Second Order Analysis for Bang-Bang Control Problems of PDEs

In this paper, we derive some sufficient second order optimality conditions for control problems of partial differential equations (PDEs) when the cost functional does not involve the usual quadratic term for the control or higher nonlinearities for it. Though not always, in this situation the optimal control is typically bang-bang. Two different control problems are studied. The second differs from the first in the presence of the $L^1$ norm of the control. This term leads to optimal controls that are sparse and usually take only three different values (we call them bang-bang-bang controls). Though the proofs are detailed in the case of a semilinear elliptic state equation, the approach can be extended to parabolic control problems. Some hints are provided in the last section to extend the results.

[1]  Fredi Tröltzsch,et al.  First- and Second-Order Optimality Conditions for a Class of Optimal Control Problems with Quasilinear Elliptic Equations , 2009, SIAM J. Control. Optim..

[2]  G. Stampacchia,et al.  Problemi al contorno ellittici, con dati discontinui, dotati di soluzioni hölderiane , 1960 .

[3]  Fredi Tröltzsch,et al.  Sufficient Second-Order Optimality Conditions for Semilinear Control Problems with Pointwise State Constraints , 2008, SIAM J. Optim..

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

[5]  Helmut Maurer,et al.  Second order optimality conditions for bang-bang control problems , 2003 .

[6]  Fredi Tröltzsch,et al.  Error Estimates for the Numerical Approximation of a Semilinear Elliptic Control Problem , 2002, Comput. Optim. Appl..

[7]  Fredi Tröltzsch,et al.  Second Order Sufficient Optimality Conditions for Some State-constrained Control Problems of Semilinear Elliptic Equations , 2000, SIAM J. Control. Optim..

[8]  Eduardo Casas,et al.  Error estimates for the numerical approximation of Neumann control problems , 2008, Comput. Optim. Appl..

[9]  Fredi Tröltzsch,et al.  Second Order Analysis for Optimal Control Problems: Improving Results Expected From Abstract Theory , 2012, SIAM J. Optim..

[10]  Eduardo Casas,et al.  Second Order Optimality Conditions for Semilinear Elliptic Control Problems with Finitely Many State Constraints , 2001, SIAM J. Control. Optim..

[11]  Roland Herzog,et al.  Optimality Conditions and Error Analysis of Semilinear Elliptic Control Problems with L1 Cost Functional , 2012, SIAM J. Optim..

[12]  Fredi Tröltzsch,et al.  Second-Order Necessary and Sufficient Optimality Conditions for Optimization Problems and Applications to Control Theory , 2002, SIAM J. Optim..

[13]  B. Goh.  Necessary Conditions for Singular Extremals Involving Multiple Control Variables , 1966 .

[14]  J. F. Bonnans,et al.  Second-Order Analysis for Control Constrained Optimal Control Problems of Semilinear Elliptic Systems , 1998 .

[15]  Helmut Maurer,et al.  Second Order Sufficient Conditions for Time-Optimal Bang-Bang Control , 2003, SIAM J. Control. Optim..

[16]  Frank H. Clarke,et al.  A New Approach to Lagrange Multipliers , 1976, Math. Oper. Res..

[17]  A. A. Mili︠u︡tin,et al.  Calculus of variations and optimal control , 1998 .

[18]  Michael Hinze,et al.  A note on the approximation of elliptic control problems with bang-bang controls , 2010, Computational Optimization and Applications.

[19]  Ursula Felgenhauer,et al.  On Stability of Bang-Bang Type Controls , 2002, SIAM J. Control. Optim..

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

[21]  Joseph C. Dunn.  Second-Order Optimality Conditions in Sets of ${\sf L}^\infty$ Functions with Range in a Polyhedron , 1995 .

[22]  Andrei V. Dmitruk,et al.  Quadratic order conditions for bang-singular extremals , 2011, 1107.0161.

[23]  Jean-Pierre Raymond,et al.  Error Estimates for the Numerical Approximation of a Distributed Control Problem for the Steady-State Navier--Stokes Equations , 2007, SIAM J. Control. Optim..

[24]  H. Maurer,et al.  Equivalence of second order optimality conditions for bang-bang control problems. Part 2 : Proofs, variational derivatives and representations , 2007 .

[25]  Jean-Pierre Raymond,et al.  ESTIMATES FOR THE NUMERICAL APPROXIMATION OF DIRICHLET BOUNDARY CONTROL FOR SEMILINEAR ELLIPTIC EQUATIONS , 2006 .

[26]  Carlos E. Kenig,et al.  The Inhomogeneous Dirichlet Problem in Lipschitz Domains , 1995 .