Search results for "optimal control problems"

showing 4 items of 4 documents

Time-Optimal Synthesis for Three Relevant Problems: The Brockett Integrator, the Grushin Plane and the Martinet Distribution

2015

We construct the time-optimal synthesis for 3 problems that are linear in the control and with polytopic constraints in the controls. Namely, the Brockett integrator, the Grushin plane, and the Martinet distribution. The main purpose is to illustrate the steps in solving an optimal control problem and in particular the use of second order conditions. The Grushin and the Martinet case are particularly important: the first is the prototype of a rank-varying distribution, the second of a non-equiregular structure.

EngineeringControl and Optimizationbusiness.industryPlane (geometry)ta111Structure (category theory)Optimal controlControl and Systems Engineering; Modeling and Simulation; Control and OptimizationModeling and simulationControl theoryControl and Systems EngineeringIntegratorModeling and SimulationTrajectoryoptimal control problemsMathematics::Metric GeometryOrder (group theory)Applied mathematics[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]businessDistribution (differential geometry)ComputingMilieux_MISCELLANEOUS
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Functional A Posteriori Error Estimates for Time-Periodic Parabolic Optimal Control Problems

2015

This article is devoted to the a posteriori error analysis of multiharmonic finite element approximations to distributed optimal control problems with time-periodic state equations of parabolic type. We derive a posteriori estimates of the functional type, which are easily computable and provide guaranteed upper bounds for the state and co-state errors as well as for the cost functional. These theoretical results are confirmed by several numerical tests that show high efficiency of the a posteriori error bounds. peerReviewed

Mathematical optimizationControl and OptimizationMathematicsofComputing_NUMERICALANALYSISFinite element approximations010103 numerical & computational mathematicsType (model theory)01 natural sciencesparabolic time-periodic optimal control problemsError analysisFOS: MathematicsApplied mathematicsMathematics - Numerical AnalysisNumerical testsfunctional a posteriori error estimates0101 mathematicsMathematics - Optimization and Control49N20 35Q61 65M60 65F08Mathematicsta113Time periodicta111Numerical Analysis (math.NA)State (functional analysis)Optimal controlComputer Science Applications010101 applied mathematicsOptimization and Control (math.OC)multiharmonic finite element methodsSignal ProcessingA priori and a posterioriAnalysisNumerical Functional Analysis and Optimization
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Error Estimates for a Class of Elliptic Optimal Control Problems

2016

In this article, functional type a posteriori error estimates are presented for a certain class of optimal control problems with elliptic partial differential equation constraints. It is assumed that in the cost functional the state is measured in terms of the energy norm generated by the state equation. The functional a posteriori error estimates developed by Repin in the late 1990s are applied to estimate the cost function value from both sides without requiring the exact solution of the state equation. Moreover, a lower bound for the minimal cost functional value is derived. A meaningful error quantity coinciding with the gap between the cost functional values of an arbitrary admissible …

Mathematical optimizationControl and OptimizationNumerical analysis010102 general mathematicsta111010103 numerical & computational mathematicsOptimal control01 natural sciencesUpper and lower boundsComputer Science ApplicationsExact solutions in general relativityElliptic partial differential equationerror estimatesNorm (mathematics)Signal ProcessingA priori and a posterioriNumerical testselliptic optimal control problems0101 mathematicsAnalysisMathematics
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Guaranteed lower bounds for cost functionals of time-periodic parabolic optimization problems

2019

In this paper, a new technique is shown for deriving computable, guaranteed lower bounds of functional type (minorants) for two different cost functionals subject to a parabolic time-periodic boundary value problem. Together with previous results on upper bounds (majorants) for one of the cost functionals, both minorants and majorants lead to two-sided estimates of functional type for the optimal control problem. Both upper and lower bounds are derived for the second new cost functional subject to the same parabolic PDE-constraints, but where the target is a desired gradient. The time-periodic optimal control problems are discretized by the multiharmonic finite element method leading to lar…

Optimization problemtime-periodic conditionmultiharmonic finite element methodDiscretizationtwo-sided boundsSystems and Control (eess.SY)010103 numerical & computational mathematicsSystem of linear equationsElectrical Engineering and Systems Science - Systems and Control01 natural sciencesUpper and lower boundsSaddle pointFOS: MathematicsFOS: Electrical engineering electronic engineering information engineeringApplied mathematicsMathematics - Numerical AnalysisBoundary value problem0101 mathematicsMathematics - Optimization and ControlMathematicsosittaisdifferentiaaliyhtälöt35Kxx 65M60 65M70 65M15 65K10parabolic optimal control problemsNumerical Analysis (math.NA)matemaattinen optimointiOptimal controlFinite element method010101 applied mathematicsComputational MathematicsComputational Theory and MathematicsOptimization and Control (math.OC)Modeling and Simulationa posteriori error analysisnumeerinen analyysiguaranteed lower boundsComputers & Mathematics with Applications
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