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This work deals with some numerical experiments regarding the distributed control of semilinear parabolic equations of the type y(t) - y(xx) + f (y) = u(Xw), in (0, 1) x (0, T), with Neumann and initial auxiliary conditions, where w is an open subset of (0, 1), f is a C-1 nondecreasing real function, a is the output control and T > 0 is (arbitrarily) fixed. Given a target state y(T) we study the associated approximate controllability problem (given epsilon > 0, find u is an element of L-2(0, T), such that parallel toy(T; u) - y(T)parallel to(L2(0,1)) less than or equal to epsilon) by passing to the limit (when k --> infinity) in the penalized optimal control problem. (find u(k) as the minimum of J(k)(u) = 1/2 parallel touparallel to(L2)(2) ((0,T)) + (k/2)parallel toy(T; u) -y(T)parallel to(L2)(2) ((0,1))). In the superlinear case (e.g., f (y) = \y\(n-1)y, n > 1) the existence of two obstruction functions Y+/-infinity shows that the approximate controllability is only possible if Y-infinity (x,T) +/- y(T)(x) less than or equal to Y-infinity(x,T) for a.e. x is an element of (0, 1). We carry out some numerical experiments showing that, for a fixed k, the "minimal cost" J(k)(u) (and the norm of the optimal control u(k)) for a superlinear function f becomes much larger when this condition is not satisfied. We also compare the values of J(k)(u) (and the norm of the optimal control u(k)) for a fixed y(T) associated with two nonlinearities: one sublinear and the other one superlinear., European project, Ministerio de Ciencia y Tecnologfa of Spain. “Ramón y Cajal, MCYT, Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

We prove the approximate controllability of several nonlinear parabolic boundary-value problems by means of two different methods: the first one can be called a Cancellation method and the second one uses the Kakutani fixed-point theorem., DGICYT (Spain), Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

We show how the approximate controllability of nonlinear parabolic problems may fail although it is a well-known property for linear equations. For the case of the obstacle problem the answer depends on the negativeness of the righ hand side term and for semilinear equations it depends on the sub or superlinear character at the infinity of the nonlinear term., Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

In this work, we study an approximate controllability problem with constraint on the control. This problem appears naturally of weak and pointwise sentinel with a small or singleton region observation. The main tool is a theorem of uniqueness of the solution of ill-posed Cauchy problem for the parabolic equation.

In this work, we study an approximate controllability problem with constraint on the control. This problem appears naturally of weak and pointwise sentinel with a small or singleton region observation. The main tool is a theorem of uniqueness of the solution of ill-posed Cauchy problem for the parabolic equation.

In this paper, we discuss the approximate controllability of the neutral stochastic differential equations of Sobolev type with unbounded delay in Hilbert Spaces. A set of sufficient conditions are established for the existence and approximate controllability of the mild solutions using Krasnoselskii-Schaefer-type fixed point theorems and stochastic analysis theory. An application involving partial differential equation with unbounded delay is addressed.

In this manuscript, we study the approximate controllability results for nonlocal impulsive fractional neutral stochastic integro-differential equations with state-dependent delay conditions in Hilbert spaces under the assumptions that the corresponding linear system is approximately controllable. The results are obtained by using fractional calculus, semigroup theory, stochastic analysis and fixed point theorem. An example is provided to show the application of our result.

In this manuscript, we study the approximate controllability results for nonlocal impulsive fractional neutral stochastic integro-differential equations with state-dependent delay conditions in Hilbert spaces under the assumptions that the corresponding linear system is approximately controllable. The results are obtained by using fractional calculus, semigroup theory, stochastic analysis and fixed point theorem. An example is provided to show the application of our result.

This paper deals with some inverse and control problems for the Navier-Stokes and related systems. We will focus on some particular aspects that have recently led to interesting (theoretical and numerical) results: geometric inverse problems, Eulerian and Lagrangian controllability and vortex reduction oriented to shape optimization.

