Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, 09/2018, Volume 339, pp. 514 - 541

In this paper, a local–global model reduction method is presented to solve stochastic optimal control problems constrained by stochastic partial differential equations (stochastic PDEs...

Local–global model reduction | Generalized multiscale finite element method | Stochastic optimal control problem | Reduced basis method | BOUNDARY CONTROL-PROBLEMS | APPROXIMATIONS | ALGORITHM | FINITE-ELEMENT METHODS | Local-global model reduction | POSTERIORI ERROR ESTIMATION | ELLIPTIC PDES | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | UNCERTAINTY | OPTIMIZATION | Models | Analysis | Methods | Differential equations | Mathematics - Numerical Analysis

Local–global model reduction | Generalized multiscale finite element method | Stochastic optimal control problem | Reduced basis method | BOUNDARY CONTROL-PROBLEMS | APPROXIMATIONS | ALGORITHM | FINITE-ELEMENT METHODS | Local-global model reduction | POSTERIORI ERROR ESTIMATION | ELLIPTIC PDES | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | UNCERTAINTY | OPTIMIZATION | Models | Analysis | Methods | Differential equations | Mathematics - Numerical Analysis

Journal Article

Computers and Mathematics with Applications, ISSN 0898-1221, 01/2018, Volume 75, Issue 2, pp. 335 - 348

This work is devoted to investigate the spectral approximation of optimal control of parabolic problems. The space–time method is used to boost high-order...

Parabolic equations | A priori error estimates | Space–time spectral method | Optimal control | MATHEMATICS, APPLIED | OPTIMIZATION PROBLEMS | APPROXIMATION | POINTWISE STATE CONSTRAINTS | GALERKIN METHOD | FINITE-ELEMENT METHODS | NAVIER-STOKES EQUATIONS | Space-time spectral method | CONSTRAINED OPTIMAL-CONTROL | ELLIPTIC-EQUATIONS

Parabolic equations | A priori error estimates | Space–time spectral method | Optimal control | MATHEMATICS, APPLIED | OPTIMIZATION PROBLEMS | APPROXIMATION | POINTWISE STATE CONSTRAINTS | GALERKIN METHOD | FINITE-ELEMENT METHODS | NAVIER-STOKES EQUATIONS | Space-time spectral method | CONSTRAINED OPTIMAL-CONTROL | ELLIPTIC-EQUATIONS

Journal Article

SIAM journal on control and optimization, ISSN 1095-7138, 2004, Volume 43, Issue 3, pp. 970 - 985

An optimal control problem for a two-dimensional (2-d) elliptic equation is investigated with pointwise control constraints...

Numerical approximation | Error estimates | Superconvergence | Linear-quadratic optimal control problems | Control constraints | Elliptic equations | linear-quadratic optimal control problems | MATHEMATICS, APPLIED | elliptic equations | control constraints | APPROXIMATION | CONSTRAINED OPTIMAL-CONTROL | error estimates | numerical approximation | superconvergence | AUTOMATION & CONTROL SYSTEMS

Numerical approximation | Error estimates | Superconvergence | Linear-quadratic optimal control problems | Control constraints | Elliptic equations | linear-quadratic optimal control problems | MATHEMATICS, APPLIED | elliptic equations | control constraints | APPROXIMATION | CONSTRAINED OPTIMAL-CONTROL | error estimates | numerical approximation | superconvergence | AUTOMATION & CONTROL SYSTEMS

Journal Article

Numerische Mathematik, ISSN 0029-599X, 2/2012, Volume 120, Issue 2, pp. 345 - 386

In this paper, a priori error estimates for space–time finite element discretizations of optimal control problems governed by semilinear parabolic PDEs and subject to pointwise control constraints are derived...

65M15 | Mathematical Methods in Physics | Numerical Analysis | Theoretical, Mathematical and Computational Physics | Appl.Mathematics/Computational Methods of Engineering | Numerical and Computational Physics | Mathematics, general | Mathematics | 65M60 | 49M25 | NUMERICAL APPROXIMATION | ELLIPTIC CONTROL-PROBLEM | MATHEMATICS, APPLIED | EQUATION | CONTROL CONSTRAINTS

65M15 | Mathematical Methods in Physics | Numerical Analysis | Theoretical, Mathematical and Computational Physics | Appl.Mathematics/Computational Methods of Engineering | Numerical and Computational Physics | Mathematics, general | Mathematics | 65M60 | 49M25 | NUMERICAL APPROXIMATION | ELLIPTIC CONTROL-PROBLEM | MATHEMATICS, APPLIED | EQUATION | CONTROL CONSTRAINTS

Journal Article

SIAM Journal on Numerical Analysis, ISSN 0036-1429, 2017, Volume 55, Issue 2, pp. 1101 - 1133

We investigate an a posteriori error analysis of adaptive finite element approximations of linear-quadratic boundary optimal control problems under bilateral box constraints, which act on a Neumann boundary control...

