arXiv subjects

Marcelo Bongarti

Publications and source records attributed to Marcelo Bongarti.

12 recordsLinked to original sources

arXiv2026

A free boundary problem driven by boundary distance in the coincidence set

We study a free boundary problem of minimising a functional containing a non-local term rewarding depth into the zero phase: for $u\geqslant 0$ on a bounded, open set $Ω\subset\mathbb R^d$, we minimise $$J(u) = \int_Ω\left(\frac12|\nabla u|^2 - fu\right) \;-\; \int_{\{u=0\}} F\big(\mathrm{dist}(x,\partial \{u=0\})\big)\;\mathrm{d}x. $$ This kind of functional arises, for example, from a two-membranes problem with an adhesive contact energy. We first address a well-definedness issue caused by the non-local, boundary-sensitive nature of the functional and prove existence of minimisers, establishing along the way a weak lower semicontinuity result for the non-local term. We then derive stationarity conditions for minimisers, including a variational (Euler--Lagrange type) inequality, a PDE on the positivity set, and, under a mild non-degeneracy assumption, a free boundary condition obtained via inner variations and a Danskin-type differentiation of the distance function.

math.AP↗

arXiv2026

State constrained convex Nash equilibrium problems coupled with linear hyperbolic PDEs

We study the existence of equilibria for state constrained, convex generalized Nash equilibrium problems (GNEPs) coupled with hyperbolic partial differential equations (PDEs). Analogous problems have been addressed for state constrained GNEPs coupled with elliptic and parabolic PDEs, respectively, but the problematic regularity of the set-valued constraint maps has been a barrier for development of an existence theory in the hyperbolic case. This is mainly due to compactness issues with the strategy-to-state maps. Beyond existence, we also provide first-order optimality conditions for a class of fairly general linear hyperbolic PDEs and show its relevance for applications such as the wave equation, advertising dynamics, and the linearized isothermal Euler system on networks.

math.OC↗

arXiv2026

Structure versus regularity of set-valued maps in convex generalized Nash equilibrium problems in Banach spaces

A generalized Nash equilibrium problem (GNEP) in Banach space consists of $N>1$ optimal control problems with couplings in both the objective functions and, most importantly, in the feasible sets. We address the existence of equilibria for convex GNEPs in Banach space. We show that the standard assumption of lower semicontinuity of the set-valued constraint maps - foundational in the current literature on GNEPs - can be replaced by graph convexity or the so-called Knaster-Kuratowski-Mazurkiewicz (KKM) property. Lower semicontinuity is often essential for obtaining upper semicontinuity of best response maps, crucial for the existence theory based on Kakutani-Fan fixed-point arguments. However, in function spaces or in settings with partial differential equation (PDE) constraints, verifying lower semicontinuity becomes much more challenging (even in convex cases), whereas graph convexity, for example, is often straightforward to check. Our results unify several existence theorems in the literature and clarify the structural role of constraint maps. We also extend Rosen's uniqueness condition to Banach spaces using a multiplier bias framework.

math.OC↗

arXiv2025

Optimal Control of a Reaction-Diffusion Epidemic Model with Noncompliance

In this paper, we consider an optimal distributed control problem for a reaction-diffusion-based SIR epidemic model with human behavioral effects. We develop a model wherein non-pharmaceutical intervention methods are implemented, but a portion of the population does not comply with them, and this noncompliance affects the spread of the disease. Drawing from social contagion theory, our model allows for the spread of noncompliance parallel to the spread of the disease. The quantities of interest for control are the reduction in infection rate among the compliant population, the rate of spread of noncompliance, and the rate at which non-compliant individuals become compliant after, e.g., receiving more or better information about the underlying disease. We prove the existence of global-in-time solutions for fixed controls and study the regularity properties of the resulting control-to-state map. The existence of optimal control is then established in an abstract framework for a fairly general class of objective functions. Necessary first--order optimality conditions are obtained via a Lagrangian based stationarity system. We conclude with a discussion regarding minimization of the size of infected and non-compliant populations and present simulations with various parameters values to demonstrate the behavior of the model.

math.AP↗

arXiv2023

Optimal boundary control of the isothermal semilinear Euler equation for gas dynamics on a network

The analysis and boundary optimal control of the nonlinear transport of gas on a network of pipelines is considered. The evolution of the gas distribution on a given pipe is modeled by an isothermal semilinear compressible Euler system in one space dimension. On the network, solutions satisfying (at nodes) the so called Kirchhoff flux continuity conditions are shown to exist in a neighborhood of an equilibrium state. The associated nonlinear optimization problem then aims at steering such dynamics to a given target distribution by means of suitable (network) boundary controls while keeping the distribution within given (state) constraints. The existence of local optimal controls is established and a corresponding Karush-Kuhn-Tucker (KKT) stationarity system with an almost surely non-singular Lagrange multiplier is derived.

math.OC↗

arXiv2022

Reducing Memory Requirements of Quantum Optimal Control

Quantum optimal control problems are typically solved by gradient-based algorithms such as GRAPE, which suffer from exponential growth in storage with increasing number of qubits and linear growth in memory requirements with increasing number of time steps. These memory requirements are a barrier for simulating large models or long time spans. We have created a nonstandard automatic differentiation technique that can compute gradients needed by GRAPE by exploiting the fact that the inverse of a unitary matrix is its conjugate transpose. Our approach significantly reduces the memory requirements for GRAPE, at the cost of a reasonable amount of recomputation. We present benchmark results based on an implementation in JAX.

quant-ph↗

arXiv2021

Boundary stabilization of the linear MGT equation with partially absorbing boundary data and degenerate viscoelasticity

The Jordan--Moore--Gibson--Thompson (JMGT) equation is a well-established and recently widely studied model for nonlinear acoustics (NLA). It is a third-order (in time) semilinear Partial Differential Equation (PDE) model with the distinctive feature of predicting the propagation of ultrasound waves at \textit{finite} speed due to heat phenomenon know as \textit{second sound} which leads to the hyperbolic character of heat propagation. In this paper, we consider the problem of stabilizability of the linear (known as) MGT--equation. We consider a special geometry that is suitable for studying the problem of controlling (from the boundary) the acoustic pressure involved in medical treatments like lithotripsy, thermotherapy, sonochemistry, or any other procedures using High Intensity Focused Ultrasound (HIFU).

math.AP↗

arXiv2021

Boundary feedback stabilization of a critical nonlinear JMGT equation with Neumann-undissipated part of the boundary

Boundary feedback stabilization of a critical, nonlinear Jordan--Moore--Gibson--Thompson (JMGT) equation is considered. JMGT arises in modeling of acoustic waves involved in medical/engineering treatments like lithotripsy, thermotherapy, sonochemistry, or any other procedures using High Intensity Focused Ultrasound (HIFU). It is a well-established and recently widely studied model for nonlinear acoustics (NLA): a third--order (in time) semilinear Partial Differential Equation (PDE) with the distinctive feature of predicting the propagation of ultrasound waves at \textit{finite} speed due to heat phenomenon know as \textit{second sound} which leads to the hyperbolic character of heat propagation. In practice, the JMGT dynamics is largely used for modeling the evolution of the acoustic velocity and, most importantly, the acoustic pressure as sound waves propagate through certain media. %Due to its sensitivity to different media, such model (or similar) is often used for medical/engineering treatments such as lithotripsy, thermotherapy, sonochemistry, or any other procedures using High Intensity Focused Ultrasound (HIFU). In this work, \emph{critical} refers to (usual) case where media--damping effects are non--existent or non--measurable and therefore cannot be relied upon for stabilization purposes. In this paper the issue of boundary stabilizability of originally unstable (JMGT) equation is resolved. Motivated by modeling aspects in HIFU technology, boundary feedback is supported only on a portion of the boundary, while the remaining part of the boundary is left free (available to control actions) . Since the boundary conditions imposed on the "free" part of the boundary fail to satisfy Lopatinski condition (unlike Dirichlet boundary conditions), the analysis of uniform stabilization from the boundary becomes very subtle and requires careful geometric considerations.

math.AP↗

arXiv2021

Alguns Teoremas do Tipo Valor Médio: de Lagrange a Malesevic

Our goal in this work is to present some mean value type theorems that are not studied in classic calculus and analysis courses. They are simple theorems yet with large applicability in mathematical analysis (for example, in the study of functional equations and integral operators), computational mathematics, economics among other areas.

math.HO↗

arXiv2020

Boundary Stabilization of the linear MGT equation with Feedback Neumann control

The Jordan-Moore-Gibson-Thompson (JMGT)\cite{christov_heat_2005,jordan_nonlinear_2008,straughan_heat_2014} equation is a benchmark model describing propagation of nonlinear acoustic waves in heterogeneous fluids at rest. This is a third-order (in time) dynamics which accounts for a finite speed of propagation of heat signals (see \cite{coulouvrat_equations_1992,crighton_model_1979,jordan_nonlinear_2008,jordan_second-sound_2014,kaltenbacher_jordan-moore-gibson-thompson_2019}). In this paper, we study a boundary stabilization of linearized version (also known as MGT-equation) in the {\it critical case}, configuration in which the smallness of the diffusion effects leads to conservative dynamics \cite{kaltenbacher_wellposedness_2011}. Through a single measurement in {\it{feedback}} form made on a non-empty, relatively open portion of the boundary under natural geometric conditions, we were able to obtain uniform exponential stability results that are, in addition, uniform with respect to the space-dependent viscoelasticity parameter which no longer needs to be assumed positive and in fact can be degenerate and taken to be zero on the whole domain. This result, of independent interest in the area of boundary stabilization of MGT equations, provides a necessary first step for the study of optimal boundary feedback control on {\it infinite horizon} \cite{bucci_feedback_2019}.

math.AP↗

arXiv2020

Vanishing relaxation time dynamics of the Jordan Moore-Gibson-Thompson equation arising in nonlinear acoustics

The (third-order in time) JMGT equation \cite{Jordan2,HCP} is a nonlinear (quasi-linear) Partial Differential Equation (PDE) model introduced to describe a nonlinear propagation of sound in an acoustic medium. The important feature is that the model avoids the infinite speed of propagation paradox associated with a classical second order in time equation referred to as Westervelt equation. Replacing Fourier's law by Maxwell-Cattaneo's law gives rise to the third order in time derivative scaled by a small parameter $τ>0$, the latter represents the thermal relaxation time parameter and is intrinsic to the medium where the dynamics occur. In this paper we provide an asymptotic analysis of the third order model when $τ\rightarrow 0 $. It is shown that the corresponding solutions converge {\it in a strong topology of the phase space } to a limit which is the solution of Westervelt equation. In addition, rate of convergence is provided for solutions displaying higher order regularity. This addresses an open question raised in \cite{kaltev2}, where a related JMGT equation has been studied and {\it weak star } convergence of the solutions when $τ\rightarrow 0$ has been established. Thus, our main contribution is showing {\it strong convergence on infinite time horizon,} along with related rates of convergence valid on a finite time horizon. The key to unlocking the difficulty owns to a tight control and propagation of the "smallness" of the initial data in carrying the estimates at three different topological levels. The rate of convergence allows one then to estimate the relaxation time needed for the signal to reach the target. The interest in studying this type of problems is motivated by a large array of applications arising in engineering and medical sciences.

math.AP↗

arXiv2020

Singular Thermal Relaxation Limit for the Moore-Gibson-Thompson Equation Arising in Propagation of Acoustic Waves

Moore-Gibson-Thompson (MGT) equations, which describe acoustic waves in a heterogeneous medium, are considered. These are the third order in time evolutions of a predominantly hyperbolic type. MGT models account for a finite speed propagation due to the appearance of thermal relaxation coefficient τ {>} {0} in front of the third order time derivative. Since the values of τ are relatively small and often negligible, it is important to understand the asymptotic behavior and characteristics of the model when τ {\to} {0}. This is a particularly delicate issue since the τ- dynamics is governed by a generator which is singular as τ {\to} {0}. It turns out that the limit dynamics corresponds to the linearized Westervelt equation which is of a parabolic type. In this paper, we provide a rigorous analysis of the asymptotics which includes strong convergence of the corresponding evolutions over infinite horizon. This is obtained by studying convergence rates along with the uniform exponential stability of the third order evolutions. Spectral analysis for the MGT-equation along with a discussion of spectral uppersemicontinuity for both equations (MGT and linearized Westervelt) will also be provided.

math.AP↗