Luciano Pandolfi

Luciano Pandolfi, Enrico Priola, Jerzy Zabczyk

We consider linear systems on a separable Hilbert space $H$, which are null controllable at some time $T_0>0$ under the action of a point or boundary control. Parabolic and hyperbolic control systems usually studied in applications are special cases. To every initial state $ y_0 \in H$ we associate the minimal "energy" needed to transfer $ y_0 $ to $ 0 $ in a time $ T \ge T_0$ ("energy" of a control being the square of its $ L^2 $ norm). We give both necessary and sufficient conditions under which the minimal energy converges to $ 0 $ for $ T\to+\infty $. This extends to boundary control systems the concept of null controllability with vanishing energy introduced by Priola and Zabczyk (Siam J. Control Optim. 42 (2003)) for distributed systems. The proofs in Priola-Zabczyk paper depend on properties of the associated Riccati equation, which are not available in the present, general setting. Here we base our results on new properties of the quadratic regulator problem with stability and the Linear Operator Inequality.

Luciano Pandolfi

In this paper we consider a viscoelastic string whose deformation is controlled at one end. We study the relations and the controllability of the couples traction/velocity and traction/deformation and we show that the first couple behaves very like as in the purely elastic case, while new phenomena appears when studying the couple of the traction and the deformation. Namely, while traction and velocity are independent (for large time), traction and deformation are related at each time but the relation is not so strict. In fact we prove that an arbitrary number of "Fourier" components of the traction and, independently, of the deformation can be assigned at any time.

Andrei Halanay, Luciano Pandolfi

Heat equations with memory of Gurtin-Pipkin type have controllability properties which strongly resemble those of the wave equation. Instead, recent counterexamples show that when the laplacian appears also out of the memory term, the control properties do not parallel those of the (memoryless) heat equation, in the sense that there are $L^2$-initial conditions which cannot be controlled to zero. The proof of this fact (presented in previous papers) consists in the construction of two quite special examples of systems with memory which cannot be controlled to zero. Here we prove that lack of controllability holds in general, for every systems with smooth memory kernel.

Luciano Pandolfi

In this paper we solve the problem of the identification of a coefficient which appears in the model of a distributed system with persistent memory encountered in linear viscoelasticity (and in diffusion processes with memory). The additional data used in the identification are subsumed in the input output map from the deformation to the traction on the boundary. We extend a dynamical approach to identification introduced by Belishev in the case of purely elastic (memoryless) bodies and based on a special equation due to Blagoveshchenskii. So, in particular, we extend Blagoveshchenskii equation to our class of systems with persistent memory.

Luciano Pandolfi

We study a general class of control systems with memory, which in particular includes systems with fractional derivatives and integrals and also the standard heat equation. We prove that the approximate controllability property of the heat equation is inherited by every system with memory in this class while controllability to zero is a singular property, which holds solely in the special case that the system indeed reduces to the standard heat equation.

Luciano Pandolfi, Daniele Triulzi

The following fact is known for large classes of distributed control systems: when the target is regular, there exists a regular steering control. This fact is important to prove convergence estimates of numerical algorithms for the approximate computation of the steering control. In this paper we extend this property to a class of systems with persistent memory (of Maxwell/Boltzmann type) and we give a variational characterization of the smooth steering control which may open the way to an extension of the numerical approach proposed by Ervedoza and Zuazua.

Luciano Pandolfi

In this paper we consider a viscoelastic three dimensional body (of Maxwell-Boltzmann type) controlled on (part of) the boundary. We assume that the material is isotropic and homogeneous. If the body is elastic (i.e. no dissipation due to past memory), controllability has been studied by several authors. We prove that the viscoelastic body inherits the controllability properties of the corresponding purely elastic system. The proof relays on cosine operator methods combined with moment theory.