A posteriori error analysis | Discontinuous Galerkin methods | Adaptive finite element methods | Control constraints | Optimal boundary control problems | CONVECTION-DIFFUSION EQUATIONS | MATHEMATICS, APPLIED | control constraints | 2ND-ORDER ELLIPTIC PROBLEMS | STRATEGY | optimal boundary control problems | FINITE-ELEMENT APPROXIMATION | POSTERIORI ERROR ANALYSIS | PARTIAL-DIFFERENTIAL-EQUATIONS | VARIATIONAL DISCRETIZATION | CONVERGENCE | CONSTRAINED OPTIMAL-CONTROL | adaptive finite element methods | discontinuous Galerkin methods | a posteriori error analysis

A posteriori error analysis | Discontinuous Galerkin methods | Adaptive finite element methods | Control constraints | Optimal boundary control problems | CONVECTION-DIFFUSION EQUATIONS | MATHEMATICS, APPLIED | control constraints | 2ND-ORDER ELLIPTIC PROBLEMS | STRATEGY | optimal boundary control problems | FINITE-ELEMENT APPROXIMATION | POSTERIORI ERROR ANALYSIS | PARTIAL-DIFFERENTIAL-EQUATIONS | VARIATIONAL DISCRETIZATION | CONVERGENCE | CONSTRAINED OPTIMAL-CONTROL | adaptive finite element methods | discontinuous Galerkin methods | a posteriori error analysis

Journal Article

SIAM Journal on Control and Optimization, ISSN 0363-0129, 2012, Volume 50, Issue 4, pp. 2355 - 2372

In this paper, we derive some sufficient second order optimality conditions for control problems of partial differential equations (PDEs...

Sparse controls | Semilinear partial differential equation | Second order optimality conditions | Optimal control | Bang-bang controls | NUMERICAL APPROXIMATION | MATHEMATICS, APPLIED | SEMILINEAR ELLIPTIC-EQUATIONS | sparse controls | optimal control | semilinear partial differential equation | second order optimality conditions | bang-bang controls | SUFFICIENT OPTIMALITY CONDITIONS | EQUIVALENCE | AUTOMATION & CONTROL SYSTEMS | Partial differential equations | Mathematical analysis | Control equipment | Proving | Norms | Nonlinearity | Optimization

Sparse controls | Semilinear partial differential equation | Second order optimality conditions | Optimal control | Bang-bang controls | NUMERICAL APPROXIMATION | MATHEMATICS, APPLIED | SEMILINEAR ELLIPTIC-EQUATIONS | sparse controls | optimal control | semilinear partial differential equation | second order optimality conditions | bang-bang controls | SUFFICIENT OPTIMALITY CONDITIONS | EQUIVALENCE | AUTOMATION & CONTROL SYSTEMS | Partial differential equations | Mathematical analysis | Control equipment | Proving | Norms | Nonlinearity | Optimization

Journal Article

Journal of applied mathematics, ISSN 1687-0042, 2019, Volume 2019, pp. 1 - 5

In this paper, a state-constrained optimal control problem governed by -Laplacian elliptic equations is studied...

Problems | Elliptic functions | Discretization | Mathematical analysis | Optimal control

Problems | Elliptic functions | Discretization | Mathematical analysis | Optimal control

Journal Article

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Full Text
A priori error estimates for optimal control problems with state and control constraints

Optimal control applications & methods, ISSN 0143-2087, 2018, Volume 39, Issue 3, pp. 1168 - 1181

Summary The Galerkin spectral method is used to approximate elliptic optimal control problems with integral state and control constraints in this paper...

spectral method | optimal control | elliptic equations | error estimates | MATHEMATICS, APPLIED | APPROXIMATION | POINTWISE CONTROL | ELLIPTIC CONTROL-PROBLEMS | STOKES EQUATIONS | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | REGULAR LAGRANGE MULTIPLIERS | GALERKIN SPECTRAL METHOD | AUTOMATION & CONTROL SYSTEMS | Galerkin method | Control theory | Optimal control | Optimization | Spectral methods

spectral method | optimal control | elliptic equations | error estimates | MATHEMATICS, APPLIED | APPROXIMATION | POINTWISE CONTROL | ELLIPTIC CONTROL-PROBLEMS | STOKES EQUATIONS | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | REGULAR LAGRANGE MULTIPLIERS | GALERKIN SPECTRAL METHOD | AUTOMATION & CONTROL SYSTEMS | Galerkin method | Control theory | Optimal control | Optimization | Spectral methods

Journal Article

Mathematics of Computation of the American Mathematical Society, ISSN 0025-5718, 09/2008, Volume 77, Issue 263, pp. 1269 - 1291

In this paper, we investigate the superconvergence property of the numerical solution of a quadratic convex optimal control problem by using rectangular mixed finite element methods...

