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New a posteriori L ∞ ( L 2 ) and L 2 ( L 2 ) -error estimates of mixed finite element methods for general nonlinear parabolic optimal control problems

Zuliang Lu (2016)

Applications of Mathematics

We study new a posteriori error estimates of the mixed finite element methods for general optimal control problems governed by nonlinear parabolic equations. The state and the co-state are discretized by the high order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. We derive a posteriori error estimates in L ∞ ( J ; L 2 ( Ω ) ) -norm and L 2 ( J ; L 2 ( Ω ) ) -norm for both the state, the co-state and the control approximation. Such estimates, which seem to be new, are an important...

New classes of analytic and Gevrey semigroups and applications

Angelo Favini, Roberto Triggiani (1993)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We consider the operator - A + i ⁢ B on a complex Hilbert space, where A is positive self-adjoint and B is self-adjoint, and where, moreover, « B is comparable to A α , α ≥ 1 », in a technical sense. Two applications are given.

Newton and conjugate gradient for harmonic maps from the disc into the sphere

Morgan Pierre (2004)

ESAIM: Control, Optimisation and Calculus of Variations

We compute numerically the minimizers of the Dirichlet energy E ( u ) = 1 2 ∫ B 2 | ∇ u | 2 d x among maps u : B 2 → S 2 from the unit disc into the unit sphere that satisfy a boundary condition and a degree condition. We use a Sobolev gradient algorithm for the minimization and we prove that its continuous version preserves the degree. For the discretization of the problem we use continuous P 1 finite elements. We propose an original mesh-refining strategy needed to preserve the degree with the discrete version of the algorithm (which is a preconditioned...

Null controllability of the heat equation with boundary Fourier conditions: the linear case

Enrique Fernández-Cara, Manuel González-Burgos, Sergio Guerrero, Jean-Pierre Puel (2006)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper, we prove the global null controllability of the linear heat equation completed with linear Fourier boundary conditions of the form ∂ y ∂ n + β y = 0 . We consider distributed controls with support in a small set and nonregular coefficients β = β ( x , t ) . For the proof of null controllability, a crucial tool will be a new Carleman estimate for the weak solutions of the classical heat equation with nonhomogeneous Neumann boundary conditions.

Numerical analysis for optimal shape design in elliptic boundary value problems

Zdeněk Kestřánek (1988)

Aplikace matematiky

Shape optimization problems are optimal design problems in which the shape of the boundary plays the role of a design, i.e. the unknown part of the problem. Such problems arise in structural mechanics, acoustics, electrostatics, fluid flow and other areas of engineering and applied science. The mathematical theory of such kind of problems has been developed during the last twelve years. Recently the theory has been extended to cover also situations in which the behaviour of the system is governed...

Numerical identification of a coefficient in a parabolic quasilinear equation

Jan Neumann (1985)

Aplikace matematiky

In the article the following optimal control problem is studied: to determine a certain coefficient in a quasilinear partial differential equation of parabolic type so that the solution of a boundary value problem for this equation would minimise a given integral functional. In addition to the design and analysis of a numerical method the paper contains the solution of the fundamental problems connected with the formulation of the problem in question (existence and uniqueness of the solution of...

Numerical procedure to approximate a singular optimal control problem

Silvia C. Di Marco, Roberto L.V. González (2007)

ESAIM: Mathematical Modelling and Numerical Analysis

In this work we deal with the numerical solution of a Hamilton-Jacobi-Bellman (HJB) equation with infinitely many solutions. To compute the maximal solution – the optimal cost of the original optimal control problem – we present a complete discrete method based on the use of some finite elements and penalization techniques.

Numerical realization of a fictitious domain approach used in shape optimization. Part I: Distributed controls

Jana Daňková, Jaroslav Haslinger (1996)

Applications of Mathematics

We deal with practical aspects of an approach to the numerical realization of optimal shape design problems, which is based on a combination of the fictitious domain method with the optimal control approach. Introducing a new control variable in the right-hand side of the state problem, the original problem is transformed into a new one, where all the calculations are performed on a fixed domain. Some model examples are presented.

Numerical resolution of an “unbalanced” mass transport problem

Jean-David Benamou (2003)

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

We introduce a modification of the Monge–Kantorovitch problem of exponent 2 which accommodates non balanced initial and final densities. The augmented lagrangian numerical method introduced in [6] is adapted to this “unbalanced” problem. We illustrate the usability of this method on an idealized error estimation problem in meteorology.

Numerical resolution of an “unbalanced” mass transport problem

Jean-David Benamou (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We introduce a modification of the Monge–Kantorovitch problem of exponent 2 which accommodates non balanced initial and final densities. The augmented Lagrangian numerical method introduced in [6] is adapted to this “unbalanced” problem. We illustrate the usability of this method on an idealized error estimation problem in meteorology.

On a shape control problem for the stationary Navier-Stokes equations

Max D. Gunzburger, Hongchul Kim, Sandro Manservisi (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

An optimal shape control problem for the stationary Navier-Stokes system is considered. An incompressible, viscous flow in a two-dimensional channel is studied to determine the shape of part of the boundary that minimizes the viscous drag. The adjoint method and the Lagrangian multiplier method are used to derive the optimality system for the shape gradient of the design functional.

On a Theorem of Ingham.

S. Jaffard, M. Tucsnak, E. Zuazua (1997)

The journal of Fourier analysis and applications [[Elektronische Ressource]]

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