Open Access
Issue
ESAIM: COCV
Volume 27, 2021
Article Number 51
Number of page(s) 36
DOI https://doi.org/10.1051/cocv/2021046
Published online 04 June 2021
  1. J.E. Ackermann, Sampled-Data Control Systems: Analysis and Synthesis, Robust System Design. Springer-Verlag Berlin Heidelberg (1985). [Google Scholar]
  2. R.P. Agarwal, V. Otero-Espinar, K. Perera and D.R. Vivero, Basic properties of Sobolev’s spaces on time scales. Adv. Differ. Equ. 14 (2006) 38121. [Google Scholar]
  3. R.P. Agarwal and M. Bohner, Basic calculus on time scales and some of its applications. Results Math. 35 (1999) 3–22. [Google Scholar]
  4. K.J. Aström, On the choice of sampling rates in optimal linear systems. IBM Res.: Eng. Stud. (1963). [Google Scholar]
  5. T. Bakir, B. Bonnard, L. Bourdin and J. Rouot, Pontryagin-type conditions for optimal muscular force response to functional electric stimulations. J. Optim. Theory Appl. 184 (2020) 581–602. [Google Scholar]
  6. B. Bamieh and J.B. Pearson, The 2 problem for sampled-data systems. Syst. Control Lett. 19 (1992) 1–12. [Google Scholar]
  7. Z. Bartosiewicz and D.F.M. Torres, Noether’s theorem on time scales. J. Math. Anal. Appl. 342 (2008) 1220–1226. [Google Scholar]
  8. M. Bohner, Calculus of variations on time scales. Dyn. Syst. Appl. 13 (2004) 339–349. [Google Scholar]
  9. M. Bohner and A. Peterson, Dynamic equations on time scales. An introduction with applications. Birkhäuser Boston Inc., Boston, MA (2001). [Google Scholar]
  10. M. Bohner and A. Peterson, Advances in dynamic equations on time scales. Birkhäuser Boston Inc., Boston, MA (2003). [Google Scholar]
  11. M. Bohner, K. Kenzhebaev, O. Lavrova and O. Stanzhytskyi, Pontryaginś maximum principle for dynamic systems on time scales. J. Differ. Equ. Appl. 3 (2017) 1161–1189. [CrossRef] [Google Scholar]
  12. V.G. Boltyanskii, Optimal control of discrete systems. John Wiley & Sons, New York-Toronto, Ont. (1978). [Google Scholar]
  13. J.F. Bonnans and C. De La Vega, Optimal control of state constrained integral equations. Set-Valued Anal. 18 (2010) 307–326. [CrossRef] [Google Scholar]
  14. B. Bonnard, L. Faubourg, G. Launay and E. Trélat, Optimal control with state constraints and the space shuttle re-entry problem. J. Dynam. Control Syst. 9 (2003) 155–199. [CrossRef] [Google Scholar]
  15. L. Bourdin, Nonshifted calculus of variations on time scales with nabla-differentiable sigma. J. Math. Anal. Appl. 411 (2014) 543–554. [CrossRef] [Google Scholar]
  16. L. Bourdin, Note on Pontryagin maximum principle with running state constraints and smooth dynamics – proof based on the Ekeland variational principle. Res. Notes (2016). [Google Scholar]
  17. L. Bourdin and G. Dhar, Continuity/constancy of the Hamiltonian function in a Pontryagin maximum principle for optimal sampled-data control problems with free sampling times. Math. Control Signals Syst. 31 (2019) 503–544. [CrossRef] [Google Scholar]
  18. L. Bourdin and G. Dhar, Optimal sampled-data controls with running inequality state constraints: Pontryagin maximum principle and bouncing trajectory phenomenon. To appear in: Math. Program., Ser. A (2020). https://doi.org/10.1007/s10107-020-01574-2. [Google Scholar]
  19. L. Bourdin and E. Trélat, General Cauchy-Lipschitz theory for Delta-Cauchy problems with Carathéodory dynamics on time scales. J. Differ. Equ. Appl. 20 (2014) 526–547. [Google Scholar]
  20. L. Bourdin and E. Trélat, Pontryagin Maximum Principle for finite dimensional nonlinear optimal control problems on time scales. SIAM J. Control Optim. 51 (2013) 3781–3813. [Google Scholar]
  21. L. Bourdin and E. Trélat, Optimal sampled-data control, and generalizations on time scales. Math. Control Relat. Fields 6 (2016) 53–94. [Google Scholar]
  22. L. Bourdin and E. Trélat, Pontryagin Maximum Principle for optimal sampled-data control problems. In proceedings of the IFAC Workshop CAO (2015). [Google Scholar]