Finite element method | Interpolation | Error rates | A posteriori knowledge | Approximation | Equations of state | Optimal control | Mathematics | Control theory | Estimation methods | Quadratic optimal control problems | Superconvergence | Rectangular partition | error estimates | Mixed finite elements | rectangular partition | ORDER | MATHEMATICS, APPLIED | APPROXIMATION | L-2 error estimates | mixed finite elements | STATE | ELLIPTIC-EQUATIONS | quadratic optimal control problems | superconvergence

Finite element method | Interpolation | Error rates | A posteriori knowledge | Approximation | Equations of state | Optimal control | Mathematics | Control theory | Estimation methods | Quadratic optimal control problems | Superconvergence | Rectangular partition | error estimates | Mixed finite elements | rectangular partition | ORDER | MATHEMATICS, APPLIED | APPROXIMATION | L-2 error estimates | mixed finite elements | STATE | ELLIPTIC-EQUATIONS | quadratic optimal control problems | superconvergence

Journal Article

Journal of mathematical analysis and applications, ISSN 0022-247X, 2014, Volume 409, Issue 1, pp. 446 - 458

We consider the minimization problem of an integral functional with integrand that is not convex in the control on solutions of a control system described by...

Fractional differential equation | Relaxation property | Feedback control | Optimal control | Nonconvex constraint | EXISTENCE | VARIATIONAL-HEMIVARIATIONAL INEQUALITIES | MATHEMATICS, APPLIED | MULTIFUNCTIONS | L(P)-CONTINUOUS EXTREME SELECTORS | DECOMPOSABLE VALUES | MATHEMATICS | BANACH-SPACES | THEOREMS | ELLIPTIC TYPE | APPROXIMATE CONTROLLABILITY | SEMILINEAR EVOLUTION INCLUSIONS | Control systems | Differential equations

Fractional differential equation | Relaxation property | Feedback control | Optimal control | Nonconvex constraint | EXISTENCE | VARIATIONAL-HEMIVARIATIONAL INEQUALITIES | MATHEMATICS, APPLIED | MULTIFUNCTIONS | L(P)-CONTINUOUS EXTREME SELECTORS | DECOMPOSABLE VALUES | MATHEMATICS | BANACH-SPACES | THEOREMS | ELLIPTIC TYPE | APPROXIMATE CONTROLLABILITY | SEMILINEAR EVOLUTION INCLUSIONS | Control systems | Differential equations

Journal Article

Computational Optimization and Applications, ISSN 0926-6003, 7/2011, Volume 49, Issue 3, pp. 549 - 566

We consider an elliptic optimal control problem with pointwise bounds on the gradient of the state...

Elliptic optimal control problem | Gradient constraints | Error estimates | Operations Research/Decision Theory | Convex and Discrete Geometry | Mathematics | Statistics, general | Operations Research, Mathematical Programming | Optimization | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | FINITE-ELEMENT APPROXIMATION | Mathematical optimization | Studies | Errors | Control theory | Estimates | Analysis | Finite element method | Approximation | Discretization | Mathematical analysis | Regularity | Standards

Elliptic optimal control problem | Gradient constraints | Error estimates | Operations Research/Decision Theory | Convex and Discrete Geometry | Mathematics | Statistics, general | Operations Research, Mathematical Programming | Optimization | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | FINITE-ELEMENT APPROXIMATION | Mathematical optimization | Studies | Errors | Control theory | Estimates | Analysis | Finite element method | Approximation | Discretization | Mathematical analysis | Regularity | Standards

Journal Article

ESAIM - Control, Optimisation and Calculus of Variations, ISSN 1292-8119, 2016, Volume 22, Issue 2, pp. 355 - 370

Optimal control problems in measure spaces governed by parabolic equations with are considered...

