Paola Loreti

Regularity of the Minimum Time Function and Minimum Energy Problems: The Linear Case

We prove new results on the continuity properties and on reachable sets of the minimum time function associated with a linear dynamical system in a separable Hilbert space H. Part of the results are new also in the finite dimensional case. The regularity results are stated thanks to the result we obtain on the connection between the minimum time function and the minimum energy.

Partial Exact Controllability for Spherical Membranes

In this paper the partial exact controllability of an elastic spherical membrane is proved. The reachability problem for the integrodifferential equation, introduced for the vibrations of the meridional displacement, is solved. The main result is a generalization of a Inghams theorem on nonharmonic Fourier series.

Expansions in Complex Bases

Beginning with a seminal paper of Renyi, expansions in noninteger real bases have been widely studied in the last forty years. They turned out to be relevant in various domains of mathematics, such as the theory of finite automata, number theory, fractals or dynamical systems. Several results were extended by Daroczy and Katai for expansions in complex bases. We introduce an adaptation of the so-called greedy algorithm to the complex case, and we generalize one of their main theorems.

Optimal Energy Decay Rate for Partially Damped Systems by Spectral Compensation

We study the stability of weakly coupled and partially damped systems by means of the Riesz basis approach in higher dimensional spaces. We propose a weaker structural damping that compensates for the behavior of the eigenvalues of the system, therefore giving the optimal polynomial energy decay rate for smooth initial data.

Partial Observability of Coupled Linear Systems

In [25] Lions proved several partial observability theorems for coupled linear distributed systems with small enough coupling parameters by a perturbation argument. He asked whether analogous results hold for large values of the parameters. Adapting an approach of Graham and Russell [7] we obtain positive results in particular cases.

A note on partial exact controllability for spherical membranes

We describe some of our (1997) recently obtained results. We consider the axially symmetric vibrations of a spherical membrane described by a pair of coupled partial differential equation in the meridional and radial displacement. Then we consider the equivalent problem of an integro-differential equation involving only the meriodional displacement, and state the partial exact controllability problem. To solve it, we follow the reachability Hilbert uniqueness method. As for the invertibility of the associate operator we derive a result concerning non-harmonic Fourier series. This result generalizes a theorem due to Ingham (1936) on trigonometric inequalities for almost period functions. The assumptions we made on this abstract theorem are motivated by the problem of partial controllability for spherical membranes. Indeed these results leads to solve the partial exact controllability problem for spherical membranes.