  23. L. Bourdin, O. Stanzhytskyi and E. Trélat, Addendum to “Pontryagin’s maximum principle for dynamic systems on time scales”. J. Differ. Equ. Appl. 23 (2017) 1760–1763. [Google Scholar]
  24. A. Bressan and B. Piccoli, Introduction to the mathematical theory of control. Vol. 2 of AIMS Series on Applied Mathematics. American Institute of Mathematical Sciences (AIMS), Springfield, MO (2007). [Google Scholar]
  25. A. Cabada and D.R. Vivero, Criterions for absolute continuity on time scales. J. Differ. Equ. Appl. 11 (2005) 1013–1028. [Google Scholar]
  26. A. Cabada and D.R. Vivero, Expression of the Lebesgue Δ-integral on time scales as a usual Lebesgue integral: application to the calculus of Δ-antiderivatives. Math. Comput. Model. 43 (2006) 194–207. [Google Scholar]
  27. T. Chen and B. Francis, Optimal sampled-data control systems. Springer-Verlag London, Ltd., London (1996). [Google Scholar]
  28. F.H. Clarke, Functional analysis, calculus of variations and optimal control. Vol. 264 of Graduate Texts in Mathematics. Springer Science & Business Media (2013). [CrossRef] [Google Scholar]
  29. O. Cots, Geometric and numerical methods for a state constrained minimum time control problem of an electric vehicle. ESAIM: COCV 23 (2017) 1715–1749. [EDP Sciences] [Google Scholar]
  30. O. Cots, J. Gergaud and D. Goubinat, Direct and indirect methods in optimal control with state constraints and the climbing trajectory of an aircraft. Opt. Control Appl. Methods 39 (2018) 281–301. [Google Scholar]
  31. A. Dmitruk, On the development of Pontryagin’s maximum principle in the works of A. Ya. Dubovitskii and A. A. Milyutin. Control Cybern. 4A (2009) 923–957. [Google Scholar]
  32. A.V. Dmitruk and N.P. Osmolovskii, Proof of the maximum principle for a problem with state constraints by the V-change of time variable. Discrete Contin. Dyn. Syst. Ser. B 24 (2019) 2189–2204. [Google Scholar]
  33. A.Y. Dubovitskii and A.A. Milyutin, Extremum problems in the presence of restrictions. USSR Comput. Math. Math. Phys. 5 (1965) 1–80. [Google Scholar]
  34. I. Ekeland, On the variational principle. J. Math. Anal. Appl. 47 (1974) 324–353. [CrossRef] [MathSciNet] [Google Scholar]
  35. H.O. Fattorini, Infinite-dimensional optimization and control theory. Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge (1999). [Google Scholar]
  36. I. Fonseca and G. Leoni, Modern methods in the calculus of variations: Lp spaces. Springer (2007). [Google Scholar]
  37. A. Fryszkowski, Fixed point theory for decomposable sets. Springer Netherlands (2004). [Google Scholar]
  38. R.V. Gamkrelidze, Optimal control processes for bounded phase coordinates. Izv. Akad. Nauk SSSR. Ser. Mat. 24 (1960) 315–356. [Google Scholar]
  39. J.C. Geromel and M. Souza, On an LMI approach to optimal sampled-data state feedback control design. Internat. J. Control 88 (2015) 2369–2379. [Google Scholar]
  40. I.V. Girsanov, Lectures on mathematical theory of extremum problems. Vol. 67 of Lecture Notes in Economics and Mathematical Systems, edited by B.T. Poljak. Translated from the Russian by D. Louvish. Springer-Verlag, Berlin-New York (1972). [CrossRef] [Google Scholar]
  41. G.S. Guseinov, Integration on time scales. J. Math. Anal. Appl. 285 (2003) 107–127. [Google Scholar]
  42. S. Hilger, Ein Maßkettenkalkül mit Anwendungen auf Zentrumsmannigfaltigkeiten. PhD thesis, Universität Würzburg (1988). [Google Scholar]
  43. R. Hilscher and V. Zeidan, Calculus of variations on time scales: weak local piecewise Crd1 solutions with variable endpoints. J. Math. Anal. Appl. 289 (2004) 143–166. [Google Scholar]
  44. R. Hilscher and V. Zeidan, First-order conditions for generalized variational problems over time scales. Comput. Math. Appl. 62 (2011) 3490–3503. [Google Scholar]
  45. R. Hilscher and V. Zeidan, Weak maximum principle and accessory problem for control problems on time scales. Nonlinear Anal. 70 (2009) 3209–3226. [Google Scholar]
  46. R. Hilscher and V. Zeidan, Time scale embedding theorem and coercivity of quadratic functionals. Analysis (Munich) 28 (2008) 1–28. [Google Scholar]
