J. Límaco

Universidade Federal Fluminense | UFF · Matematica Aplicada

This paper discusses the application of the Nash strategy to a quasi-linear parabolic equation with a semilinear term. First, we demonstrate the existence of a Nash quasi-equilibrium for both equations using the fixed point method. Subsequently, we establish that the functionals are convex, ensuring that the Nash quasiequilibrium is, in fact, a Na...

In this paper, we deal with the local null controllability of an initial boundary value problem for a thermistor equation. The control is distributed, locally in space. The main ingredients of the proof are suitable Carleman estimates for an adjoint system and Liusternik’s Inverse Mapping Theorem in Hilbert spaces. As a complement to the theoretica...

In this paper, we prove the well‐posedness of a nonlinear wave equation coupled with boundary conditions of Dirichlet and acoustic type imposed on disjoints open boundary subsets. The proposed nonlinear equation models small vertical vibrations of an elastic medium with weak internal damping and a general nonlinear term. We also prove the exponenti...

We investigate Pareto equilibria for bi-objective optimal control problems. Our framework comprises the situation in which an agent acts with a distributed control in a portion of a given domain, and aims to achieve two distinct (possibly conflicting) targets. We analyze systems governed by linear and semilinear heat equations and also systems with...

This paper deals with the hierarchical control of a nonlinear parabolic equation in one dimension. The novelty in this work is the appearance of the spatial derivative of the solution instead of considering only the solution in the quasilinear term (nonlinearity), here lies the difficulty of approaching said equation. We use Stackelberg–Nash strate...

In this work, we prove a Carleman estimate, which allows us to obtain the local null-controllability for a class of strongly degenerate parabolic equations with nonlocal terms, naturally extending the main result of Demarque R et al. (Nonlin Anal: Real World Appl 43:523 – 547, 2018). We present a theoretical approach, applying Lyusternik’s Inverse...

In this paper, we prove the well-posedness of a nonlinear wave equation coupled with boundary conditions of Dirichlet and acoustic type imposed on disjoints open boundary subsets. The proposed nonlinear equation models small vertical vibrations of an elastic medium with weak internal damping and a general nonlinear term. We also prove the exponenti...

This paper concerns the null control of quasi-linear parabolic systems where the diffusion coefficient depends on the gradient of the state variable. In our main theoretical result, with some assumptions on the regularity and growth of the diffusion coefficient and regular initial data, we prove that local null controllability holds. To this purpos...

This paper deals with the analysis of the internal control of a free-boundary problem for the 1D heat equation with local and nonlocal nonlinearities. We prove a local null controllability result with distributed controls, locally suported in space. The proof is based on Schauder’s fixed point theorem combined with some appropriate specific estimat...

We investigate the null controllability property of systems that mathematically describe the dynamics of some non-Newtonian incompressible viscous flows. The principal model we study was proposed by O. A. Ladyzhenskaya, although the techniques we develop here apply to other fluids having a shear-dependent viscosity. Taking advantage of the Pontryag...

MSC: 35B37 35L15 93B05 93C10 93C20 Keywords: Hyperbolic system Nonlocal terms Exact controllability Observability inequality a b s t r a c t This paper deals with the internal and boundary exact controllability of some nonlinear hyperbolic systems with local and nonlocal nonlinearities in dimension one. Nonlocal terms in the space and time variable...

This paper deals with the application of Stackelberg–Nash strategies for the controllability of parabolic partial differential equations, with nonlinear diffusion terms in the spatial variable with dimension N (here N is an any positive integer). We consider three cases: in the first one, the main control (the leader) acts in the interior of the do...

This paper is devoted to the theoretical and numerical analysis of the null controllability of a quasi-linear parabolic equation. First, we establish a local controllability result. The proof relies on an appropriate inverse function argument. Then, we formulate an iterative algorithm for the computation of the null control and we prove a convergen...

In this paper, we establish a local null controllability result for a nonlinear parabolic PDE with local and nonlocal nonlinearities in a domain whose boundary moves in time by a control force with a multiplicative part acting on a prescribed subdomain. We prove that, if the initial data is sufficiently small and the linearized system at zero satis...

In this work, we prove a Carleman estimate for a parabolic problem which has a dissipative degenerate term. The prove relies on choose a suitable weight function that change of sign inside the control domain.

This paper deals with the numerical implementation of a systematic method for solv- ing bi-objective optimal control problems for wave equations. More precisely, we look for Nash and Pareto equilibria which respectively correspond to appropriate noncooperative and cooperative strategies in multi-objective optimal control. The numerical methods desc...

This paper deals with the hierarchical control of the parabolic equation.We use Stackelberg{Nash strategies. As usual, we consider one leader and two followers. To each leader we associate a Nash equilibrium corresponding to a bi-objective optimal control problem, then, we look for a leader that solves null controllability e with trajectories probl...

This work deals with the null controllability of an initial boundary value problem for a parabolic-elliptic coupled system with nonlinear terms of local and non-local kinds. The control is distributed, locally in space and appears only in one PDE. We first prove that, if the initial data is sufficiently small and the linearized system at zero satis...

This paper deals with the analysis of the internal control of a parabolic PDE with nonlinear diffusion, nonlocal in space. In our main result, we prove the local exact controllability to the trajectories with distributed controls , locally supported in space. The main ingredients of the proof are a compactness-uniqueness argument and Kakutani's Fix...

This paper deals with the analysis of the internal control of a parabolic PDE with nonlinear diffusion, nonlocal in space. In our main result, we prove the local exact controllability to the trajectories with distributed controls , locally supported in space. The main ingredients of the proof are a compactness-uniqueness argument and Kakutani's Fix...

Nonlinear electromagneto-elasticity with moving boundary is presented in this work. We introduce a diffeomorphism operator that transforms the moving boundary system into an equivalent one with fixed boundary and we apply the Galerkin's method and results of compactness for obtain the existence and uniqueness of solution.

The original version of this article unfortunately contained a mistake. On page 653, Equation (52) should be

In this paper, we are concerned with the internal control of a class of one-dimensional nonlinear parabolic systems with nonlocal and weakly degenerate diffusion coefficients. Our main theorem stablishes a local null controllability result with only one internal control for a system of two equations. The proof, based on the ideias developed by Furs...

This paper deals with the null controllability of a differential turbulence model of the Ladyzhenskaya-Smagorinsky kind. In the equations, we find local and nonlocal nonlinearities: the usual transport terms and a turbulent viscosity that depends on the global in space energy dissipated by the mean flow. We prove that the N-systems are locally null...

In this paper, we investigate the controllability for the one-dimensional plate equation in intervals with a moving boundary. This equation models the vertical displacement of a point x at time t in a bar with uniform cross section. We assume the ends of the bar with small and uniform variations. More precisely, we have introduced functions α(t) an...

In this paper, we are concerned with the internal control of a class of one-dimensional nonlinear parabolic systems with nonlocal and weakly degenerate diffusion coefficients. Our main theorem establishes a local null controllability result with only one internal control for a system of two equations. The proof, based on the ideias developed by Fur...

The purpose of this article is to give a new proof of a null controllability result for a 1D free-boundary problem of the Stefan kind for a heat PDE. We introduce a method based on local inversion that, in contrast with other previous arguments, does not rely on any compactness property and can be generalized to higher dimensions.

The purpose of this article is to give a new proof of a null controllability result for a 1D free-boundary problem of the Stefan kind for a heat PDE. We introduce a method based on local inversion that, in contrast with other previous arguments, does not rely on any compactness property and can be generalized to higher dimensions.

The purpose of this article is to give a new proof of a null controllability result for a 1D free-boundary problem of the Stefan kind for a heat PDE. We introduce a method based on local inversion that, in contrast with other previous arguments, does not rely on any compactness property and can be generalized to higher dimensions.

We establish a local null controllability result for following nonlinear parabolic equation: ut−(b(x,∫10u)ux)x+f(t,x,u)=hχω,(t,x)∈(0,T)×(0,1) where b(x,r)=ℓ(r)a(x) is a function with separated variables that defines an operator which degenerates at x=0 and has a nonlocal term. Our approach relies on an application of Liusternik’s inverse mapping t...

This paper deals with the optimal control of a mathematical model for the evolution of a low-grade glioma (LGG). We will consider a model of the Fischer-Kolmogorov kind for two compartments of tumor cells, using ideas from Galochkina, Bratus and Perez-Garcia [10] and Perez-Garcia [17]. The controls are of the form (t(1),..., t(n);d(1),..., d(n)), w...

