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Optimal Control of a Linear Elliptic Equation With a SupremumNorm Functional
, 2000
"... We consider an optimal control problem of a linear elliptic equation with a functional containing a supremumnorm term. The control acts on the boundary. Necessary first order optimality conditions are derived for problems with pointwise control and state constraints. For this purpose the original p ..."
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Cited by 9 (0 self)
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We consider an optimal control problem of a linear elliptic equation with a functional containing a supremumnorm term. The control acts on the boundary. Necessary first order optimality conditions are derived for problems with pointwise control and state constraints. For this purpose the original problem is substituted by an equivalent problem with a differentiable functional. In a second part we discuss a numerical approach to such problems. The control problem is transformed into a linear (resp. quadratic) programming problem. In a particular situation we can compare the numerical results with the analytic solutions.
A Level Set Approach For The Solution Of A StateConstrained Optimal Control Problem
 Num. Math
, 2004
"... State constrained optimal control problems for linear elliptic partial differential equations are considered. The corresponding first order optimality conditions in primaldual form are analyzed and linked to a free boundary problem resulting in a novel algorithmic approach with the boundary (interf ..."
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Cited by 6 (2 self)
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State constrained optimal control problems for linear elliptic partial differential equations are considered. The corresponding first order optimality conditions in primaldual form are analyzed and linked to a free boundary problem resulting in a novel algorithmic approach with the boundary (interface) between the active and inactive sets as optimization variable. The new algorithm is based on the level set methodology. The speed function involved in the level set equation for propagating the interface is computed by utilizing techniques from shape optimization. Encouraging numerical results attained by the new algorithm are reported on.
A New Computational Method for Optimal Control of a Class of Constrained Systems Governed by Partial Differential Equations
 in Proc. of the 15th IFAC World Congress
, 2002
"... A computationally e#cient technique for the numerical solution of constrained optimal control problems governed by onedimensional partial di#erential equations is considered in this paper. This technique utilizes inversion to map the optimal control problem to a lower dimensional space. Results ..."
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Cited by 5 (1 self)
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A computationally e#cient technique for the numerical solution of constrained optimal control problems governed by onedimensional partial di#erential equations is considered in this paper. This technique utilizes inversion to map the optimal control problem to a lower dimensional space. Results are presented using the Nonlinear Trajectory Generation software package (NTG) showing that realtime implementation may be possible. Copyright c 2002 IFAC Keywords: Optimal Control, Partial Di#erential Equations, Inversion, Realtime, Computational methods.
Verification of SecondOrder Sufficient Optimality Conditions for Semilinear Elliptic and Parabolic Control Problems
"... We study optimal control problems for semilinear parabolic equations subject to control constraints and for semilinear elliptic equations subject to control and state constraints. We quote known secondorder sufficient optimality conditions (SSC) from the literature. Both problem classes, the parabo ..."
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Cited by 4 (2 self)
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We study optimal control problems for semilinear parabolic equations subject to control constraints and for semilinear elliptic equations subject to control and state constraints. We quote known secondorder sufficient optimality conditions (SSC) from the literature. Both problem classes, the parabolic one with boundary control and the elliptic one with boundary or distributed control, are discretized by a finite difference method. The discrete SSC are stated and numerically verified in all cases providing evidence of optimality where only necessary conditions had been studied before.
Sufficient Optimality for Discretized Parabolic and Elliptic Control Problems
"... We study optimal control problems for semilinear parabolic and elliptic equations subject to control and state constraints. We quote known secondorder sufficient optimality conditions (SSC) from the literature. Both problem classes are discretized by a finite difference method. The discrete SSC are ..."
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Cited by 2 (1 self)
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We study optimal control problems for semilinear parabolic and elliptic equations subject to control and state constraints. We quote known secondorder sufficient optimality conditions (SSC) from the literature. Both problem classes are discretized by a finite difference method. The discrete SSC are stated and numerically verified for fine discretizations with the help of sparse linear algebra techniques. This confirms initial results reported earlier for such discretized control problems. In order to relate these results to optimality for the underlying continuous problems corresponding theoretical, especially convergence, results are still unavailable at present.
Boundary Control for an Industrial UnderActuated Tubular
, 2005
"... Several control strategies are presented and studied for an industrial underactuated tubular chemical reactor. This work presents a casestudy of the performance of a decentralized versus centralized control strategy. The tubular reactor under consideration is characterized by nonlinear kinetic law ..."
