Catálogo de publicaciones - libros

We consider a boundary stabilization problem for the plate equation in a square. The feedback law gives the bending moment on a part of the boundary as function of the velocity field of the plate. The main result of the paper asserts that the obtained closed loop system is exponentially stable if and only if the controlled part of the boundary contains a vertical and a horizontal part of non-zero length (the geometric optics condition introduced by Bardos, Lebeau and Rauch in [] for the wave equation is thus not necessary in this case). The proof of the main result uses the methodology introduced in Ammari and Tucsnak [], where the exponential stability for the closed loop problem is reduced to an observability estimate for the corresponding uncontrolled system combined to a boundedness property of the transfer function of the associated open loop system. The second essential ingredient of the proof is an observability inequality recently proved by Ramdani, Takahashi, Tenenbaum and Tucsnak []

We review recent results on the boundary and interior feedback stabilization of Navier-Stokes equations, = 2, 3, and provide new ones.

We develop an adaptive finite element method for a class of distributed optimal control problems with control constraints. The method is based on a residual-type a posteriori error estimator and incorporates data oscillations. The analysis is carried out for conforming P1 approximations of the state and the co-state and elementwise constant approximations of the control and the co-control. We prove convergence of the error in the state, the co-state, the control, and the co-control. Under some additional non-degeneracy assumptions on the continuous and the discrete problems, we then show that an error reduction property holds true at least asymptotically. The analysis uses the reliability and the discrete local efficiency of the a posteriori estimator as well as quasi-orthogonality properties as essential tools. Numerical results illustrate the performance of the adaptive algorithm.

In active flood hazard mitigation, lateral flow withdrawal is used to reduce the impact of flood waves in rivers. Through emergency side channels, lateral outflow is generated. The optimal outflow controls the flood in such a way that the cost of the created damage is minimized. The flow is governed by a networked system of nonlinear hyperbolic partial differential equations, coupled by algebraic node conditions. Two types of integrals appear in the objective function of the corresponding optimization problem: Boundary integrals (for example, to measure the amount of water that flows out of the system into the floodplain) and distributed integrals.

For the evaluation of the derivative of the objective function, we introduce an adjoint backwards system. For the numerical solution we consider a discretized system with a consistent discretization of the continuous adjoint system, in the sense that the discrete adjoint system yields the derivatives of the discretized objective function. Numerical examples are included.

The present article is concerned with boundary control stabilization of semi-linear parabolic systems which are unstable if uncontrolled. A particular emphasis is put on Dirichlet control in that context. We investigate two different numerical approaches to solve these problems. The first approach relies on the extension method proposed by A.V. Fursikov where the considered partial differential equations are first solved on an extended domain with suitable initial value leading a stable solution. The needed control is then de-fined as an appropriate trace of this solution. The second approach relies on the formulation of the stabilization problem as an optimization problem with constraints based on partial differential equations. We address the numerical issues related to both class of approaches toward a comparison of their specific stabilization properties. The considered methodology is applied to the solution of test parabolic problems assuming linear and nonlinear models.

Efficient numerical techniques for the solution of constrained optimal control problems for the nonlinear heat equation are considered. The nonlinearity in the governing equation is due to the boundary conditions which cover the Boltzmann radiation boundary condition. With respect to numerical algorithms, variants of semismooth Newton methods are proposed which allow a convergence analysis in function space. For the latter aspect the concept of generalized (Newton, or slant) differentiability is invoked. The paper ends with a comparison of the proposed algorithms among each other and with a sequential quadratic programming method.

We discuss optimal and model predictive control techniques applied to the Boussinesq approximation of the Navier-Stokes system. We focus on mathematical modeling, discuss possible control scenarios, and provide a concise description of the numerical implementation. Furthermore, several numerical examples are provided.

This paper discusses semi-smooth Newton methods for solving nonlinear non-smooth equations in Banach spaces. Such investigations are motivated by complementarity problems, variational inequalities and optimal control problems with control or state constraints, for example. The function () for which we desire to find a root is typically Lipschitz continuous but not regular. The primal-dual active set strategy for the optimization with the inequality constraints is formulated as a semi-smooth Newton method. Sufficient conditions for global convergence assuming diagonal dominance are established. Globalization strategies are also discussed assuming that the merit function |()| has appropriate descent directions.

In this paper we consider the problem of determining parameters in nonlinear partial differential equations of hyperbolic type from boundary measurements. In order to investigate the qualitative behavior of this class of identification problems, we analyze the model problem of identifying in the nonlinear wave equation − (()) = 0 and discuss stability and identifiability for this problem. Moreover, we derive applicability of these results to material parameter identification in piezoelectricity and provide numerical reconstruction results.

We consider a sequential quadratic programming (SQP) method for the solution of an optimal control problem governed by a semilinear elliptic equation with nonlocal interface conditions. These conditions arise from conductive-radiative heat transfer in non-convex domains. After stating firstand second-order optimality conditions, we introduce the SQP algorithm that uses an active set method to solve the linear quadratic subproblems arising in each step. The corresponding optimality systems are discretized by linear finite elements, using a partly exact summarized midpoint rule for the discretization of the nonlocal radiation interface conditions. The paper ends with some numerical results demonstrating the efficiency of the proposed method.