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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 characterization of C 1 , 1 functions via lower directional derivatives

Dušan Bednařík, Karel Pastor (2009)

Mathematica Bohemica

The notion of ˜ -stability is defined using the lower Dini directional derivatives and was introduced by the authors in their previous papers. In this paper we prove that the class of ˜ -stable functions coincides with the class of C 1 , 1 functions. This also solves the question posed by the authors in SIAM J. Control Optim. 45 (1) (2006), pp. 383–387.

A complete characterization of invariant jointly rank-r convex quadratic forms and applications to composite materials

Vincenzo Nesi, Enrico Rogora (2007)

ESAIM: Control, Optimisation and Calculus of Variations

The theory of compensated compactness of Murat and Tartar links the algebraic condition of rank-r convexity with the analytic condition of weak lower semicontinuity. The former is an algebraic condition and therefore it is, in principle, very easy to use. However, in applications of this theory, the need for an efficient classification of rank-r convex forms arises. In the present paper, we define the concept of extremal 2-forms  and characterize them in the rotationally invariant jointly...

A deterministic affine-quadratic optimal control problem

Yuanchang Wang, Jiongmin Yong (2014)

ESAIM: Control, Optimisation and Calculus of Variations

A deterministic affine-quadratic optimal control problem is considered. Due to the nature of the problem, optimal controls exist under some very mild conditions. Further, it is shown that under some assumptions, the optimal control is unique which leads to the differentiability of the value function. Therefore, the value function satisfies the corresponding Hamilton–Jacobi–Bellman equation in the classical sense, and the optimal control admits a state feedback representation. Under some additional...

A duality-based approach to elliptic control problems in non-reflexive Banach spaces

Christian Clason, Karl Kunisch (2011)

ESAIM: Control, Optimisation and Calculus of Variations

Convex duality is a powerful framework for solving non-smooth optimal control problems. However, for problems set in non-reflexive Banach spaces such as L1(Ω) or BV(Ω), the dual problem is formulated in a space which has difficult measure theoretic structure. The predual problem, on the other hand, can be formulated in a Hilbert space and entails the minimization of a smooth functional with box constraints, for which efficient numerical methods exist. In this work, elliptic control problems with...

A duality-based approach to elliptic control problems in non-reflexive Banach spaces*

Christian Clason, Karl Kunisch (2011)

ESAIM: Control, Optimisation and Calculus of Variations

Convex duality is a powerful framework for solving non-smooth optimal control problems. However, for problems set in non-reflexive Banach spaces such as L1(Ω) or BV(Ω), the dual problem is formulated in a space which has difficult measure theoretic structure. The predual problem, on the other hand, can be formulated in a Hilbert space and entails the minimization of a smooth functional with box constraints, for which efficient numerical methods exist. In this work, elliptic control problems with...

A Haar-Rado type theorem for minimizers in Sobolev spaces

Carlo Mariconda, Giulia Treu (2011)

ESAIM: Control, Optimisation and Calculus of Variations

Let be a minimum for where f is convex, is convex for a.e. x. We prove that u shares the same modulus of continuity of ϕ whenever Ω is sufficiently regular, the right derivative of g satisfies a suitable monotonicity assumption and the following inequality holds This result generalizes the classical Haar-Rado theorem for Lipschitz functions.

A Haar-Rado type theorem for minimizers in Sobolev spaces

Carlo Mariconda, Giulia Treu (2011)

ESAIM: Control, Optimisation and Calculus of Variations

Let be a minimum for where f is convex, is convex for a.e. x. We prove that u shares the same modulus of continuity of ϕ whenever Ω is sufficiently regular, the right derivative of g satisfies a suitable monotonicity assumption and the following inequality holds This result generalizes the classical Haar-Rado theorem for Lipschitz functions.

A least-squares method for the numerical solution of the Dirichlet problem for the elliptic monge − ampère equation in dimension two

Alexandre Caboussat, Roland Glowinski, Danny C. Sorensen (2013)

ESAIM: Control, Optimisation and Calculus of Variations

We address in this article the computation of the convex solutions of the Dirichlet problem for the real elliptic Monge − Ampère equation for general convex domains in two dimensions. The method we discuss combines a least-squares formulation with a relaxation method. This approach leads to a sequence of Poisson − Dirichlet problems and another sequence of low dimensional algebraic eigenvalue problems of a new type. Mixed finite element approximations with a smoothing procedure are used for the...

A method for constructing ε-value functions for the Bolza problem of optimal control

Jan Pustelnik (2005)

International Journal of Applied Mathematics and Computer Science

The problem considered is that of approximate minimisation of the Bolza problem of optimal control. Starting from Bellman's method of dynamic programming, we define the ε-value function to be an approximation to the value function being a solution to the Hamilton-Jacobi equation. The paper shows an approach that can be used to construct an algorithm for calculating the values of an ε-value function at given points, thus approximating the respective values of the value function.

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