Optimal control for elliptic systems with pointwise euclidean norm constraints on the controls

Optimal control for an elliptic system with pointwise Euclidean norm constraints on the control variables is investigated. First order optimality conditions are derived in a manner that is amenable for numerical realisation. An efficient semismooth Newton algorithm is proposed based on this optimality system. Numerical examples are given to validate the superlinear convergence of the semismooth Newton algorithm.

[1]  Michael Ulbrich,et al.  Semismooth Newton Methods for Operator Equations in Function Spaces , 2002, SIAM J. Optim..

[2]  Daniel Wachsmuth.  Sufficient second-order optimality conditions for convex control constraints , 2006 .

[3]  G. Burton.  Sobolev Spaces , 2013 .

[4]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[5]  John E. Dennis,et al.  Numerical methods for unconstrained optimization and nonlinear equations , 1983, Prentice Hall series in computational mathematics.

[6]  Mauro Giudici,et al.  Systems, Control, Modeling and Optimization , 2006 .

[7]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[8]  Arnd Rösch,et al.  Error Estimates for Parabolic Optimal Control Problems with Control Constraints , 2004 .

[9]  M. Heinkenschloss,et al.  Global Convergence of Trust-Region Interior-Point Algorithms for Infinite-Dimensional Nonconvex Mini , 1999 .

[10]  L. Zajícek.  Control and Cybernetics , 2007 .

[11]  Daniel Wachsmuth.  Numerical Solution of Optimal Control Problems with Convex Control Constraints , 2006, Systems, Control, Modeling and Optimization.

[12]  Kazufumi Ito,et al.  Lagrange multiplier approach to variational problems and applications , 2008, Advances in design and control.

[13]  Boris Vexler,et al.  Adaptive Finite Elements for Elliptic Optimization Problems with Control Constraints , 2008, SIAM J. Control. Optim..

[14]  E. Lieb,et al.  Analysis, Second edition , 2001 .

[15]  Michael Hintermüller,et al.  A SQP-Semismooth Newton-type Algorithm applied to Control of the instationary Navier--Stokes System Subject to Control Constraints , 2006, SIAM J. Optim..

[16]  Daniel Wachsmuth.  Optimal Control Problems with Convex Control Constraints , 2007 .

[17]  Marion Kee,et al.  Analysis , 2004, Machine Translation.

[18]  Kazufumi Ito,et al.  The Primal-Dual Active Set Method for Nonlinear Optimal Control Problems with Bilateral Constraints , 2004, SIAM J. Control. Optim..

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

[20]  W. Rudin.  Principles of mathematical analysis , 1964 .

[21]  Michael Ulbrich,et al.  Constrained optimal control of Navier-Stokes flow by semismooth Newton methods , 2003, Syst. Control. Lett..

[22]  Kazufumi Ito,et al.  The Primal-Dual Active Set Strategy as a Semismooth Newton Method , 2002, SIAM J. Optim..

[23]  Michael Ulbrich,et al.  Semismooth Newton Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces , 2011, MOS-SIAM Series on Optimization.

[24]  Karl Kunisch,et al.  Optimal control for an elliptic system with convex polygonal control constraints , 2013 .