This paper deals with some inverse and control problems for the Navier-Stokes and related systems. We will focus on some particular aspects that have recently led to interesting (theoretical and numerical) results: geometric inverse problems, Eulerian and Lagrangian controllability and vortex reduction oriented to shape optimization.

This paper deals with some inverse and control problems for the Navier-Stokes and related systems. We will focus on some particular aspects that have recently led to interesting (theoretical and numerical) results: geometric inverse problems, Eulerian and Lagrangian controllability and vortex reduction oriented to shape optimization.

This paper deals with some inverse and control problems for the Navier-Stokes and related systems. We will focus on some particular aspects that have recently led to interesting (theoretical and numerical) results: geometric inverse problems, Eulerian and Lagrangian controllability and vortex reduction oriented to shape optimization.

This work deals with some numerical experiments regarding the distributed control of semilinear parabolic equations of the type y(t) - y(xx) + f (y) = u(Xw), in (0, 1) x (0, T), with Neumann and initial auxiliary conditions, where w is an open subset of (0, 1), f is a C-1 nondecreasing real function, a is the output control and T > 0 is (arbitrarily) fixed. Given a target state y(T) we study the associated approximate controllability problem (given epsilon > 0, find u is an element of L-2(0, T), such that parallel toy(T; u) - y(T)parallel to(L2(0,1)) less than or equal to epsilon) by passing to the limit (when k --> infinity) in the penalized optimal control problem. (find u(k) as the minimum of J(k)(u) = 1/2 parallel touparallel to(L2)(2) ((0,T)) + (k/2)parallel toy(T; u) -y(T)parallel to(L2)(2) ((0,1))). In the superlinear case (e.g., f (y) = \y\(n-1)y, n > 1) the existence of two obstruction functions Y+/-infinity shows that the approximate controllability is only possible if Y-infinity (x,T) +/- y(T)(x) less than or equal to Y-infinity(x,T) for a.e. x is an element of (0, 1). We carry out some numerical experiments showing that, for a fixed k, the "minimal cost" J(k)(u) (and the norm of the optimal control u(k)) for a superlinear function f becomes much larger when this condition is not satisfied. We also compare the values of J(k)(u) (and the norm of the optimal control u(k)) for a fixed y(T) associated with two nonlinearities: one sublinear and the other one superlinear., European project, Ministerio de Ciencia y Tecnologfa of Spain. “Ramón y Cajal, MCYT, Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

Two problems arising in Environment are considered. The first one concerns a conjecture possed by von Neumann in 1955 on the possible modification of the albedo in order to control the Earth surface temperature. The second one is related to the approximate controllability of Stackelberg-Nash strategies for some optimization problems as, for instance, the pollution control in a lake. The results of thesecondpart were obtained in collaboration with Jacques-LouisLions., DGES(Spain), Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

Two problems arising in Environment are considered. The first one concerns a conjecture possed by von Neumann in 1955 on the possible modification of the albedo in order to control the Earth surface temperature. The second one is related to the approximate controllability of Stackelberg-Nash strategies for some optimization problems as, for instance, the pollution control in a lake. The results of thesecondpart were obtained in collaboration with Jacques-LouisLions., DGES(Spain), Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

The authors obtain an approximate controllability result for the nonlinear equation $y'+Ay+F(t,y)y=h(t)$ with the initial condition $y(0)=x$, where $x$ is the control and $A$ generates a $C_0$-semigroup in a Hilbert space. The Schauder fixed point theorem is the main tool. Then they apply their result to get approximate controllability for some classes of elliptic and parabolic problems with dynamic boundary conditions. There are 72 references, many of them referring to mathematical models in various areas where the results could be applied., DGICYT, CNCSU/CNFIS, Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

The authors obtain an approximate controllability result for the nonlinear equation $y'+Ay+F(t,y)y=h(t)$ with the initial condition $y(0)=x$, where $x$ is the control and $A$ generates a $C_0$-semigroup in a Hilbert space. The Schauder fixed point theorem is the main tool. Then they apply their result to get approximate controllability for some classes of elliptic and parabolic problems with dynamic boundary conditions. There are 72 references, many of them referring to mathematical models in various areas where the results could be applied., DGICYT, CNCSU/CNFIS, Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

This paper presents some known results on the approximate and null controllability of the Navier–Stokes equations. All of them can be viewed as partial answers to a conjecture of J.-L. Lions.