  47. R.F. Hartl, S.P. Sethi and R.G. Vickson, A survey of the maximum principles for optimal control problems with state constraints. SIAM Rev. 37 (1995) 181–218. [Google Scholar]
  48. J.M. Holtzman and H. Halkin, Directional convexity and the maximum principle for discrete systems. SIAM J. Control 4 (1966) 263–275. [Google Scholar]
  49. A. Huseynov, The Riesz representation theorem on time scales. Math. Comput. Model. 55 (2012) 1570–1579. [Google Scholar]
  50. D.H. Jacobson, M.M. Lele and J.L. Speyer, New necessary conditions of optimality for control problems with state-variable inequality constraints. J. Math. Anal. Appl. 35 (1971) 255–284. [Google Scholar]
  51. I.D. Landau, Digital Control Systems. Springer (2006). [Google Scholar]
  52. E.B. Lee and L. Markus, Foundations of optimal control theory. Robert E. Krieger Publishing Co., Inc., Melbourne, FL, second edition (1986). [Google Scholar]
  53. A. Levis, R. Schlueter and M. Athans, On the behavior of optimal linear sampled-data regulators. Int. J. Control 13 (1971) 343–361. [Google Scholar]
  54. X. Li and J. Yong, Optimal control theory for infinite dimensional systems. Birkhäuser Boston (1995). [Google Scholar]
  55. H. Maurer, On optimal control problems with bounded state variables and control appearing linearly. SIAM J. Control Optim. 15 (1977) 345–362. [Google Scholar]
  56. H. Maurer, J.R. Kim and G. Vossen, On a state-constrained control problem in optimal production and maintenance. In Optimal Control and Dynamic Games. Springer (2005) 289–308. [Google Scholar]
  57. S.M. Melzer and B.C. Kuo, Sampling period sensitivity of the optimal sampled data linear regulator. Automatica J. IFAC 7 (1971) 367–370. [Google Scholar]
  58. R.H. Middleton and G.C. Goodwin, Digital control and estimation: A unifiedapproach (1990). [Google Scholar]
  59. B. Mordukhovich, Variational Analysis and Generalized Differentiation I. Springer-Verlag, Berlin Heidelberg (2006). [Google Scholar]
  60. D. Nešić and A.R. Teel, Sampled-data control of nonlinear systems: an overview of recent results. Perspect. Robust Control 268 (2001) 221–239. [Google Scholar]
  61. L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze and E.F. Mishchenko, The mathematical theory of optimal processes. John Wiley & Sons, Inc. (1962). [Google Scholar]
  62. B.N. Pshenichnyi, Necessary conditions for an extremum. Marcel Dekker, Inc., New York (1971). [Google Scholar]
  63. A. Puchkova, V. Rehbock and K.L. Teo, Closed-form solutions of a fishery harvesting model with state constraint. Optimal Control Appl. Methods 35 (2014) 395–411. [Google Scholar]
  64. W. Rudin, Real and complex analysis. 3rd ed. McGraw-Hill, New York, NY. (1987). [Google Scholar]
  65. M. Salgado, R. Middleton and G.C. Goodwin, Connection between continuous and discrete Riccati equations with applications to Kalman filtering. Proc. IEE-D 135 (1988) 28–34. [Google Scholar]
  66. S.P. Sethi and G.L. Thompson, Optimal Control Theory. Applications to Management Science and Economics. Kluwer Academic Publishers, Boston, MA, second edition (2000). [Google Scholar]
  67. W. Sierpinski, Sur les fonctions d’ensemble additives et continues. Fundam. Math. 3 (1922) 240–246. [Google Scholar]
  68. M. Souza, G.W.G. Vital and J.C. Geromel, Optimal sampled data state feedback control of linear systems. In Proceedings of the 19th World Congress The International Federation of Automatic Control (2014). [Google Scholar]
  69. E. Trélat, Contrôle optimal : théorie & applications. Vuibert Paris (2005). [Google Scholar]
  70. T. van Keulen, J. Gillot, B. de Jager and M. Steinbuch, Solution for state constrained optimal control problems applied to power split control for hybrid vehicles. Automatica J. IFAC 50 (2014) 187–192. [Google Scholar]
  71. R.B. Vinter, Optimal Control. Birkhaüser, Boston (2000). [Google Scholar]

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