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A boundary-value problem for linear PDAEs

Wiesław Marszałek, Zdzisław Trzaska (2002)

International Journal of Applied Mathematics and Computer Science

We analyze a boundary-value problem for linear partial differential algebraic equations, or PDAEs, by using the method of the separation of variables. The analysis is based on the Kronecker-Weierstrass form of the matrix pencil[A,-λ_n B]. A new theorem is proved and two illustrative examples are given.

A Carleman estimates based approach for the stabilization of some locally damped semilinear hyperbolic equations

Louis Tebou (2008)

ESAIM: Control, Optimisation and Calculus of Variations

First, we consider a semilinear hyperbolic equation with a locally distributed damping in a bounded domain. The damping is located on a neighborhood of a suitable portion of the boundary. Using a Carleman estimate [Duyckaerts, Zhang and Zuazua, Ann. Inst. H. Poincaré Anal. Non Linéaire (to appear); Fu, Yong and Zhang, SIAM J. Contr. Opt. 46 (2007) 1578–1614], we prove that the energy of this system decays exponentially to zero as the time variable goes to infinity. Second, relying on another Carleman...

A Carleman estimates based approach for the stabilization of some locally damped semilinear hyperbolic equations

Louis Tebou (2007)

ESAIM: Control, Optimisation and Calculus of Variations

First, we consider a semilinear hyperbolic equation with a locally distributed damping in a bounded domain. The damping is located on a neighborhood of a suitable portion of the boundary. Using a Carleman estimate [Duyckaerts, Zhang and Zuazua, Ann. Inst. H. Poincaré Anal. Non Linéaire (to appear); Fu, Yong and Zhang, SIAM J. Contr. Opt.46 (2007) 1578–1614], we prove that the energy of this system decays exponentially to zero as the time variable goes to infinity. Second, relying on another Carleman...

A certified reduced basis method for parametrized elliptic optimal control problems

Mark Kärcher, Martin A. Grepl (2014)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper, we employ the reduced basis method as a surrogate model for the solution of linear-quadratic optimal control problems governed by parametrized elliptic partial differential equations. We present a posteriori error estimation and dual procedures that provide rigorous bounds for the error in several quantities of interest: the optimal control, the cost functional, and general linear output functionals of the control, state, and adjoint variables. We show that, based on the assumption...

A comparison of two FEM-based methods for the solution of the nonlinear output regulation problem

Branislav Rehák, Sergej Čelikovský, Javier Ruiz, Jorge Orozco-Mora (2009)

Kybernetika

The regulator equation is the fundamental equation whose solution must be found in order to solve the output regulation problem. It is a system of first-order partial differential equations (PDE) combined with an algebraic equation. The classical approach to its solution is to use the Taylor series with undetermined coefficients. In this contribution, another path is followed: the equation is solved using the finite-element method which is, nevertheless, suitable to solve PDE part only. This paper...

A general transfer function representation for a class of hyperbolic distributed parameter systems

Krzysztof Bartecki (2013)

International Journal of Applied Mathematics and Computer Science

Results of transfer function analysis for a class of distributed parameter systems described by dissipative hyperbolic partial differential equations defined on a one-dimensional spatial domain are presented. For the case of two boundary inputs, the closed-form expressions for the individual elements of the 2×2 transfer function matrix are derived both in the exponential and in the hyperbolic form, based on the decoupled canonical representation of the system. Some important properties of the transfer...

A level set method in shape and topology optimization for variational inequalities

Piotr Fulmański, Antoine Laurain, Jean-Francois Scheid, Jan Sokołowski (2007)

International Journal of Applied Mathematics and Computer Science

The level set method is used for shape optimization of the energy functional for the Signorini problem. The boundary variations technique is used in order to derive the shape gradients of the energy functional. The conical differentiability of solutions with respect to the boundary variations is exploited. The topology modifications during the optimization process are identified by means of an asymptotic analysis. The topological derivatives of the energy shape functional are employed for the topology...