This dissertation presents theoretical and numerical results for some Partial Differential Equations (PDEs). For the distinction of the topics addressed during the development of this work, we have divided its goals into four parts. Precisely, in the first two chapters, we focus on the development of some results associated with the k–ε model, and...

n this brief communication we present a new integral transform, so far un- known, which is applicable, for instance, to studying the kinetic theory of natural eigenmodes or transport excited in plasmas with bounded distribution functions such as in Q machines/plasma diodes or in the scrap-off layer of Tokamak fusion plasmas. The results are valid f...

We establish a local null controllability result for following the nonlinear parabolic equation: $$u_t-\left(b\left(x,\int_0^1u \ \right)u_x \right)_x+f(t,x,u)=h\chi_\omega,\ (t,x)\in (0,T)\times (0,1) $$ where $b(x,r)=\ell(r)a(x)$ is a function with separated variables that defines an operator which degenerates at $x=0$ and has a nonlocal term. Ou...

In this paper, we prove controllability results for some linear and semilinear systems where we find two parabolic PDEs and one elliptic PDE and we act through one locally supported in space scalar control. The arguments rely on a careful analysis of the linear case and an application of an inverse function theorem. The facts that we act through a...

This paper deals with the local null control of a free-boundary problem for the classical 1D heat equation with distributed controls, locally supported in space. In the main result we prove that, if the final time T is fixed and the initial state is sufficiently small, there exist controls that drive the state exactly to rest at time t=T.

In this article, the authors establish the local null controllability property for semilinear parabolic systems in a domain whose boundary moves in time by a single control force acting on a prescribed subdomain. The proof is based on Kakutani’s fixed point theorem combined with observability estimates for the associated linearized system.

This paper deals with the control of a differential turbulence model of the Ladyzhenskaya–Smagorinsky kind. In the equations we find local and nonlocal nonlinearities: the usual transport terms and a turbulent viscosity that depends on the global in space energy dissipated by the mean flow. We prove that the system is locally null-controllable, wit...

In this article the authors investigated the existence of solutions for the linear thennoelastic system in a noncylindrical domain ...

This paper deals with the null controllability of an initial-boundary value problem for a parabolic coupled system with nonlinear terms of local and nonlocal kinds. The control is distributed in space and time and is exerted through one scalar function whose support can be arbitrarily small. We first prove that, if the initial data are sufficiently...

In this paper, we prove the null controllability of some parabolic-elliptic systems. The control is distributed, locally supported in space and appears only in one PDE. The arguments rely on fixed-point reformulation and suitable Carleman estimates for the solutions to the adjoint system. Under appropriate assumptions, we also prove that the soluti...

We investigate the existence and uniqueness of weak solution for a mixed problem for wave operator of the type: L(u) = ∂ 2u ∂t 2 - δu + |u|ρ- f, ρ> 1. The operator is defined for real functions u = u(x, t) and f = f(x, t) where (x, t) ε Q a bounded cylinder of R n+1. The nonlinearity |u|ρbrings serious difficulties to obtain global a priori estimat...

In this article, we prove the existence of solutions for an hyperbolic equation known as the Benjamin-Bona-Mahony equation. Our study involves increasing , decreasing, and mixed non-cylindrical domains and for this analysis, our main tools are the change of variable technique, the Galerkin and penalization method.

In this paper, we analyze from the mathematical point of view a model for small vertical vibrations of an elastic string with weak internal damping and quadratic term, coupled with mixed boundary conditions of Dirichlet type and acoustic type. Our goal is to extend some of the results of Frota-Goldstein work in the sense of considering a weaker int...

This paper is concerned with a detailed exposition on the Carleman inequality for a parabolic equation. Specifically, it represents only a part of the work of A. V. Fursikov & O. Yu Imanovilov [7] for the particular model p t − ∆p + f (p) = h of the heat equation. Moreover, we study the null controllability employing fixe points for multi-valued ma...

In this paper we investigate the existence and uniqueness of global solutions, and a rate stability for the energy related with a Cauchy problem to the viscous Burgers equation in unbounded domain $\mathbb{R}\times(0,\infty)$. Some aspects associated with a Cauchy problem are presented in order to employ the approximations of Faedo-Galerkin in whol...

In this work we present the existence, the uniqueness and numerical solutions for a mathematical model associated with equations of Benjamin–Bona–Mahony type in a domain with moving boundary. We apply the Galerkin method, multiplier techniques, energy estimates and compactness results to obtain the existence and uniqueness. For numerical solutions,...

