References
- Bonnans JF, Shapiro A. Perturbation analysis of optimization problems. New York, NY: Springer; 2000.
- Fiacco AV. Introduction to sensitivity and stability analysis in nonlinear programming. New York, NY: Academic Press; 1983.
- Agrachev A, Stefani G, Zezza PL. Strong optimality for a bang-bang trajectory. SIAM J. Control Optim. 2002;41:991–1014.
- Milyutin AA, Osmolovskii NP. Calculus of variations and optimal control. Providence, RI: Amer. Mathem. Soc; 1998.
- Noble J, Schättler H. Sufficient conditions for relative minima of broken extremals in optimal control theory. J. Math. Anal. Appl. 2002;269:98–128.
- Osmolovskii NP. Second-order conditions for broken extremals. In: Ioffe A, Reich S, Shafrir J, editors. Calculus of variations and optimal control. Boca Raton, FL: Chapman & Hall/CRC. Res. Notes Math. 2000;411:198–216.
- Osmolovskii NP. Quadratic extremality conditions for broken extremals in the general problem of the calculus of variations. Optimal control and dynamical systems. J. of Mathem. Sci. 2004;123:3987–4122.
- Osmolovskii NP, Lempio F. Transformation of quadratic forms to perfect squares for broken extremals. Set-Valued Anal. 2002;10:209–232.
- Poggiolini L, Stefani G. State-local optimality of a bang-bang trajectory: a Hamiltonian approach. Systems Control Lett. 2004;53:269–279.
- Schättler H, Ledzewicz U. Geometric optimal control. Theory, methods and examples. New York, NY: Springer; 2012.
- Felgenhauer U, Poggiolini L, Stefani G. Optimality and stability result for bang-bang optimal controls with simple and double switch behavior. Control & Cybern. 2009;38:1305–1325.
- Poggiolini L, Spadini M. Sufficient optimality conditions for a bang-bang trajectory. SIAM J. Control Optim. 2011;49:140–161.
- Maurer H, Osmolovskii NP. Equivalence of second-order optimality conditions for bang-bang control problems. Control & Cybernetics. 2005;34:927–950.
- Maurer H, Osmolovskii NP. Equivalence of second order optimality conditions for bang-bang control problems. II. Proofs, variational derivatives and representations. Control & Cybernetics. 2007;36:5–45.
- Osmolovskii NP. Quadratic conditions for nonsingular extremals in optimal control (A theoretical treatment). Russian J. of Mathem. Physics. 1995;2:487–512.
- Felgenhauer U. On stability of bang-bang type controls. SIAM J. Control Optim. 2003;41:1843–1867.
- Kim JR, Maurer H. Sensitivity analysis of optimal ontrol problems with bang-bang controls. 42th IEEE Conference on Decision Control. Maui, Hawaii: Decision. Control. 2003;4:3281–3286.
- Felgenhauer U. Directional sensitivity differentials for parametric bang-bang control problems. In: Large-Scale Scientific Computing, Lirkov I, Margenov S, Wasniewski J, editors. Lecture notes comp. sci. Vol. 5910. Berlin: Springer. 2010; p. 264–271.
- Alt W, Baier R, Lempio F, Gerdts M. Approximations of linear control problems with bang-bang solutions. Optimization. 2013;62:9–32.
- Alt W, Seydenschwanz M. Regularization and discretization of linear-quadratic control problems. Control & Cybernetics. 2012;40:903–920.
- Klötzler R. On a general conception of duality in optimal control. In: Equadiff IV, Fabera J, editors. Lect. notes math. Vol. 703. New York, NY: Springer; 1979. p. 189–196.
- Schättler H, Ledzewicz U, Dekhordi SM, Reisi M. A geometric analysis of bang-bang extremals in optimal control problems for combination cancer chemotherapy, 51st IEEE Conf. Maui, Hawaii: Decision Control; 2012.
- Maurer H, Osmolovskii NP. Second-order sufficient optimality conditions for time-optimal bang-bang control. SIAM J. Control Optim. 2004;42:2239–2263.
- Maurer H, Pickenhain S. Second order sufficient conditions for optimal control problems with mixed control-state constraints. J. Optim. Theor. Appl. 1995;86:649–667.
- Felgenhauer U. The shooting approach in analyzing bang-bang extremals with simultaneous control switches. Control & Cybernetics. 2008;37:307–327.
- Maurer H, Büskens C, Kim J-HR, Kaya CY. Optimization methods for the verification of second-order sufficient conditions for bang-bang controls. Optim. Contr. Appl. Meth. 2005;26:129–156.
- Felgenhauer U. Optimality properties of controls with bang-bang components in problems with semilinear state equation. Control & Cybernetics. 2005;34:763–785.
- Adams RA. Sobolev spaces. London: Academic Press; 1978.
- Agrachev AA, Sachkov YuL. Control theory from the geometric viewpoint, encyclopaedia of mathematical sciences, 87, control theory and optimization II. Berlin: Springer; 2004.
- Ledzewicz U, Nowakowski A, Schättler H. Stratifiable families of extremals and sufficient conditions for optimality in optimal control problems. J. Optim. Theory Appl. 2004;122:345–370.
- Schättler H. Local fields of extremals for optimal control problems with state constraints of relative degree 1. J. Dyn. Control Syst. 2006;12:563–599.