J. Límaco

Theoretical and numerical local null controllability for a parabolic system with local and nonlocal nonlinearities

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 small and the linearized system at zero satisfies an appropriate coupling condition, the equations can be driven exactly to zero. We also present an iterative algorithm of the quasi-Newton kind for the computation of the control and we prove a convergence result. The behavior of this algorithm is illustrated with some numerical experiments.

On the controllability of a free-boundary problem for the 1D heat equation

Abstract 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 .

Null controllability for a parabolic equation with nonlocal nonlinearities

Abstract In this paper, we establish a local null controllability result for a nonlinear parabolic PDE with nonlocal nonlinearities. The result relies on the (global) null controllability of similar linear equations and a fixed point argument. We also analyze other similar controllability problems and we present several open questions.

Analysis and numerical solution of Benjamin-Bona-Mahony equation with moving boundary

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, we shall employ the finite element method together with the Crank-Nicolson method. Some numerical experiments are presented to show the moving boundary for the problem.

Local null controllability for degenerate parabolic equations with nonlocal term

Abstract We establish a local null controllability result for following nonlinear parabolic equation: u t − b x , ∫ 0 1 u u x x + f ( t , x , u ) = h χ ω , ( 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 theorem that demands the proof of a suitable Carleman estimate.

On a Problem Connected with Navier-stokes Equations in Non Cylindrical Domains

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.

Finite approximate controllability for semilinear heat equations in noncylindrical domains

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.