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Minimax optimal control problems. Numerical analysis of the finite horizon case

Silvia C. Di Marco, Roberto L.V. González (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper we consider the numerical computation of the optimal cost function associated to the problem that consists in finding the minimum of the maximum of a scalar functional on a trajectory. We present an approximation method for the numerical solution which employs both discretization on time and on spatial variables. In this way, we obtain a fully discrete problem that has unique solution. We give an optimal estimate for the error between the approximated solution and the optimal cost function...

New regularity results and improved error estimates for optimal control problems with state constraints

Eduardo Casas, Mariano Mateos, Boris Vexler (2014)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper we are concerned with a distributed optimal control problem governed by an elliptic partial differential equation. State constraints of box type are considered. We show that the Lagrange multiplier associated with the state constraints, which is known to be a measure, is indeed more regular under quite general assumptions. We discretize the problem by continuous piecewise linear finite elements and we are able to prove that, for the case of a linear equation, the order of convergence...

Non-parametric approximation of non-anticipativity constraints in scenario-based multistage stochastic programming

Jean-Sébastien Roy, Arnaud Lenoir (2008)

Kybernetika

We propose two methods to solve multistage stochastic programs when only a (large) finite set of scenarios is available. The usual scenario tree construction to represent non-anticipativity constraints is replaced by alternative discretization schemes coming from non-parametric estimation ideas. In the first method, a penalty term is added to the objective so as to enforce the closeness between decision variables and the Nadaraya–Watson estimation of their conditional expectation. A numerical application...

Numerical analysis of some optimal control problems governed by a class of quasilinear elliptic equations

Eduardo Casas, Fredi Tröltzsch (2011)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper, we carry out the numerical analysis of a distributed optimal control problem governed by a quasilinear elliptic equation of non-monotone type. The goal is to prove the strong convergence of the discretization of the problem by finite elements. The main issue is to get error estimates for the discretization of the state equation. One of the difficulties in this analysis is that, in spite of the partial differential equation has a unique solution for any given control, the uniqueness...

Numerical analysis of some optimal control problems governed by a class of quasilinear elliptic equations*

Eduardo Casas, Fredi Tröltzsch (2011)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper, we carry out the numerical analysis of a distributed optimal control problem governed by a quasilinear elliptic equation of non-monotone type. The goal is to prove the strong convergence of the discretization of the problem by finite elements. The main issue is to get error estimates for the discretization of the state equation. One of the difficulties in this analysis is that, in spite of the partial differential equation has a unique solution for any given control, the uniqueness...

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 study of discretizations of multistage stochastic programs

Petri Hilli, Teemu Pennanen (2008)

Kybernetika

This paper presents a numerical study of a deterministic discretization procedure for multistage stochastic programs where the underlying stochastic process has a continuous probability distribution. The discretization procedure is based on quasi-Monte Carlo techniques originally developed for numerical multivariate integration. The solutions of the discretized problems are evaluated by statistical bounds obtained from random sample average approximations and out-of-sample simulations. In the numerical...

On the worst scenario method: Application to a quasilinear elliptic 2D-problem with uncertain coefficients

Petr Harasim (2011)

Applications of Mathematics

We apply a theoretical framework for solving a class of worst scenario problems to a problem with a nonlinear partial differential equation. In contrast to the one-dimensional problem investigated by P. Harasim in Appl. Math. 53 (2008), No. 6, 583–598, the two-dimensional problem requires stronger assumptions restricting the admissible set to ensure the monotonicity of the nonlinear operator in the examined state problem, and, as a result, to show the existence and uniqueness of the state solution....

Optimal control and numerical adaptivity for advection–diffusion equations

Luca Dede', Alfio Quarteroni (2005)

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

We propose a general approach for the numerical approximation of optimal control problems governed by a linear advection–diffusion equation, based on a stabilization method applied to the lagrangian functional, rather than stabilizing the state and adjoint equations separately. This approach yields a coherently stabilized control problem. Besides, it allows a straightforward a posteriori error estimate in which estimates of higher order terms are needless. Our a posteriori estimates stems from splitting...

Optimal control and numerical adaptivity for advection–diffusion equations

Luca Dede', Alfio Quarteroni (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We propose a general approach for the numerical approximation of optimal control problems governed by a linear advection–diffusion equation, based on a stabilization method applied to the Lagrangian functional, rather than stabilizing the state and adjoint equations separately. This approach yields a coherently stabilized control problem. Besides, it allows a straightforward a posteriori error estimate in which estimates of higher order terms are needless. Our a posteriori estimates stems from...

Optimal control of delay systems with differential and algebraic dynamic constraints

Boris S. Mordukhovich, Lianwen Wang (2005)

ESAIM: Control, Optimisation and Calculus of Variations

This paper concerns constrained dynamic optimization problems governed by delay control systems whose dynamic constraints are described by both delay-differential inclusions and linear algebraic equations. This is a new class of optimal control systems that, on one hand, may be treated as a specific type of variational problems for neutral functional-differential inclusions while, on the other hand, is related to a special class of differential-algebraic systems with a general delay-differential...

Optimal control of delay systems with differential and algebraic dynamic constraints

Boris S. Mordukhovich, Lianwen Wang (2010)

ESAIM: Control, Optimisation and Calculus of Variations

This paper concerns constrained dynamic optimization problems governed by delay control systems whose dynamic constraints are described by both delay-differential inclusions and linear algebraic equations. This is a new class of optimal control systems that, on one hand, may be treated as a specific type of variational problems for neutral functional-differential inclusions while, on the other hand, is related to a special class of differential-algebraic systems with a general delay-differential...

Optimal control of fluid flow in soil 1. Deterministic case.

Youcef Kelanemer (1998)

Revista Matemática Complutense

We study the numerical aspect of the optimal control of problems governed by a linear elliptic partial differential equation (PDE). We consider here the gas flow in porous media. The observed variable is the flow field we want to maximize in a given part of the domain or its boundary. The control variable is the pressure at one part of the boundary or the discharges of some wells located in the interior of the domain. The objective functional is a balance between the norm of the flux in the observation...

Optimal design in small amplitude homogenization

Grégoire Allaire, Sergio Gutiérrez (2007)

ESAIM: Mathematical Modelling and Numerical Analysis

This paper is concerned with optimal design problems with a special assumption on the coefficients of the state equation. Namely we assume that the variations of these coefficients have a small amplitude. Then, making an asymptotic expansion up to second order with respect to the aspect ratio of the coefficients allows us to greatly simplify the optimal design problem. By using the notion of H-measures we are able to prove general existence theorems for small amplitude optimal design and to provide...

Optimal design of cylindrical shells

Peter Nestler, Werner H. Schmidt (2010)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

The present paper studies an optimization problem of dynamically loaded cylindrical tubes. This is a problem of linear elasticity theory. As we search for the optimal thickness of the tube which minimizes the displacement under forces, this is a problem of shape optimization. The mathematical model is given by a differential equation (ODE and PDE, respectively); the mechanical problem is described as an optimal control problem. We consider both the stationary (time independent) and the transient...

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