Space-time finite element approximation of parabolic optimal control problems

Abstract In this paper we investigate a space-time finite element approximation of parabolic optimal control problems. The first order optimality conditions are transformed into an elliptic equation of fourth order in space and second order in time involving only the state or the adjoint state in the space-time domain. We derive a priori and a posteriori error estimates for the time discretization of the state and the adjoint state. Furthermore, we also propose a space-time mixed finite element discretization scheme to approximate the space-time elliptic equations, and derive a priori error estimates for the state and the adjoint state. Numerical examples are presented to illustrate our theoretical findings and the performance of our approach.

[1]  M. Hinze,et al.  A space-time multigrid solver for distributed control of the time-dependent Navier-Stokes system , 2008 .

[2]  Boris Vexler,et al.  A Priori Error Estimates for Space-Time Finite Element Discretization of Parabolic Optimal Control Problems , 2012, Constrained Optimization and Optimal Control for Partial Differential Equations.

[3]  Wolfgang Hackbusch,et al.  A numerical method for solving parabolic equations with opposite orientations , 1978, Computing.

[4]  Alfio Borzì,et al.  Multigrid Methods for PDE Optimization , 2009, SIAM Rev..

[5]  Boris Vexler,et al.  Adaptive Space-Time Finite Element Methods for Parabolic Optimization Problems , 2007, SIAM J. Control. Optim..

[6]  Fredi Tröltzsch,et al.  ON REGULARIZATION METHODS FOR THE NUMERICAL SOLUTION OF PARABOLIC CONTROL PROBLEMS WITH POINTWISE STATE CONSTRAINTS , 2009 .

[7]  N. Yan.  Superconvergence analysis and a posteriori error estimation of a finite element method for an optimal control problem governed by integral equations , 2009 .

[8]  S. Balasundaram,et al.  A mixed finite element method for fourth order elliptic equations with variable coefficients , 1984 .

[9]  Tao Tang,et al.  A Posteriori Error Estimates for Discontinuous Galerkin Time-Stepping Method for Optimal Control Problems Governed by Parabolic Equations , 2004, SIAM J. Numer. Anal..

[10]  Ningning Yan,et al.  A posteriori error estimates for optimal control problems governed by parabolic equations , 2003, Numerische Mathematik.

[11]  K. Deckelnick,et al.  VARIATIONAL DISCRETIZATION OF PARABOLIC CONTROL PROBLEMS IN THE PRESENCE OF POINTWISE STATE CONSTRAINTS , 2010 .

[12]  Wei Gong,et al.  Error estimates for parabolic optimal control problems with control and state constraints , 2013, Comput. Optim. Appl..

[13]  Boris Vexler,et al.  A Priori Error Estimates for Space-Time Finite Element Discretization of Parabolic Optimal Control Problems , 2019, Constrained Optimization and Optimal Control for Partial Differential Equations.

[14]  W. Hackbusch.  On the Fast Solving of Parabolic Boundary Control Problems , 1979 .

[15]  J. Lions.  Optimal Control of Systems Governed by Partial Differential Equations , 1971 .

[16]  M. Chipot.  Finite Element Methods for Elliptic Problems , 2000 .

[17]  Thomas Slawig,et al.  A Smooth Regularization of the Projection Formula for Constrained Parabolic Optimal Control Problems , 2011 .

[18]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[19]  Karl Kunisch,et al.  Semismooth Newton Methods for Optimal Control of the Wave Equation with Control Constraints , 2011, SIAM J. Control. Optim..

[20]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[21]  Stefan Ulbrich,et al.  Optimization with PDE Constraints , 2008, Mathematical modelling.

[22]  Lars B. Wahlbin.  On Maximum Norm Error Estimates for Galerkin Approximations to One-Dimensional Second Order Parabolic Boundary Value Problems , 1975 .

[23]  T. Geveci,et al.  On the approximation of the solution of an optimal control problem governed by an elliptic equation , 1979 .

[24]  Guido Büttner.  Ein Mehrgitterverfahren zur optimalen Steuerung parabolischer Probleme , 2004 .

[25]  P. Clément.  Approximation by finite element functions using local regularization , 1975 .

[26]  L. R. Scott,et al.  Finite element interpolation of nonsmooth functions satisfying boundary conditions , 1990 .

[27]  Ronald H. W. Hoppe,et al.  Adaptive Space-Time Finite Element Approximations of Parabolic Optimal Control Problems , 2011 .

[28]  J. Lions,et al.  Non-homogeneous boundary value problems and applications , 1972 .