Ademir F. Pazoto

Stabilization of a Boussinesq system of KdV–KdV type

A family of Boussinesq systems has recently been proposed by Bona, Chen, and Saut in [J.L. Bona, M. Chen, J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I. Derivation and linear theory, J. Nonlinear Sci. 12 (4) (2002) 283-318] to describe the two-way propagation of small-amplitude gravity waves on the surface of water in a canal. In this paper, we investigate the boundary stabilization of the Boussinesq system of KdV-KdV type posed on a bounded domain. We design a two-parameter family of feedback laws for which the solutions issuing from small data are globally defined and exponentially decreasing in the energy space.

ON THE CONTROLLABILITY OF A COUPLED SYSTEM OF TWO KORTEWEG–DE VRIES EQUATIONS

This paper proves the local exact boundary controllability property of a nonlinear system of two coupled Korteweg–de Vries equations which models the interactions of weakly nonlinear gravity waves (see [10]). Following the method in [24], which combines the analysis of the linearized system and the Banachs fixed point theorem, the controllability problem is reduced to prove a nonstandard unique continuation property of the eigenfunctions of the corresponding differential operator.

Inverse problem for the heat equation and the Schrödinger equation on a tree

In this paper, we establish global Carleman estimates for the heat and Schrodinger equations on a network. The heat equation is considered on a general tree and the Schrodinger equation on a star-shaped tree. The Carleman inequalities are used to prove the Lipschitz stability for an inverse problem consisting in retrieving a stationary potential in the heat (resp. Schrodinger) equation from boundary measurements.

Stabilization of the Gear–Grimshaw system on a periodic domain

This paper is devoted to the study of a nonlinear coupled system of two Korteweg–de Vries equations in a periodic domain under the effect of an internal damping term. The system was introduced by Gear and Grimshaw to model the interactions of two-dimensional, long, internal gravity waves propagation in a stratified fluid. Designing a time-varying feedback law and using a Lyapunov approach, we establish the exponential stability of the solutions in Sobolev spaces of any positive integral order.

Parameters of connectivity in (a,b)-linear graphs

Abstract For some a and b positive rational numbers, a simple graph with n vertices and m = a n − b edges is an ( a , b ) -linear graph, when n > 2 b . We characterize non-empty classes of ( a , b ) -linear graphs and determine those which contain connected graphs. For non-empty classes, we build sequences of ( a , b ) -linear graphs and sequences of connected ( a , b ) -linear graphs. Furthermore, for each of these sequences where every graph is bounded by a constant, we show that its correspondent sequence of diameters diverges, while its correspondent sequence of algebraic connectivities converges to zero.

UNIFORM STABILIZATION OF A PLATE EQUATION WITH NONLINEAR LOCALIZED DISSIPATION

We study the existence and uniqueness of a plate equation in a bounded domain of Rⁿ, with a dissipative nonlinear term, localized in a neighborhood of part of the boundary of the domain. We use techniques from control theory, the unique continuation property and Nakao method to prove the uniform stabilization of the energy of the system with algebraic decay rates depending on the order of the nonlinearity of the dissipative term.

Stabilization of a Boussinesq system with generalized damping

Abstract A family of Boussinesq systems was proposed by J. L. Bona, M. Chen and J.-C. Saut to describe the two-way propagation of small amplitude gravity waves on the surface of water in a canal. Our work considers a class of these Boussinesq systems which couples two Benjamin–Bona–Mahony with periodic boundary conditions. We study the stability properties of the resulting system when generalized damping operators are introduced in each equation. By means of spectral analysis and Fourier expansion, we prove that the solutions of the linearized system decay uniformly or not to zero, depending on the parameters of the damping operators. In the uniform decay case, we show that the same property holds for the nonlinear system.

Control of a Boussinesq system of KdV-KdV type on a bounded interval

We consider a Boussinesq system of KdV-KdV type introduced by J. Bona, M. Chen and J.-C. Saut as a model for the motion of small amplitude long waves on the surface of an ideal fluid. This system of two equations can describe the propagation of waves in both directions, while the single KdV equation is limited to unidirectional waves. We are concerned here with the exact controllability of the Boussinesq system by using some boundary controls. By reducing the controllability problem to a spectral problem which is solved by using the Paley-Wiener method introduced by the third author for KdV, we determine explicitly all the critical lengths for which the exact controllability fails for the linearized system, and give a complete picture of the controllability results with one or two boundary controls of Dirichlet or Neumann type. The extension of the exact controllability to the full Boussinesq system is derived in the energy space in the case of a control of Neumann type. It is obtained by incorporating a boundary feedback in the control in order to ensure a global Kato smoothing effect.

Boundary controllability of a nonlinear coupled system of two Korteweg–de Vries equations with critical size restrictions on the spatial domain

AbstractThis article is dedicated to improve the controllability results obtained by Cerpa and Pazoto (Commun Contemp Math 13:183–189, 2011) and by Micu et al. (Commun Contemp Math 11(5):779–827, 2009) for a nonlinear coupled system of two Korteweg–de Vries equations posed on a bounded interval. Initially, Micu et al. (2009) proved that the nonlinear system is exactly controllable by using four boundary controls without any restriction on the length L of the interval. Later on, in Cerpa and Pazoto (2011), two boundary controls were considered to prove that the same system is exactly controllable for small values of the length L and large time of control T. Here, we use the ideas contained in Capistrano-Filho et al. (Z Angew Math Phys 67(5):67–109, 2016) to prove that, with another configuration of four controls, it is possible to prove the existence of the so-called critical length phenomenon for the linear system, i.e., whether the system is controllable depends on the length of the spatial domain. In addition, when we consider only one control input, the boundary controllability still holds for suitable values of the length L and time of control T. In both cases, the control spaces are sharp due a technical lemma which reveals a hidden regularity for the solution of the adjoint system.