A model-based fault detection and diagnosis scheme for distributed parameter systems : a learning systems approach

Michael A. Demetriou (2002)

ESAIM: Control, Optimisation and Calculus of Variations

In this note, fault detection techniques based on finite dimensional results are extended and applied to a class of infinite dimensional dynamical systems. This special class of systems assumes linear plant dynamics having an abrupt additive perturbation as the fault. This fault is assumed to be linear in the (unknown) constant (and possibly functional) parameters. An observer-based model estimate is proposed which serves to monitor the system’s dynamics for unanticipated failures, and its well...

A Model-Based Fault Detection and Diagnosis Scheme for Distributed Parameter Systems: A Learning Systems Approach

Michael A. Demetriou (2010)

ESAIM: Control, Optimisation and Calculus of Variations

In this note, fault detection techniques based on finite dimensional results are extended and applied to a class of infinite dimensional dynamical systems. This special class of systems assumes linear plant dynamics having an abrupt additive perturbation as the fault. This fault is assumed to be linear in the (unknown) constant (and possibly functional) parameters. An observer-based model estimate is proposed which serves to monitor the system's dynamics for unanticipated failures, and its well...

A multi-model approach to Saint-Venant equations: A stability study by LMIs

Valérie Dos Santos Martins, Mickael Rodrigues, Mamadou Diagne (2012)

International Journal of Applied Mathematics and Computer Science

This paper deals with the stability study of the nonlinear Saint-Venant Partial Differential Equation (PDE). The proposed approach is based on the multi-model concept which takes into account some Linear Time Invariant (LTI) models defined around a set of operating points. This method allows describing the dynamics of this nonlinear system in an infinite dimensional space over a wide operating range. A stability analysis of the nonlinear Saint-Venant PDE is proposed both by using Linear Matrix Inequalities...

A Posteriori Error Estimation for Reduced Order Solutions of Parametrized Parabolic Optimal Control Problems

Mark Kärcher, Martin A. Grepl (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We consider the efficient and reliable solution of linear-quadratic optimal control problems governed by parametrized parabolic partial differential equations. To this end, we employ the reduced basis method as a low-dimensional surrogate model to solve the optimal control problem and develop a posteriori error estimation procedures that provide rigorous bounds for the error in the optimal control and the associated cost functional. We show that our approach can be applied to problems involving...

A singular controllability problem with vanishing viscosity

Ioan Florin Bugariu, Sorin Micu (2014)

ESAIM: Control, Optimisation and Calculus of Variations

The aim of this paper is to answer the question: Do the controls of a vanishing viscosity approximation of the one dimensional linear wave equation converge to a control of the conservative limit equation? The characteristic of our viscous term is that it contains the fractional power α of the Dirichlet Laplace operator. Through the parameter α we may increase or decrease the strength of the high frequencies damping which allows us to cover a large class of dissipative mechanisms. The viscous term,...

A singular perturbation problem in exact controllability of the Maxwell system

John E. Lagnese (2001)

ESAIM: Control, Optimisation and Calculus of Variations

This paper studies the exact controllability of the Maxwell system in a bounded domain, controlled by a current flowing tangentially in the boundary of the region, as well as the exact controllability the same problem but perturbed by a dissipative term multiplied by a small parameter in the boundary condition. This boundary perturbation improves the regularity of the problem and is therefore a singular perturbation of the original problem. The purpose of the paper is to examine the connection,...

A Singular Perturbation Problem in Exact Controllability of the Maxwell System

John E. Lagnese (2010)

ESAIM: Control, Optimisation and Calculus of Variations

This paper studies the exact controllability of the Maxwell system in a bounded domain, controlled by a current flowing tangentially in the boundary of the region, as well as the exact controllability the same problem but perturbed by a dissipative term multiplied by a small parameter in the boundary condition. This boundary perturbation improves the regularity of the problem and is therefore a singular perturbation of the original problem. The purpose of the paper is to examine the connection, for...

Absolute stability results for well-posed infinite-dimensional systems with applications to low-gain integral control

Hartmut Logemann, Ruth F. Curtain (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We derive absolute stability results for well-posed infinite-dimensional systems which, in a sense, extend the well-known circle criterion to the case that the underlying linear system is the series interconnection of an exponentially stable well-posed infinite-dimensional system and an integrator and the nonlinearity ϕ satisfies a sector condition of the form (ϕ(u),ϕ(u) - au) ≤ 0 for some constant a>0. These results are used to prove convergence and stability properties of low-gain integral...

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