Error estimates for parabolic optimal control problems with control and state constraints

The numerical approximation to a parabolic control problem with control and state constraints is studied in this paper. We use standard piecewise linear and continuous finite elements for the space discretization of the state, while the dG(0) method is used for time discretization. A priori error estimates for control and state are obtained by an improved maximum error estimate for the corresponding discretized state equation. Numerical experiments are provided which support our theoretical results.

[1]  Kenneth Eriksson,et al.  Adaptive finite element methods for parabolic problems IV: nonlinear problems , 1995 .

[2]  Ricardo H. Nochetto,et al.  Convergence Past Singularities for a Fully Discrete Approximation of Curvature-Drive Interfaces , 1997 .

[3]  E. Casas.  Boundary control of semilinear elliptic equations with pointwise state constraints , 1993 .

[4]  E. Casas.  Control of an elliptic problem with pointwise state constraints , 1986 .

[5]  E. Casas.  Pontryagin's Principle for State-Constrained Boundary Control Problems of Semilinear Parabolic Equations , 1997 .

[6]  Fredi Tröltzsch,et al.  Optimality conditions for state-constrained PDE control problems with time-dependent controls , 2008 .

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

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

[9]  C. Meyer.  Error estimates for the finite-element approximation of an elliptic control problem with pointwise state and control constraints , 2008 .

[10]  J. Nitsche,et al.  L∞-convergence of finite element approximations , 1977 .

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

[12]  Rolf Rannacher,et al.  Some Optimal Error Estimates for Piecewise Linear Finite Element Approximations , 1982 .

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

[14]  Ilio Galligani,et al.  Mathematical Aspects of Finite Element Methods , 1977 .

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

[16]  Wei,et al.  A NEW FINITE ELEMENT APPROXIMATION OF A STATE-CONSTRAINED OPTIMAL CONTROL PROBLEM , 2009 .

[17]  Rolf Rannacher,et al.  A Priori Error Estimates for Finite Element Discretizations of Parabolic Optimization Problems with Pointwise State Constraints in Time , 2011, SIAM J. Control. Optim..

[18]  Andreas Günther,et al.  Hamburger Beiträge zur Angewandten Mathematik Finite element approximation of elliptic control problems with constraints on the gradient , 2007 .

[19]  J. Frédéric Bonnans,et al.  Optimal Control of a Parabolic Equation with Time-Dependent State Constraints , 2010, SIAM J. Control. Optim..

[20]  GengshengBB Wang,et al.  ERROR ESTIMATES FOR AN OPTIMAL CONTROL PROBLEM GOVERNED BY THE HEAT EQUATION WITH STATE AND CONTROL CONSTRAINTS , 2009 .

[21]  Donald A. French,et al.  Analysis of a robust finite element approximation for a parabolic equation with rough boundary data , 1993 .

[22]  Wei Gong,et al.  A Mixed Finite Element Scheme for Optimal Control Problems with Pointwise State Constraints , 2011, J. Sci. Comput..

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

[24]  Alfred H. Schatz,et al.  Pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids: Part I. Global estimates , 1998, Math. Comput..

[25]  Michael Hinze,et al.  A Variational Discretization Concept in Control Constrained Optimization: The Linear-Quadratic Case , 2005, Comput. Optim. Appl..

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

[27]  O. Ladyženskaja.  Linear and Quasilinear Equations of Parabolic Type , 1968 .

[28]  Michael Hinze,et al.  Convergence of a Finite Element Approximation to a State-Constrained Elliptic Control Problem , 2007, SIAM J. Numer. Anal..

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