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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 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 deterministic LQ tracking problem: parametrisation of the controller

Ľuboš Čirka, Ján Mikleš, Miroslav Fikar (2002)

Kybernetika

The article discusses an optimal Linear Quadratic (LQ) deterministic control problem when the Youla–Kučera parametrisation of controller is used. We provide a computational procedure for computing a deterministic optimal single-input single-output (SISO) controller from any stabilising controller. This approach allows us to calculate a new optimal LQ deterministic controller from a previous one when the plant has changed. The design based on the Youla –Kučera parametrisation approach is compared...

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 quadratic optimal control problem for a class of linear discrete distributed systems

Mostafa Rachik, Mustapha Lhous, Ouafa El Kahlaoui (2006)

International Journal of Applied Mathematics and Computer Science

A linear quadratic optimal control problem for a class of discrete distributed systems is analyzed. To solve this problem, we introduce an adequate topology and establish that optimal control can be determined though an inversion of the appropriate isomorphism. An example and a numerical approach are given.

A simple solution to the finite-horizon LQ problem with zero terminal state

Lorenzo Ntogramatzidis (2003)

Kybernetika

This short paper deals with the classical finite-horizon linear-quadratic regulator problem with the terminal state constrained to be zero, for both continuous and discrete-time systems. Closed-form expressions for the optimal state and costate trajectories of the Hamiltonian system, as well as the corresponding control law, are derived through the solutions of two infinite- horizon LQ problems, thus avoiding the use of the Riccati differential equation. The computation of the optimal value of the...

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...

An application of the Fourier transform to optimization of continuous 2-D systems

Vitali Dymkou, Michael Dymkov (2003)

International Journal of Applied Mathematics and Computer Science

This paper uses the theory of entire functions to study the linear quadratic optimization problem for a class of continuous 2D systems. We show that in some cases optimal control can be given by an analytical formula. A simple method is also proposed to find an approximate solution with preassigned accuracy. Some application to the 1D optimization problem is presented, too. The obtained results form a theoretical background for the design problem of optimal controllers for relevant processes.

An SQP trust region method for solving the discrete-time linear quadratic control problem

El-Sayed M.E. Mostafa (2012)

International Journal of Applied Mathematics and Computer Science

In this paper, a sequential quadratic programming method combined with a trust region globalization strategy is analyzed and studied for solving a certain nonlinear constrained optimization problem with matrix variables. The optimization problem is derived from the infinite-horizon linear quadratic control problem for discrete-time systems when a complete set of state variables is not available. Moreover, a parametrization approach is introduced that does not require starting a feasible solution...

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