This article concerns the initial-boundary value problem for the nonlinear Boussinesq equations on time dependent domains in $mathbb{R}^n$ with $1leq n leq 4$. Global solvability, uniqueness of solutions and the exponential decay to the energy are established provided the initial data are bounded in some sense.

This paper contains results about hierarchic control in domains of Rn+1 with moving boundaries. It develops the Stackelberg–Nash strategy for this type of domains. It also contains a proof of weak and strong solutions for an optimality system.

In this article, we present results concerning the existence, uniqueness and the asymptotic behavior of solutions for a beam evolution equation with variable coefficients in noncylindrical domains.

In this paper we analyze from the mathematical point of view a model for small vertical vibrations of an elastic string with fixed ends and the density of the material being not constant. We employ techniques of functional analysis, mainly a theorem of compactness for the analysis of the approximation of Faedo–Galerkin method. We obtain strong glob...

In this article, we present results concerning with the existence of global solutions and a rate decay estimate for energy associated with an initial and boundary value problem for a beam evolution equation with variable coefficients in non-cylindrical domains. Copyright © 2007 John Wiley & Sons, Ltd.

We investigate the initial–boundary value problem for the one-dimensional nonlinear Boussinesq equation inside domains with moving ends having both small increasing and decreasing displacements. Global solvability, uniqueness of solutions and the exponential decay to the energy are established provided the initial data are bounded in some sense.

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In this article we consider the quasi linear evolution Carrier equa-tion when the non-linear term depends on the position at the point x, elapsed time t and of the transverse deformation of the segment of the string on the action of a linear internal damping. The Carrier equation is studied first in a cylindrical domain with objective of establishi...

We investigate an initial-boundary value problem for equations of Benjamin–Bona–Mahony (BBM) type in two different physical situations. In the first, the mixed problem is considered on a cylinder domain Q of Rn×Rt. In the second one, the mixed problem is studied inside of an increasing noncylindrical domain of Rn×Rt. In both situations we show the...

A nonlinear partial differential equation of the following form is considered: u'-\div\Big(a(u)\nabla u\Big)+ b(u)\;\vert\nabla u\vert^2=0, which arises from the heat conduction problems with strong temperature-dependent material parameters, such as mass density, specific heat and heat conductivity. Existence, uniqueness and asymptotic behavior of...

In this work, we are interested in obtaining existence, uniqueness of the solution and an approximate numerical solution for the model of linear thermoelasticity with moving boundary. We apply finite element method with finite difference for evolution in time to obtain an approximate numerical solution. Some numerical experiments were presented to...

A nonlinear partial differential equation of the form u ' - div a ( u ) ∇ u+b(u)|∇u| 2 =0 is considered, which arises from heat conduction problems with strong temperature-dependent material parameters, such as mass density, specific heat and heat conductivity. Existence, uniqueness and asymptotic behavior of solutions to initial-boundary value pro...

In this article we are concerned with the existence and uniqueness of global weak solutions of a mixed problem associated with one-dimensional damped elastic stretched string equation when the supports of the ends have small displacements. In addition, we show that the energy decays exponentially. In previous investigations about string equation in...

In this paper we investigate a model of Kirchhoff type for vibrations of elastic bodies represented by bounded open sets Ot of JR.n when the boundaries ft are moving with the time t. With restrictions on the rest position and the initial velocity we prove global existence and uniqueness of solutions, in the Sobolev class, for a certain mixed proble...

We investigate the following initial-boundary value problem for the nonlinear beam equation with variable coefficients on the action of a linear internal damping: (a(x,t)u ' (x,t)) ' +Δ(b(x,t)Δu(x,t))-Mx , t , ∫ Ω |∇u(x,t)| 2 d xΔu(x,t)+δu ' (x,t)=0inQ, u(x,t)=∂u ∂ν(x,t)=0onΣ, u(x,0)=u 0 (x),u ' (x,0)=u 1 (x)inΩ, where Ω is a non-empty bounded open...

In this work we are concerned with the existence and uniqueness of strong global solutions and exponential decay of the total energy for an initial-boundary value problem associated with the Kirchhoff equation with variable coefficients on the action of a nonlinear internal damping.

In this study we showed the existence of weak solutions of equations that represent flows of a non-homogeneous viscous incompressible fluids in a non cylindrical domain in R3 . The classical Navier-stokes equation is a particular case of the equations here considered.

