arXiv subjects

Francesca Bucci

Publications and source records attributed to Francesca Bucci.

At least 19 recordsLinked to original sources

arXiv2026

Well-posedness and stability for a thermoelectromagnetic system

In this article we consider a system of coupled partial differential equations that mirrors the interconnection between the evolution of an electromagnetic field -- described by the Maxwell's system -- and that of the temperature distribution in a bounded region in three-dimension. A discussion of the mathematical model is provided. We establish the well-posedness of the initial-boundary value problems associated with the thermoelectromagnetic system, in an appropriate functional-analytic framework. Then, our investigation and main results pertain to the long-time behaviour of the solutions. It is shown that either exponential stability or convergence to stationary solutions hold true, according as the conductivity is positive definite or semidefinite, respectively; in the latter case, strong stability is attained in specific topological or analytical settings. This complements and expands earlier results obtained for the uncoupled Maxwell's system.

math.AP↗

arXiv2026

Reducing the Virgo site infrastructure noise in preparation of the O4 observing run

The heating, ventilation and air conditioning systems serving the experimental halls of the Virgo gravitational wave interferometer generate low-frequency noise - namely below 100 Hz - of seismic, acoustic, and electromagnetic origin. Such disturbances have repeatedly affected the interferometer sensitivity throughout its operational history, with particularly notable impacts during the third observing run. In preparation for the fourth run, a comprehensive investigation was carried out to identify the most critical noise sources within this infrastructure and to trace their transmission paths into the experimental areas. This manuscript presents the methodology and results of the noise characterization campaign, together with the design, implementation and assessment of targeted mitigation measures. The technical solutions adopted, along with the operational best practices developed, provide valuable guidance for the design of low-noise environments in future gravitational-wave observatories.

astro-ph.IM↗

arXiv2025

Linear quadratic control of parabolic-like evolutions with memory of the inputs

A study of the linear quadratic (LQ) control problem on a finite time interval for a model equation in Hilbert spaces which comprehends the memory of the inputs was performed recently by the authors. The outcome included a closed-loop representation of the unique optimal control, along with the derivation of a related coupled system of three quadratic (operator) equations which is shown to be well-posed. Notably, in the absence of memory the above elements -- namely, formula and system -- reduce to the known feedback formula and single differential Riccati equation, respectively. In this work we take the next natural step, and prove the said results for a class of evolutions where the control operator is no longer bounded. These findings appear to be the first ones of their kind; furthermore, they extend the classical theory of the LQ problem and Riccati equations for parabolic partial differential equations.

math.OC↗

arXiv2024

An optimal control problem for Maxwell's equations

This article is concerned with the optimal boundary control of the Maxwell system. We consider a Bolza problem, where the quadratic functional to be minimized penalizes the electromagnetic field at a given final time. Since the state is weighted in the energy space topology -- a physically realistic choice --, the property that the optimal cost operator does satisfy the Riccati equation (RE) corresponding to the optimization problem is missed, just like in the case of other significant hyperbolic partial differential equations; however, we prove that this Riccati operator as well as the optimal solution can be recovered by means of approximating problems for which the optimal synthesis holds via proper differential Riccati equations. In the case of zero conductivity, an explicit representation of the optimal pair is valid which does not demand the well-posedness of the RE, instead.

math.OC↗

arXiv2024

Optimal synthesis control for evolution equations subject to nonlocal inputs

We consider the linear quadratic (LQ) optimal control problem for a class of evolution equations in infinite dimensions, in the presence of distributed and nonlocal inputs. Following the perspective taken in our previous research work on the LQ problem for integro-differential equations, where the memory term -- here involving the control actions -- is seen as a component of the state, we offer a full (closed-loop, Riccati-like) solution to the optimization problem.

math.OC↗

arXiv2023

Riccati-based solution to the optimal control of linear evolution equations with finite memory

In this article we study the optimal control problem with quadratic functionals for a linear Volterra integro-differential equation in Hilbert spaces. With the finite history seen as an (additional) initial datum for the evolution, following the variational approach utilized in the study of the linear-quadratic problem for memoryless infinite dimensional systems, we attain a closed-loop form of the unique optimal control via certain operators that are shown to solve a coupled system of quadratic differential equations. This result provides a first extension to the partial differential equations realm of the Riccati-based theory recently devised by L. Pandolfi in a finite dimensional context.