This paper presents some known results on the approximate and null controllability of the Navier–Stokes equations. All of them can be viewed as partial answers to a conjecture of J.-L. Lions.

This paper presents some known results on the approximate and null controllability of the Navier–Stokes equations. All of them can be viewed as partial answers to a conjecture of J.-L. Lions.

This paper presents some known results on the approximate and null controllability of the Navier–Stokes equations. All of them can be viewed as partial answers to a conjecture of J.-L. Lions.

We prove the approximate controllability of several nonlinear parabolic boundary-value problems by means of two different methods: the first one can be called a Cancellation method and the second one uses the Kakutani fixed-point theorem., DGICYT (Spain), Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

We study the approximate controllability property for y(t) - Delta phi(y) = u chi(omega), on Omega x (0, T), where Omega is a bounded open set of R-N and omega subset of Omega. First, we show some negative results for the case phi(s) = \s\(m-1)s, 0 < m < 1, by means of an obstruction phenomenon. In a second part, we obtain a positive answer on the space H-1-gamma(Omega), for any gamma > 0, for a class of functions phi essentially linear at infinity, by using a higher order vanishing viscosity argument., Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

Wt,give some negative and positive results on the approximate controllability of the Stokes system formulated on a cylinder Omega = G x R of R-3 when the control is a density of external unidirectional forces. We distinguish the case where the direction of the controls e is parallel to the cylinder generatrix (e = e(3)) from the one where e is orthogonal to this generatrix (e = e(1)). A negative result in the case of e = e(3) is proved for periodic boundary conditions on x(3), and homogeneous Dirichlet conditions on partial derivative G x R where G is a general set of R-2. In contrast to that, the approximate controllability is proved for homogeneous Dirichlet conditions on partial derivative Omega (i.e. zero on partial derivative G x R and solutions in (L-2(G x R))(3) for any t), when G is a rectangle and e = e(1) is orthogonal to the cylinder generatrix., DGICYT (Spain), Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

The main goal of this paper is to present a new result concerning controllability of the time-dependent Navier-Stokes equations. Here, the control variable is the trace of the velocity field on a "small" part of the boundary. The main result states that the linear space spanned by final states is dense in the L space of admissible fields. For the proof, one uses a duality argument that is suggested by the linear theory. This reduces the task to an existence/regularity result for a nonlinear problem.

The main goal of this paper is to present a new result concerning controllability of the time-dependent Navier-Stokes equations. Here, the control variable is the trace of the velocity field on a "small" part of the boundary. The main result states that the linear space spanned by final states is dense in the L space of admissible fields. For the proof, one uses a duality argument that is suggested by the linear theory. This reduces the task to an existence/regularity result for a nonlinear problem.

The main goal of this paper is to present a new result concerning controllability of the time-dependent Navier-Stokes equations. Here, the control variable is the trace of the velocity field on a "small" part of the boundary. The main result states that the linear space spanned by final states is dense in the L space of admissible fields. For the proof, one uses a duality argument that is suggested by the linear theory. This reduces the task to an existence/regularity result for a nonlinear problem.

The main goal of this paper is to present a new result concerning controllability of the time-dependent Navier-Stokes equations. Here, the control variable is the trace of the velocity field on a "small" part of the boundary. The main result states that the linear space spanned by final states is dense in the L space of admissible fields. For the proof, one uses a duality argument that is suggested by the linear theory. This reduces the task to an existence/regularity result for a nonlinear problem.

We show how the approximate controllability of nonlinear parabolic problems may fail although it is a well-known property for linear equations. For the case of the obstacle problem the answer depends on the negativeness of the righ hand side term and for semilinear equations it depends on the sub or superlinear character at the infinity of the nonlinear term., Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub