Vilmos Komornik

Boundary Observability, Controllability, and Stabilization of Linear Elastodynamic Systems

In 1988 Lions obtained observability and exact controllability results for linear homogeneous isotropic elastodynamic systems [ SIAM Rev., 30 (1988), pp. 1--68]. Applying some new identities we extend his theorems to nonisotropic systems. In 1991 Lagnese obtained uniform stabilizability results for two-dimensional linear homogeneous isotropic systems by applying a somewhat artificial feedback [Nonlinear Anal., 16 (1991), pp. 35--54]. Then he asked whether analogous results hold for a natural and physically implementable boundary feedback. Using some new identities and applying a method introduced in 1987 by Zuazua and the second author [ J. Math. Pures. Appl., 69 (1990), pp. 33--54], we give an affirmative answer to this question in all dimensions and also for nonisotropic systems. Moreover, we obtain good decay estimates. Finally, applying a recent general method of uniform stabilization, we construct boundary feedbacks leading to arbitrarily large energy decay rates.

Rapid Boundary Stabilization of Linear Distributed Systems

We prove that under rather general assumptions an exactly controllable problem is uniformly stabilizable with arbitrarily prescribed decay rates. Our approach is direct and constructive and avoids many of the technical difficulties associated with the usual methods based on Riccati equations. We give several applications for the wave equation and for Petrovsky systems.

Ingham-Beurling type theorems with weakened gap conditions

Completing a series of works begun by Wiener [34], Paley and Wiener [28] and Ingham [9], a far-reaching generalization of Parsevals identity was obtained by Beurling [4] for nonharmonic Fourier series whose exponents satisfy a uniform gap condition. Later this gap condition was weakened by Ullrich [33], Castro and Zuazua [5], Jaffard, Tucsnak and Zuazua [11] and then in [2] in some particular cases. In this paper we prove a general theorem which contains all previous results. Furthermore, applying a different method, we prove a variant of this theorem for nonharmonic Fourier series with vector coefficients. This result, partly motivated by control-theoretical applications, extends several earlier results obtained in [15] and [2]. Finally, applying these results we obtain an optimal simultaneous observability theorem concerning a system of vibrating strings.

Developments in non-integer bases

AbstractWe prove various theorems concerning the developments in non-integer bases. We mention two of them here, which answer some questions formulated several years ago. First fix a real number q> 1 and consider the increasing sequence 0 = yo < y1 < y2 < ¨ of those real numbers y which have at least one representation of the form y = ε0 + ε1q + ¨ + εnqn with some integer n ≧ 0 and coefficients ε, Ε {0, 1}. Then the difference sequence yk+1-yk tends to 0 for all q, sufficiently close to 1.Secondly, for each q, sufficiently close to 1, there exists a sequence (εi) of zeroes and ones, satisfying

Ingham-Type Theorems for Vector-Valued Functions and Observability of Coupled Linear Systems

\Sigma _{i = 1}^\infty\in_{iq^{ - i} }=1

Estimation of point sources and applications to inverse problems

= 1 and containing all possible finite variations of the digits 0 and 1.

On the nonlinear boundary stabilization of Kirchhoff plates

The β-expansions, i.e., greedy expansions with respect to non-integer bases q>1, were introduced by Réenyi and then investigated by many authors. Some years ago, Erdős, Horváth and Joó found the surprising fact that there exist infinitely many numbers 1<q<2 for which the β-expansion of 1 is the unique possible expansion with coefficients 0 or 1. Subsequently, the unique expansions were characterized in [9] and this characterization led to the determination (in [17]) of the smallest number q having this curious property. It is intimately related to the classical Thue-Morse sequence. Allouche and Cosnard recently proved that this q is transcendental. The purpose of this paper is to extend the previous results for expansions in arbitrary non-integer bases q>1. We also determine the smallest q having the corresponding uniqueness property in each case, and we prove that all of them are transcendental. We will also obtain some probably new properties of the Thue-Morse sequence. In the last section we answer a question concerning the existence of universal expansions, a notion introduced in [12].

Expansions in Complex Bases

We propose a new approach to study the observability of coupled linear distributed systems. It is based on a generalization of some classical theorems of nonharmonic analysis to vector-valued functions. Applying this method we answer some questions of J.-L. Lions [ Controlabilite exacte et stabilisation de systems distribues, Vol. 2, Masson, 1988] on the boundary observability of coupled linear distributed systems in particular cases.