estimates, applicable to the strong scattering case.
Semion Gutman, Michael Klibanov

Decay Estimates for the Wave Equation

Using the multiplier method and a special integral inequality we obtain sharp energy decay rate estimates for the wave equation in the presence of nonlinear distributed or boundary feedbacks. For simplicity we restrict ourselves to the wave equation, but the same approach applies for other systems as well.
Vilmos Komornik

Asymptotic Behavior and Attractors for Nonlinear von Kármán Plate Equations with Boundary Dissipation

We consider the problem of asymptotic behavior of the solutions to a fully nonlinear von Kármán plate with boundary damping. The results presented depend on whether a forcing term (representing a load or force attached to the plate) appears in the equation. In the absence of this term, the uniform decay rates for the solutions are established. In the second case, when a constant load is attached to the plate, the existence of finite dimensional attractor is demonstrated. The role of geometric conditions and its relation to the presence of “light” interior damping is also discussed.
Irena Lasiecka

Stabilization of the Korteweg-de Vries Equation on a Periodic Domain

We study solutions of the Korteweg-de Vries (KdV) equations
$$ {u_t} + u{u_x} + {u_{xxx}} = f $$
$$ {u_t} + u{u_x} + {u_{xxx}} = 0 $$
for t ≥ 0 and 0 ≤ x ≤ 1 where the subscripts denote partial derivatives. hi the first case, periodic boundary conditions are imposed at 0 and 1, and the distributed control f is assumed to be generated by a linear feedback control law conserving the “volume” or “mass” ∫ 0 1 u(x, t)dx which monotonically reduces the “energy” ∫ 0 1 u(x, t)2 dx. For the second equation a feedback boundary control is applied having the same properties. In both cases we obtain uniform exponential decay of the solutions to a constant state.
David L. Russell, Bing-Yu Zhang

A Note Concerning Boundary Effects and Long Time Vibrations of Layered Media

In this work, we study the vibration of, and wave propagation in, a periodically laminated composite slab. Special emphasis is given to the long time behavior of the vibration. It is found that a phase shift phenomenon occurs as a result of the interaction of the disturbance with the microstructure and the boundaries of the slab. We provide an asymptotic analysis to explain this phenomenon which we also illustrate by numerical calculations.
Fadil Santosa, Michael Vogelius

On the Linearised Dynamics of Linked Mechanical Structures

In this paper we take a bird’s eye view of the linear models which have recently been used by various authors in the area of boundary control of partial differential equations to describe the dynamics of mechanical structures which link together elastic and rigid obects. We will illustrate the modeling procedure with various models, old and new. We provide a general existence theorem for the very complicated systems involved in this modeling by using a familiar variational approach in a functional analytic setting.
E. J. P. Georg Schmidt
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