arXiv subjects

Luciano Pandolfi

Publications and source records attributed to Luciano Pandolfi.

17 recordsLinked to original sources

arXiv2024

The quadratic tracking problem for systems with persistent memory in $\zzr^d$

The classical quadratic regulator problem has rarely been studied for systems with persistent memory until recent times. In this paper we study the quadratic tracking problem on a \emph{ finite time horizon} for a system described by a controlled linear Volterra integrodifferential equation in $\zzr^d$. We use the Fredholm equation approach and we derive the synthesis of the optimal control in terms of a Riccati differential equation, independent of the reference signal, and of two equations which instead depend on the reference signal. In the final section we introduce a representation of the system in a state space \emph{of finite memory } (as the tracking problem under study) and we show that the equations used in the synthesis can be formulated as a differential system in this space. This fact has to be contrasted with the semigroup approach which requires that the system be recasted in a space of infinite memory.

math.OC↗

arXiv2023

The Lebesgue Integral Via The Tonelli Method

Leonida Tonelli devised an interesting and efficient method to introduce the Lebesgue integral. The details of this method can only be found in the original Tonelli paper and in an old italian course and solely for the case of the functions of one variable. We believe that this method is worth knowing and here we present a complete account for functions of every number of variables.

math.CA↗

arXiv2020

On the fourth order Cattaneo equation of heat conduction with memory

The well known heat equation with finite speed of propagation proposed by Cattaneo is obtained by a more general fourth order PDE when a certain (small) parameters is put equal to zero. It seems that this fourth order equation has been essentially overlooked in the literature, in particular when it is subject to non homogeneous boundary conditions. In this paper we examine its well posedness and the asymptotic behavior when certain coefficients tend to singular values.

math.AP↗

arXiv2019

Controllability properties for equations with memory of fractional type

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.

math.OC↗

arXiv2018

Controllability of a linear system with persistent memory via boundary traction

We consider a linear viscoelastic system of Maxwell-Boltzmann type. Hence, viscosity contributes a memory term to the elastic equation. The system is controlled via the traction exerted on a part $Γ_1$ of the boundary of the body. We prove that if the associated elastic system (i.e. the elastic system without memory) is exactly controllable then the viscoelastic system is exactly controllable too. This is similar to the known result when the boundary deformation is controlled, but the proof is far more delicate since controllability under boundary traction corresponds to the fact that a certain sequence of functions is a Riesz-Fisher sequence, but not a Riesz sequence.

math.OC↗

arXiv2017

On the regularity of solutions to the Moore-Gibson-Thompson equation: a perspective via wave equations with memory

We undertake a regularity analysis of the solutions to initial/boundary value problems for the (third-order in time) Moore-Gibson-Thompson (MGT) equation. The key to the present investigation is that the MGT equation falls within a large class of systems with memory, with affine term depending on a parameter. For this model equation a regularity theory is provided, which is of also independent interest; it is shown in particular that the effect of boundary data that are square integrable (in time and space) is the same displayed by wave equations. Then, a general picture of the (interior) regularity of solutions corresponding to homogeneous boundary conditions is specifically derived for the MGT equation in various functional settings. This confirms the gain of one unity in space regularity for the time derivative of the unknown, a feature that sets the MGT equation apart from other PDE models for wave propagation. The adopted perspective and method of proof enables us to attain as well the (sharp) regularity of boundary traces.

math.AP↗

arXiv2017

Joint identification via deconvolution of the flux and energy relaxation kernels of the Gurtin-Pipkin model in thermodynamics with memory

In this paper we present a linear method for the identification of both the energy and flux relaxation kernels in the equation of thermodynamics with memory proposed by M.E. Gurtin and A.G. Pipkin. The method reduces the identification of the two kernels to the solution of two (linear) deconvolution problems. The energy relaxation kernel is reconstructed by means of energy measurements as the solution of a Volterra integral equation of the first kind which does not depend on the still unknown flux relaxation kernel. Then, flux measurements are used to identify the flux relaxation kernel.

math.OC↗

arXiv2017

Controllability and lack of controllability with smooth controls in viscoelasticity via moment methods

In this paper we study controllability of a linear equation with persistent memory when the control belongs to $ H^k_0(0,T;L^2(\ZOMq)) $. In the case the memory is zero, our equation is reduced to the wave equation and a result due to Everdoza and Zuazua informally states that smoother targets can be reached by using smoother controls. In this paper we prove that this result can be partially extended to systems with memory, but that the memory is an obstruction to a complete extensions.

math.OC↗

arXiv2017

Identification of a space varying coefficient of a linear viscoelastic string of Maxwell-Boltzman type

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.

math.OC↗

arXiv2015

Controllability of isotropic viscoelastic bodies of Maxwell-Boltzmann type

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.

math.OC↗

arXiv2015

Regularity of the steering control for systems with persistent memory

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.

math.OC↗

arXiv2015

Identification of a relaxation kernel using two boundary measures

We consider a distributed system with persistent memory of a type which is often encountered in viscoelasticity or in the study of diffusion processes with memory. The relaxation kernel, i.e. the kernel of the memory term, is scarcely known from first principles, and it has to be inferred from experiments taken on samples of the material. We prove that two boundary measures give a linear Volterra integral equation of the first kind for the unknown kernel. Hence, with two measures, the identification of the kernel, which in principle is a nonlinear problem, is reduced to the solution of a mildly ill posed but linear problem.

math.DS↗

arXiv2013

Sharp control time for viscoelastic bodys

It is now well understood that equations of viscoelasticity can be seen as perturbation of wave type equations. This observation can be exploited in several different ways and it turns out that it is a usefull tool when studying controllability. Here we compare a viscoelastic system which fills a surface of a solid region (the string case has already been studied) with its memoryless counterpart (which is a generalized telegraph equation) in order to prove exact controllability of the viscoelastic body at precisely the same times at which the telegraph equation is controllable. The comparison is done using a moment method approach to controllability and we prove, using the perturbations theorems of Paley-Wiener and Bari, that a new sequence derived from the viscoelastic system is a Riesz sequence, a fact that implies controllability of the viscoelastic system. The results so obtained generalize existing controllability results and furthermore show that the ``sharp'' control time for the telegraph equation and the viscoelastic system coincide.

eess.SY↗

arXiv2013

Lack of controllability of thermal systems with memory

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.

eess.SY↗

arXiv2012

Traction, deformation and velocity of deformation in a viscoelastic string

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.

math-ph↗

arXiv2012

Boundary controllability and source reconstruction in a viscoelastic string under external traction

Treatises on vibrations devote large space to study the dynamical behavior of an elastic system subject to known external tractions. In fact, usually a "system" is not an isolated body but it is part of a chain of mechanisms which disturb the "system" for example due to the periodic rotation of shafts. This kind of problem has been rarely studied in control theory. In the specific case we shall study, the case of a viscoelastic string, the effect of such external action is on the horizontal component of the traction, and so it affects the coefficients of the corresponding wave type equation, which will be time dependent. The usual methods used in controllability are not naturally adapted to this case. For example at first sight it might seem that moment methods can only be used in case of coefficients which are constant in time. Instead, we shall see that moment methods can be extended to study controllability of a viscoelastic string subject to external traction and in particular we shall study a controllability problem which is encountered in the solution of the inverse problem consisting in the identification of a distributed disturbance source.

math.OC↗

arXiv2011

Linear Operator Inequality and Null Controllability with Vanishing Energy for unbounded control systems

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.

math.OC↗