Second-Order Analysis and Numerical Approximation for Bang-Bang Bilinear Control Problems

We consider bilinear optimal control problems, whose objective functionals do not depend on the controls. Hence, bang-bang solutions will appear. We investigate sufficient second-order conditions for bang-bang controls, which guarantee local quadratic growth of the objective functional in $L^1$. In addition, we prove that for controls that are not bang-bang, no such growth can be expected. Finally, we study the finite-element discretization, and prove error estimates of bang-bang controls in $L^1$-norms.

[1]  G. Stampacchia,et al.  Inverse Problem for a Curved Quantum Guide , 2012, Int. J. Math. Math. Sci..

[2]  J. Douglas,et al.  The stability inLq of theL2-projection into finite element function spaces , 1974 .

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

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

[5]  Monique Dauge,et al.  Neumann and mixed problems on curvilinear polyhedra , 1992 .

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

[7]  Gerd Wachsmuth,et al.  Sufficient Second-Order Conditions for Bang-Bang Control Problems , 2017, SIAM J. Control. Optim..

[8]  F. Tröltzsch.  Optimal Control of Partial Differential Equations: Theory, Methods and Applications , 2010 .

[9]  H SchatzAlfred.  Pointwise Error Estimates and Asymptotic Error Expansion Inequalities for the Finite Element Method on Irregular Grids , 2000 .

[10]  Daniel Wachsmuth,et al.  Robust error estimates for regularization and discretization of bang–bang control problems , 2015, Comput. Optim. Appl..

[11]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[12]  R. Rannacher,et al.  Zur L∞-Konvergenz linearer finiter Elemente beim Dirichlet-Problem , 1976 .

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

[14]  P. Grisvard.  Elliptic Problems in Nonsmooth Domains , 1985 .

[15]  J. M. Thomas,et al.  Introduction à l'analyse numérique des équations aux dérivées partielles , 1983 .

[16]  J. Frédéric Bonnans,et al.  Optimal control of infinite dimensional bilinear systems: application to the heat and wave equations , 2016, Mathematical Programming.

[17]  Jean-Pierre Raymond,et al.  Boundary control of semilinear elliptic equations with discontinuous leading coefficients and unbounded controls , 1997 .

[18]  Helmut Maurer,et al.  Equivalence of second order optimality conditions for bang-bang control problems. Part 1: Main results , 2005 .

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

[20]  Eduardo Casas,et al.  UNIFORM CONVERGENCE OF THE FEM. APPLICATIONS TO STATE CONSTRAINED CONTROL PROBLEMS , 2002 .

[21]  C. Meyer,et al.  Uniqueness Criteria for the Adjoint Equation in State-Constrained Elliptic Optimal Control , 2011 .

[22]  Eduardo Casas.  Error Estimates for the Numerical Approximation of Semilinear Elliptic Control Problems with Finitely Many State Constraints , 2002 .

[23]  J. Frédéric Bonnans,et al.  Second order analysis of control-affine problems with scalar state constraint , 2014, Math. Program..

[24]  Alfred H. Schatz,et al.  Pointwise Error Estimates and Asymptotic Error Expansion Inequalities for the Finite Element Method on Irregular Grids: Part II. Interior Estimates , 2000, SIAM J. Numer. Anal..

[25]  Sophia Th. Kyritsi-Yiallourou,et al.  Handbook of Applied Analysis , 2009 .

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

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