math.OC↗

arXiv2021

On the infinitesimal generator of an optimal state semigroup

In this article we fully describe the domain of the infinitesimal generator of the optimal state semigroup which arises in the theory of the linear-quadratic problem for a specific class of boundary control systems. This represents an improvement over earlier work of the authors, joint with Lasiecka, where a set inclusion was established, but not an equality. The novel part of the proof of this result developes through appropriate asymptotic estimates that take advantage of the regularity analysis carried out in the study of the optimization problem, while the powers of positive operators and interpolation are still key tools. We also attest to the validity of an assumed relation between two significant parameters in the case of distinct systems of coupled hyperbolic-parabolic partial differential equations which are pertinent to the underlying framework.

math.OC↗

arXiv2020

Uniqueness for Riccati equations with unbounded operator coefficients

In this article we address the issue of uniqueness for differential and algebraic operator Riccati equations, under a distinctive set of assumptions on their unbounded coefficients. The class of boundary control systems characterized by these assumptions encompasses diverse significant physical interactions, all modeled by systems of coupled hyperbolic/parabolic partial differential equations. The proofs of uniqueness provided tackle and overcome the obstacles raised by the peculiar regularity properties of the composite dynamics. These results supplement the theories of the finite and infinite time horizon linear-quadratic problem devised by the authors jointly with Lasiecka, as the unique solution to the Riccati equation enters the closed loop form of the optimal control.

math.OC↗

arXiv2020

Improved boundary regularity for a Stokes-Lam\'e system

This paper recalls a partial differential equations system, which is the linearization of a recognized fluid-elasticity interaction three-dimensional model. A collection of regularity results for the traces of the fluid variable on the interface between the body and the fluid is established, in the case a suitable boundary dissipation is present. These regularity estimates -- in time and space, of local and global nature -- are geared toward ensuring the well-posedness of the algebraic Riccati equations which arise from the associated optimal boundary control problems on an infinite time horizon. The theory of operator semigroups and interpolation provide the main tools.

math.AP↗

arXiv2020

The Cauchy-Dirichlet problem for the Moore-Gibson-Thompson equation

The Cauchy-Dirichlet problem for the Moore-Gibson-Thompson equation is analyzed. With the focus on non-homogeneous boundary data, two approaches are offered: one is based on the theory of hyperbolic equations, while the other one uses the theory of operator semigroups. This is a mixed hyperbolic problem with a characteristic spatial boundary. Hence, the regularity results exhibit some deficiencies when compared with the non-characteristic case.

math.AP↗

arXiv2018

Feedback control of the acoustic pressure in ultrasonic wave propagation

Classical models for the propagation of ultrasound waves are the Westervelt equation, the Kuznetsov and the Khokhlov-Zabolotskaya-Kuznetsov equations. The Jordan-Moore-Gibson-Thompson equation is a prominent example of a Partial Differential Equation (PDE) model which describes the acoustic velocity potential in ultrasound wave propagation, where the paradox of infinite speed of propagation of thermal signals is eliminated; the use of the constitutive Cattaneo law for the heat flux, in place of the Fourier law, accounts for its being of third order in time. Aiming at the understanding of the fully quasilinear PDE, a great deal of attention has been recently devoted to its linearization -- referred to in the literature as the Moore-Gibson-Thompson equation -- whose mathematical analysis is also of independent interest, posing already several questions and challenges. In this work we consider and solve a quadratic control problem associated with the linear equation, formulated consistently with the goal of keeping the acoustic pressure close to a reference pressure during ultrasound excitation, as required in medical and industrial applications. While optimal control problems with smooth controls have been considered in the recent literature, we aim at relying on controls which are just $L^2$ in time; this leads to a singular control problem and to non-standard Riccati equations. In spite of the unfavourable combination of the semigroup describing the free dynamics that is not analytic, with the challenging pattern displayed by the dynamics subject to boundary control, a feedback synthesis of the optimal control as well as well-posedness of operator Riccati equations are established.

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↗

arXiv2014

Exponential decay properties of a mathematical model for a certain fluid-structure interaction

In this work, we derive a result of exponential stability for a coupled system of partial differential equations (PDEs) which governs a certain fluid-structure interaction. In particular, a three-dimensional Stokes flow interacts across a boundary interface with a two-dimensional mechanical plate equation. In the case that the PDE plate component is rotational inertia-free, one will have that solutions of this fluid-structure PDE system exhibit an exponential rate of decay. By way of proving this decay, an estimate is obtained for the resolvent of the associated semigroup generator, an estimate which is uniform for frequency domain values along the imaginary axis. Subsequently, we proceed to discuss relevant point control and boundary control scenarios for this fluid-structure PDE model, with an ultimate view to optimal control studies on both finite and infinite horizon. (Because of said exponential stability result, optimal control of the PDE on time interval $(0,\infty)$ becomes a reasonable problem for contemplation.)

math.AP↗

arXiv2013

Spectral analysis and rational decay rates of strong solutions to a fluid-structure PDE system

