Approximations of linear control problems with bang-bang solutions

References

• Alt , W . 1997 . “ Discretization and mesh-independence of Newton's method for generalized equations ” . In Mathematical Programming with Data Perturbations V , Edited by: Fiacco , AV . 1 – 30 . New York : Lecture Notes in Pure Applied Mathematics, 195, Marcel Dekker .
   Google Scholar
• Alt , W and Mackenroth , U . 1989 . Convergence of finite element approximations to state constrained convex parabolic boundary control problems . SIAM J. Control Optim. , 27 : 718 – 736 .
   Web of Science ®Google Scholar
• Baier , R , Chahma , IA and Lempio , F . 2007 . Stability and convergence of Euler's method for state-constrained differential inclusions . SIAM J. Optim. , 18 : 1004 – 1026 .
   Web of Science ®Google Scholar
• Deckelnick , K and Hinze , M . A note on the approximation of elliptic control problems with bang-bang controls . Comput. Optim. Appl. , (to appear)
   Google Scholar
• Dhamo , V and Tröltzsch , F . Some aspects of reachability for parabolic boundary control problems with control constraints . Comput. Optim. Appl. , (to appear)
   Google Scholar
• Dontchev , AL and Farkhi , EM . 1989 . Error estimates for discretized differential inclusions . Computing , 41 : 349 – 358 .
   Web of Science ®Google Scholar
• Dontchev , AL and Hager , WW . 1993 . Lipschitzian stability in nonlinear control and optimization . SIAM J. Control Optim. , 31 : 569 – 603 .
   Web of Science ®Google Scholar
• Dontchev , AL and Hager , WW . 2001 . The Euler approximation in state constrained optimal control . Math. Comput. , 70 : 173 – 203 .
   Web of Science ®Google Scholar
• Dontchev , AL , Hager , WW and Malanowski , K . 2000 . Error bounds for Euler approximation of a state and control constrained optimal control problem . Numer. Funct. Anal. Optim. , 21 : 653 – 682 .
   Web of Science ®Google Scholar
• Dontchev , AL , Hager , WW and Veliov , VM . 2000 . Second-order Runge-Kutta approximations in constrained optimal control . SIAM J. Numer. Anal. , 38 : 202 – 226 .
   Web of Science ®Google Scholar
• Ekeland , I and Temam , R . 1976 . Convex Analysis and Variational Problems , Amsterdam, Oxford : North Holland .
   Google Scholar
• Felgenhauer , U . 2003 . On stability of bang-bang type controls . SIAM J. Control Optim. , 41 : 1843 – 1867 .
   Web of Science ®Google Scholar
• Felgenhauer , U . 2008 . The shooting approach in analyzing bang-bang extremals with simultaneous control switches . Control Cybernet. , 37 : 307 – 327 .
   Web of Science ®Google Scholar
• Felgenhauer , U . 2010 . “ Directional sensitivity differentials for parametric bang-bang control problems ” . In Lecture Notes Computer Science , Edited by: Lirkov , I , Margenov , S and Wasnievski , J . Vol. 5910 , 264 – 271 . Berlin, Heidelberg : Springer .
   Google Scholar
• Felgenhauer , U , Poggiolini , L and Stefani , G . 2009 . Optimality and stability result for bang-bang optimal controls with simple and double switch behaviour . Control Cybernet. , 38 : 1305 – 1325 .
   Web of Science ®Google Scholar
• Lenhart , S and Workman , JT . 2007 . Optimal Control Applied to Biological Models , Boca Raton, London, New York : Chapman & Hall/CRC .
   Google Scholar
• Malanowski , K , Büskens , C and Maurer , H . 1997 . “ Convergence of approximations to nonlinear optimal control problems ” . In Mathematical Programming with Data Perturbations V , Edited by: Fiacco , AV . 253 – 284 . New York : Lecture Notes in Pure and Applied Mathematics 195, Marcel Dekker .
   Google Scholar
• Malanowski , K and Maurer , H . 1996 . Sensitivity analysis for parametric control problems with control-state constraints . Comput. Optim. Appl. , 5 : 253 – 283 .
   Google Scholar
• Maurer , H , Büskens , C , Kim , J-HR and Kaya , CY . 2005 . Optimization methods for the verification of second-order sufficient conditions for bang-bang controls . Optimal Control Appl. Meth. , 26 : 129 – 156 .
   Web of Science ®Google Scholar
• Maurer , H and Pesch , HJ . 1994 . Solution differentiability for parametric nonlinear control problems . Control Cybernet. , 23 : 201 – 227 .
   Google Scholar
• Maurer , H and Pesch , HJ . 1994 . Solution differentiability for parametric nonlinear control problems with control-state constraints . SIAM J. Control Optim. , 32 : 1542 – 1554 .
   Web of Science ®Google Scholar
• Osmolovskii , NP and Maurer , H . 2005 . Equivalence of second order optimality conditions for bang-bang control problems. Part 1: Main results . Control Cybernet. , 34 : 927 – 950 .
   Web of Science ®Google Scholar
• Osmolovskii , NP and Maurer , H . 2007 . Equivalence of second order optimality conditions for bang-bang control problems. Part 2 : Proofs, variational derivatives and representations . Control Cybernet. , 36 : 5 – 45 .
   Web of Science ®Google Scholar
• Sendov , B and Popov , VA . 1988 . The Averaged Moduli of Smoothness , Chichester, New York, Brisbane, Toronto, Singapore : Wiley-Interscience .
   Google Scholar
• Stoer , J and Bulirsch , R . 1980 . Introduction to Numerical Analysis , New York : Springer .
   Google Scholar
• Veliov , VM . 1997 . On the time-discretization of control systems . SIAM J. Control Optim. , 35 : 1470 – 1486 .
   Web of Science ®Google Scholar
• Veliov , VM . 2005 . Error analysis of discrete approximations to bang-bang optimal control problems: the linear case . Control Cybernet. , 34 : 967 – 982 .
   Web of Science ®Google Scholar