The objective of this paper is to establish existence, uniqueness and regularity of solutions for a mixed problem associated with equations of Benjamin-Bona-Mahony type in a domain Q̂ with moving boundary. The technique, to show the existence of solutions, consists in transforming Q̂ into a cylinder Q by using a diffeomorphism ft and to apply in Q...

We investigate finite approximate controllability for semilinear heat equation in noncylindrical domains. First we study the linearized problem and then by an application of the fixed point result of Leray-Schauder we obtain the finite approximate controllability for the semilinear state equation.

In this article, we prove the existence of solutions for a hyperbolic equation known as the the Rosenau and Benjamin-Bona-Mahony equations. We study increasing, decreasing, and mixed non-cylindrical domains. Our main tools are the Galerkin method, multiplier techniques, and energy estimates.

We investigate in this article the null controllability for the semilinear heat operator u − ∆u + f (u) in a domain which boundary is moving with the time t.

This work studies the problem of the exact controlability in the boundary of the equation u tt + u xxxx = 0 in a domain with moving boundary.

We investigate a boundary value problem for a nonlinear evolution biharmonic operator motivated by flexion of fully clamped beam in two different physical situations. In the first, the supports of the ends of the beam are fixed and in the second one, the supports of the ends of the beam have small displacements.

The motion for small-amplitude of a thin elastic planel with both linear-weak and linear-strong damping are investigated in one-dimensional domain with moving boundary. Existence and uniqueness of global solutions are shown. Asymptotic behavior of energies are also established.

Dedicated to Professor Jacque-Louis Lions on the occasion of his 70th birthday We consider a mixed problem for the operator [^(L)]u( x,t ) = \frac¶2 u¶t2 - ( a( t ) + b( t )òa( t )b( t ) ( \frac¶u¶x ) 2 dx )\frac¶2 u¶x2 \hat Lu\left( {x,t} \right) = \frac{{\partial ^2 u}}{{\partial t^2 }} - \left( {a\left( t \right) + b\left( t \right)\int_{\alpha...

This paper is divided into two parts. Part One Chapter 1 contains the deduction of the model for vertical vibrations of elastic strings with moving ends and its relations to the models of d'Alembert and Kirchhoff-Carrier. Chapter 2 contains the results of investigations related to Kirchhoff-Carrier model during the last twenty years and an almost c...

We consider the linear heat equation with potentials in an n-dimensional domain of the form Ω ε =Ω×(0,ε), where Ω is a bounded, smooth open set of ℝ n-1 , with n≥2 and ε a small parameter. We study the null controllability problem when the control acts in a cylindrical subdomain ω ε =ω×(0,ε), where ω⊂Ω is an open and non empty subset of Ω. We prove...

A model is deduced for the small deformations of an elastic circular membrane with moving boundary. This model is analyzed for the membrane identified with a unit disk. The surface tension is investigated as the deformation of the disk.

We prove the existence and uniqueness of the global solution to a Cauchy problem for the nonlinear hyperbolic equation u '' -M(∥|u|∥ 2 )Δu+ρ(t,u ' )=0· We also establish the exponential decay of the energy when t→+∞.

The existence and uniqueness of local and global solutions for the Kirchhofi{Carrier nonlinear model for the vibrations of elastic strings in noncylindrical domains are investigated by means of the Galerkin Method. The asymptotic behaviour of the energy is also studied.

We study the existence of weak solutions to the Navier-Stokes equation defined in a noncylindrical domain Q ^=∪ 0≤t≤T Ω t ×{t},Ω t =K ( t ) y , y ∈ Ω,Ω⊂ℝ n , and Q ^ is not necessarily increasing or decreasing in time. Regularity and uniqueness of solutions for the case n=2 are also considered.

Applications of the method of elliptic regularization to the Navier{Stokes system in a bounded domain of R 3 with moving boundary are given. We follow the ideas of J. L. Lions 7] and Temam 13]. reziume. naSromSi moKvanil ia R 3-Si moTavsebul moZr avsazG vr ian SemosazG vr ul areSi navie-stoqsis sist emisaTvis eliPsu r i regu l ar-izaciis meTod is z...

In this work, we are interested in obtaining the existence, uniqueness of the solution and an approximated numerical solution for the model of linear thermoelasticity with moving boundary. We apply the finite element method with a finite dierence method to obtain an approximated numerical solution. Some numerical experiments were presented to show...