Non-degenerate maximum principle for optimal control problems with discontinuous right-hand side

References

• Utkin VI. Discontinuous control systems: state of art in theory and applications. IFAC Proc Vol. 1987;20(5):25–44.
   Google Scholar
• Buhite JL, Owen DR. An ordinary differential equation from the theory of plasticity. Arch Ration Mech Anal. 1979;71(4):357–383.
   Web of Science ®Google Scholar
• Filippov AF. Differential equations with discontinuous right-hand sides. Berlin: Springer Science & Business Media; 2013.
   Google Scholar
• Utkin VI. Sliding modes in control and optimization. Berlin: Springer Science & Business Media; 2013.
   Google Scholar
• Pisano A, Usai E. Sliding mode control: a survey with applications in math. Math Comput Simul. 2011;81(5):954–979.
   Web of Science ®Google Scholar
• Wang J. Asymptotic stabilization of continuous-time linear systems with quantized state feedback. Automatica. 2018;88:83–90.
   Web of Science ®Google Scholar
• Dmitruk AV, Kaganovich AM. The hybrid maximum principle is a consequence of pontryagin maximum principle. Syst Control Lett. 2008;57(11):964–970.
   Web of Science ®Google Scholar
• Sussmann HJ. A maximum principle for hybrid optimal control problems. Proceedings of the 38th IEEE Conference on Decision and Control; 1999 Dec 7–10; Phoenix, USA. Vol. 1; IEEE; 1999. p. 425–430.
   Google Scholar
• Akhmet M, Kirane M, Tleubergenova M, et al. Control and optimal response problems for quasilinear impulsive integrodifferential equations. Eur J Oper Res. 2006;169(3):1128–1147.
   Web of Science ®Google Scholar
• Garavello M, Piccoli B. Hybrid necessary principles: an application to a car with gears. IFAC Proc Vol. 2003;36(6):253–258.
   Google Scholar
• Xu X, Antsaklis PJ. Optimal control of hybrid autonomous systems with state jumps. Proceedings of the 2003 American Control Conference; 2003 Jun 4–6; Denver, USA. Vol. 6; IEEE; 2003. p. 5191–5196.
   Google Scholar
• Ivashkevych A, Kovalchuk T. Existence of an optimal control for systems of differential equations with pulse action at nonfixed times. J Math Sci. 2018;228:1–18.
   Google Scholar
• Dieci L, Lopez L. A survey of numerical methods for ivps of odes with discontinuous right-hand side. J Comput Appl Math. 2012;236(16):3967–3991.
   Web of Science ®Google Scholar
• Zelikin M, Lokutsievskii L, Hildebrand R. Typicality of chaotic fractal behavior of integral vortices in hamiltonian systems with discontinuous right hand side. J Math Sci. 2017;221(1):1–136.
   Google Scholar
• Stewart DE, Anitescu M. Optimal control of systems with discontinuous differential equations. Numer Math. 2010;114(4):653–695.
   Web of Science ®Google Scholar
• Kirches C, Sager S, Bock HG, et al. Time-optimal control of automobile test drives with gear shifts. Optimal Control Appl Methods. 2010;31(2):137–153.
   Web of Science ®Google Scholar
• Bock HG, Kirches C, Meyer A, et al. Numerical solution of optimal control problems with explicit and implicit switches. Optim Methods Softw. 2018;33(3):450–474.
   Web of Science ®Google Scholar
• Danca MF, Codreanu S. On a possible approximation of discontinuous dynamical systems. Chaos Solitons Fractals. 2002;13(4):681–691.
   Web of Science ®Google Scholar
• Kugushev E. Necessary optimality conditions for systems described by equations with discontinuous right-hand side. Vestn Mosk Univ Mat Mekh. 1974;2:83–90.
   Google Scholar
• Ashchepkov LT. Optimal control of discontinuous systems. Nauka: Sibirsk. Otdel., Novosibirsk; 1987.
   Google Scholar
• Rosendahl R. Sufficient optimality conditions for nonsmooth optimal control problems [Ph.D thesis]. Hamburg: Informatik und Naturwissenschaften der Universität Hamburg; 2009.
   Google Scholar
• Oberle HJ, Rosendahl R. Numerical computation of a singular-state subarc in an economic optimal control model. Optimal Control Appl Methods. 2006;27(4):211–235.
   Web of Science ®Google Scholar
• Oberle HJ, Rosendahl R. On singular arcs in nonsmooth optimal control. Control Cybernet. 2008;37(2):429–450.
   Google Scholar
• Arutyunov AV, Karamzin DY. Non-degenerate necessary optimality conditions for the optimal control problem with equality-type state constraints. J Global Optim. 2016;64(4):623–647.
   Web of Science ®Google Scholar
• Donchev T, Farkhi E, Mordukhovich B. Discrete approximations, relaxation, and optimization of one-sided lipschitzian differential inclusions in hilbert spaces. J Differ Equ. 2007;243:301–328.
   Web of Science ®Google Scholar
• Donchev T, Nosheen A. Value function and optimal control of differential inclusions. Ann Alexandru Ioan Cuza Univ Math. 2015;61(1):181–193.
   Google Scholar
• Mordukhovich BS. Variational analysis and generalized differentiation. Vol. 2: applications. Berlin: Springer; 2006.
   Google Scholar
• Kostina E, Kostyukova O, Schmidt W. New necessary conditions for optimal control problems in discontinuous dynamic systems. IFIP Conference on System Modeling and Optimization; Sep 12–16. Berlin: Springer; 2011. p. 122–135.
   Google Scholar
• Morozov SF, Sumin MI. Optimal control of sliding modes of discontinuous dynamical systems. Izvestiya Vysshikh Uchebnykh Zavedenii Matematika. 1990;1:53–61.
   Google Scholar
• Hartl RF, Sethi SP, Vickson RG. A survey of the maximum principles for optimal control problems with state constraints. SIAM Rev. 1995;37(2):181–218.
   Web of Science ®Google Scholar
• Dmitruk AV, Kaganovich AM. Maximum principle for optimal control problems with intermediate constraints. Comput Math Model. 2011;22(2):180–215.
   Google Scholar