In this paper, we consider the problem of obtaining rational decay for a particular time-evolving fluid-structure model, the type of which has been considered in Chueshov and Ryzhkova (2013). In particular, this partial differential equation (PDE) system is composed of a three-dimensional Stokes flow which evolves within a three dimensional cavity. Moreover, on a (fixed) portion of the cavity wall, $Ω$ say, a fourth order plate equation is invoked so as to describe the displacements along $Ω$. Contact between these respective fluid and structure dynamics is established through the boundary interface $Ω$. Our main result of decay is as follows: The PDE solutions of this fluid-structure PDE, corresponding to smooth initial data, decay at the rate of $O(1/t)$. Our method of proof hinges upon the appropriate invocation of a relatively recent resolvent criterion for rational decays for linear strongly continuous semigroups.

math.AP↗

arXiv2012

A theory of the infinite horizon LQ-problem for composite systems of PDEs with boundary control

We study the infinite horizon Linear-Quadratic problem and the associated algebraic Riccati equations for systems with unbounded control actions. The operator-theoretic context is motivated by composite systems of Partial Differential Equations (PDE) with boundary or point control. Specific focus is placed on systems of coupled hyperbolic/parabolic PDE with an overall `predominant' hyperbolic character, such as, e.g., some models for thermoelastic or fluid-structure interactions. While unbounded control actions lead to Riccati equations with unbounded (operator) coefficients, unlike the parabolic case solvability of these equations becomes a major issue, owing to the lack of sufficient regularity of the solutions to the composite dynamics. In the present case, even the more general theory appealing to estimates of the singularity displayed by the kernel which occurs in the integral representation of the solution to the control system fails. A novel framework which embodies possible hyperbolic components of the dynamics has been introduced by the authors in 2005, and a full theory of the LQ-problem on a finite time horizon has been developed. The present paper provides the infinite time horizon theory, culminating in well-posedness of the corresponding (algebraic) Riccati equations. New technical challenges are encountered and new tools are needed, especially in order to pinpoint the differentiability of the optimal solution. The theory is illustrated by means of a boundary control problem arising in thermoelasticity.

math.OC↗

arXiv2010

Finite dimensional attractor for a composite system of wave/plate equations with localised damping

The long-term behaviour of solutions to a model for acoustic-structure interactions is addressed; the system is comprised of coupled semilinear wave (3D) and plate equations with nonlinear damping and critical sources. The questions of interest are: existence of a global attractor for the dynamics generated by this composite system, as well as dimensionality and regularity of the attractor. A distinct and challenging feature of the problem is the geometrically restricted dissipation on the wave component of the system. It is shown that the existence of a global attractor of finite fractal dimension -- established in a previous work by Bucci, Chueshov and Lasiecka (Comm. Pure Appl. Anal., 2007) only in the presence of full interior acoustic damping -- holds even in the case of localised dissipation. This nontrivial generalization is inspired by and consistent with the recent advances in the study of wave equations with nonlinear localised damping.

math.DS↗

arXiv2009

Regularity of boundary traces for a fluid-solid interaction model

We consider a mathematical model for the interactions of an elastic body fully immersed in a viscous, incompressible fluid. The corresponding composite PDE system comprises a linearized Navier-Stokes system and a dynamic system of elasticity; the coupling takes place on the interface between the two regions occupied by the fluid and the solid, respectively. We specifically study the regularity of boundary traces (on the interface) for the fluid velocity field. The obtained trace regularity theory for the fluid component of the system -- of interest in its own right -- establishes, in addition, solvability of the associated optimal (quadratic) control problems on a finite time interval, along with well-posedness of the corresponding operator Differential Riccati equations. These results complement the recent advances in the PDE analysis and control of the Stokes-Lamé system.

math.AP↗

arXiv2009

Optimal boundary control with critical penalization for a PDE model of fluid-solid interactions

We study the finite-horizon optimal control problem with quadratic functionals for an established fluid-structure interaction model. The coupled PDE system under investigation comprises a parabolic (the fluid) and a hyperbolic (the solid) dynamics; the coupling occurs at the interface between the regions occupied by the fluid and the solid. We establish several trace regularity results for the fluid component of the system, which are then applied to show well-posedness of the Differential Riccati Equations arising in the optimization problem. This yields the feedback synthesis of the unique optimal control, under a very weak constraint on the observation operator; in particular, the present analysis allows general functionals, such as the integral of the natural energy of the physical system. Furthermore, this work confirms that the theory developed in Acquistapace et al. [Adv. Differential Equations, 2005] -- crucially utilized here -- encompasses widely differing PDE problems, from thermoelastic systems to models of acoustic-structure and, now, fluid-structure interactions.

math.OC↗