optimal control | first order optimality conditions | sparsity | numerical approximation | Space-time measure controls | parabolic equations | ELLIPTIC CONTROL-PROBLEMS | MATHEMATICS, APPLIED | DIRECTIONAL SPARSITY | AUTOMATION & CONTROL SYSTEMS | Time measurement | Spacetime | Optimal control | Optimization

optimal control | first order optimality conditions | sparsity | numerical approximation | Space-time measure controls | parabolic equations | ELLIPTIC CONTROL-PROBLEMS | MATHEMATICS, APPLIED | DIRECTIONAL SPARSITY | AUTOMATION & CONTROL SYSTEMS | Time measurement | Spacetime | Optimal control | Optimization

Journal Article

SIAM Journal on Control and Optimization, ISSN 0363-0129, 2012, Volume 50, Issue 4, pp. 1735 - 1752

Optimal control problems in measure spaces governed by elliptic equations are considered for distributed and Neumann boundary control, which are known to promote sparse solutions...

Boundary control | Measure controls | Convergence estimates | Sparsity | Elliptic partial differential equation | Optimal control | MATHEMATICS, APPLIED | elliptic partial differential equation | DIRICHLET PROBLEM | optimal control | convergence estimates | sparsity | boundary control | measure controls | AUTOMATION & CONTROL SYSTEMS

Boundary control | Measure controls | Convergence estimates | Sparsity | Elliptic partial differential equation | Optimal control | MATHEMATICS, APPLIED | elliptic partial differential equation | DIRICHLET PROBLEM | optimal control | convergence estimates | sparsity | boundary control | measure controls | AUTOMATION & CONTROL SYSTEMS

Journal Article

SIAM JOURNAL ON CONTROL AND OPTIMIZATION, ISSN 0363-0129, 2019, Volume 57, Issue 4, pp. 3021 - 3045

We investigate full Lipschitzian and full Holderian stability for a class of control problems governed by semilinear elliptic partial differential equations, where the cost functional, the state...

MATHEMATICS, APPLIED | 2ND-ORDER OPTIMALITY CONDITIONS | semilinear elliptic partial differential equations | PARAMETRIC OPTIMAL-CONTROL | combined second-order subdifferential | full Holderian stability | coderivative | BANG CONTROL-PROBLEMS | LIPSCHITZ STABILITY | full Lipschitzian stability | REGULARIZATION | AUTOMATION & CONTROL SYSTEMS | perturbed control problem

MATHEMATICS, APPLIED | 2ND-ORDER OPTIMALITY CONDITIONS | semilinear elliptic partial differential equations | PARAMETRIC OPTIMAL-CONTROL | combined second-order subdifferential | full Holderian stability | coderivative | BANG CONTROL-PROBLEMS | LIPSCHITZ STABILITY | full Lipschitzian stability | REGULARIZATION | AUTOMATION & CONTROL SYSTEMS | perturbed control problem

Journal Article

Optimal control applications & methods, ISSN 0143-2087, 2019, Volume 40, Issue 2, pp. 241 - 264

Summary We analyze both a priori and a posteriori error analysis of finite‐element method for elliptic optimal control problems with measure data in a bounded convex domain in Rd (d = 2or3...

finite‐element approximations | elliptic optimal control problem | a priori and a posteriori error estimates | measure data | finite-element approximations | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | EQUATIONS | CONVERGENCE | AUTOMATION & CONTROL SYSTEMS | Variables | Lower bounds | Economic models | Error analysis | Equations of state | Upper bounds | Optimal control | Regularity | State variable

finite‐element approximations | elliptic optimal control problem | a priori and a posteriori error estimates | measure data | finite-element approximations | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | EQUATIONS | CONVERGENCE | AUTOMATION & CONTROL SYSTEMS | Variables | Lower bounds | Economic models | Error analysis | Equations of state | Upper bounds | Optimal control | Regularity | State variable

Journal Article

SIAM Journal on Control and Optimization, ISSN 0363-0129, 2016, Volume 54, Issue 5, pp. 2568 - 2593

This paper deals with a class of stochastic optimal control problems (SOCPs) in the presence of state constraints...