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Several control strategies are presented and studied for an industrial underactuated tubular chemical reactor. This work presents a casestudy of the performance of a decentralized versus centralized control strategy. The tubular reactor under consideration is characterized by nonlinear kinetic laws, and it has some structural constraints on the location of the heat exchangers and of the sensors. For this system, a set of PI controllers is considered and a multivariable LQR controller is constructed to optimally choose the gains. The performance of these control strategies is studied. Finally, a direct numerical treatment of optimal control of the partial di#erential equations is presented. Industrial results are given for the linear controllers. Simulations emphasize the possible relevance of a direct numerical treatment of the nonlinear partial di#erential equations.
ÉquipesProjets Commands
"... apport de recherche ISSN 02496399 ISRN INRIA/RR7126FR+ENGinria00436768, version 1 27 Nov 2009Asymptotic expansion for the solution of a penalized control constrained semilinear elliptic problems ..."
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apport de recherche ISSN 02496399 ISRN INRIA/RR7126FR+ENGinria00436768, version 1 27 Nov 2009Asymptotic expansion for the solution of a penalized control constrained semilinear elliptic problems
Report no. 08/10 Optimal solvers for PDEConstrained Optimization
"... Optimization problems with constraints which require the solution of a partial differential equation arise widely in many areas of the sciences and engineering, in particular in problems of design. The solution of such PDEconstrained optimization problems is usually a major computational task. Here ..."
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Optimization problems with constraints which require the solution of a partial differential equation arise widely in many areas of the sciences and engineering, in particular in problems of design. The solution of such PDEconstrained optimization problems is usually a major computational task. Here we consider simple problems of this type: distributed control problems in which the 2 and 3dimensional Poisson problem is the PDE. The large dimensional linear systems which result from discretization and which need to be solved are of saddlepoint type. We introduce two optimal preconditioners for these systems which lead to convergence of symmetric Krylov subspace iterative methods in a number of iterations which does not increase with the dimension of the discrete problem. These preconditioners are block structured and involve standard multigrid cycles. The optimality of the preconditioned iterative solver is proved theoretically and verified computationally in several test cases. The theoretical proof indicates that these approaches may have much broader applicability for other partial differential equations. Key words and phrases: Saddlepoint problems, PDEconstrained optimization, preconditioning, optimal control, linear systems, allatonce methods
Nonconvex Constrained Optimization
, 2007
"... Fast nonlinear programming methods following the allatonce approach usually employ Newton’s method for solving linearized KarushKuhnTucker (KKT) systems. In nonconvex problems, the Newton direction is only guaranteed to be a descent direction if the Hessian of the Lagrange function is positive d ..."
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Fast nonlinear programming methods following the allatonce approach usually employ Newton’s method for solving linearized KarushKuhnTucker (KKT) systems. In nonconvex problems, the Newton direction is only guaranteed to be a descent direction if the Hessian of the Lagrange function is positive definite on the nullspace of the active constraints, otherwise some modifications to Newton’s method are necessary. This condition can be verified using the signs of the KKT’s eigenvalues (inertia), which are usually available from direct solvers for the arising linear saddle point problems. Iterative solvers are mandatory for very largescale problems, but in general do not provide the inertia. Here we present a preconditioner based on a multilevel incomplete LBL T factorization, from which an approximation of the inertia can be obtained. The suitability of the heuristics for application in optimization methods is verified on an interior point method applied to the CUTE and COPS test problems, on largescale 3D PDEconstrained optimal control problems, as well as 3D
A Robust Implementation of a Sequential Quadratic Programming Algorithm with Successive Error Restoration Address:
, 2010
"... We consider sequential quadratic programming (SQP) methods for solving constrained nonlinear programming problems. It is generally believed that SQP methods are sensitive to the accuracy by which partial derivatives are provided. One reason is that differences of gradients of the Lagrangian function ..."
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We consider sequential quadratic programming (SQP) methods for solving constrained nonlinear programming problems. It is generally believed that SQP methods are sensitive to the accuracy by which partial derivatives are provided. One reason is that differences of gradients of the Lagrangian function are used for updating a quasiNewton matrix, e.g., by the BFGS formula. The purpose of this paper is to show by numerical experimentation that the method can be stabilized substantially. Even in case of large random errors leading to partial derivatives with at most one correct digit, termination subject to an accuracy of 10−7 can be achieved, at least in 90 % of 306 problems of a standard test suite. The algorithm is stabilized by a nonmonotone line search and, in addition, by internal and external restarts in case of errors when computing the search direction due to inaccurate derivatives. Additional safeguards are implemented to overcome the situation that an intermediate feasible iterate might get an objective function value smaller than the final stationary point. In addition, we show how initial and periodic scaled restarts improve the efficiency in a situation with very slow convergence.