Viscosity notion | Hamilton-Jacobi equations | Stochastic optimal control | Stochastic target problems | State constraints | stochastic optimal control | EXISTENCE | VISCOSITY SOLUTIONS | MATHEMATICS, APPLIED | APPROXIMATION | TIMES | viscosity notion | TARGET PROBLEMS | state constraints | FRONTS | NONLINEAR ELLIPTIC-EQUATIONS | SETS | stochastic target problems | CONTROLLABILITY | AUTOMATION & CONTROL SYSTEMS | Analysis of PDEs | Mathematics | Optimization and Control

Viscosity notion | Hamilton-Jacobi equations | Stochastic optimal control | Stochastic target problems | State constraints | stochastic optimal control | EXISTENCE | VISCOSITY SOLUTIONS | MATHEMATICS, APPLIED | APPROXIMATION | TIMES | viscosity notion | TARGET PROBLEMS | state constraints | FRONTS | NONLINEAR ELLIPTIC-EQUATIONS | SETS | stochastic target problems | CONTROLLABILITY | AUTOMATION & CONTROL SYSTEMS | Analysis of PDEs | Mathematics | Optimization and Control

Journal Article

IMA Journal of Numerical Analysis, ISSN 0272-4979, 10/2019, Volume 39, Issue 4, pp. 1985 - 2015

Abstract This paper concerns the adaptive finite element method for elliptic Dirichlet boundary control problems in the energy space...

Dirichlet boundary control | adaptive finite element method | MATHEMATICS, APPLIED | optimal control problem | energy space | APPROXIMATION | convergence | POSTERIORI ERROR ANALYSIS | elliptic equation

Dirichlet boundary control | adaptive finite element method | MATHEMATICS, APPLIED | optimal control problem | energy space | APPROXIMATION | convergence | POSTERIORI ERROR ANALYSIS | elliptic equation

Journal Article

Computers and Mathematics with Applications, ISSN 0898-1221, 08/2017, Volume 74, Issue 4, pp. 714 - 726

In this paper, we investigate error estimates and superconvergence of a mixed finite element method for elliptic optimal control problems...

Mixed finite element methods | Superconvergence | Optimal control problems | Elliptic equations | Error estimate | MATHEMATICS, APPLIED | APPROXIMATION | EQUATIONS | Finite element method | Methods

Mixed finite element methods | Superconvergence | Optimal control problems | Elliptic equations | Error estimate | MATHEMATICS, APPLIED | APPROXIMATION | EQUATIONS | Finite element method | Methods

Journal Article

Computational optimization and applications, ISSN 1573-2894, 2019, Volume 74, Issue 1, pp. 225 - 258

We propose a local regularization of elliptic optimal control problems which involves the nonconvex $$L^q...

DCA | Elliptic PDE | DC programming | Mathematics | 90C26 | Statistics, general | Optimization | 49K20 | 90C46 | 49J20 | Operations Research/Decision Theory | Convex and Discrete Geometry | Optimal control | Operations Research, Management Science | Nonconvex | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | PARABOLIC CONTROL-PROBLEMS | DUALITY | Analysis | Algorithms | Differential equations | Problems | Economic models | Numerical methods | Nonlinear programming | Regularization | Convex analysis

DCA | Elliptic PDE | DC programming | Mathematics | 90C26 | Statistics, general | Optimization | 49K20 | 90C46 | 49J20 | Operations Research/Decision Theory | Convex and Discrete Geometry | Optimal control | Operations Research, Management Science | Nonconvex | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | PARABOLIC CONTROL-PROBLEMS | DUALITY | Analysis | Algorithms | Differential equations | Problems | Economic models | Numerical methods | Nonlinear programming | Regularization | Convex analysis

Journal Article

Mathematical Models and Methods in Applied Sciences, ISSN 0218-2025, 2014, Volume 24, Issue 8, pp. 1457 - 1493

... (variational inequalities and quasilinear elliptic equations) and optimal control problems governed by linear elliptic partial differential equations...

Mimetic finite differences | control problems | nonlinear problems | CONVECTION-DIFFUSION PROBLEMS | MATHEMATICS, APPLIED | BOUNDARY-VALUE-PROBLEMS | POLYHEDRAL MESHES | DISCONTINUOUS GALERKIN METHOD | ELEMENT APPROXIMATION | CONTROL CONSTRAINTS | VARIATIONAL-INEQUALITIES | POSTERIORI ERROR ESTIMATORS | ELLIPTIC OBSTACLE PROBLEMS | NEWTONIAN STOKES FLOWS

Mimetic finite differences | control problems | nonlinear problems | CONVECTION-DIFFUSION PROBLEMS | MATHEMATICS, APPLIED | BOUNDARY-VALUE-PROBLEMS | POLYHEDRAL MESHES | DISCONTINUOUS GALERKIN METHOD | ELEMENT APPROXIMATION | CONTROL CONSTRAINTS | VARIATIONAL-INEQUALITIES | POSTERIORI ERROR ESTIMATORS | ELLIPTIC OBSTACLE PROBLEMS | NEWTONIAN STOKES FLOWS

Journal Article

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