EDP Sciences logo
Subscriber Authentication Point
Search
Menu
All issues Volume 30 (2024) ESAIM: COCV, 30 (2024) 39 References
Open Access
Issue
ESAIM: COCV
Volume 30, 2024
Article Number 39
Number of page(s) 26
DOI https://doi.org/10.1051/cocv/2024028
Published online 03 May 2024
  1. M.H.A. Biswas and M.d.R. de Pinho, A maximum principle for optimal control problems with state and mixed constraints. ESAIM: Control Optim. Calc. Var. 21 (2015) 939–957. [CrossRef] [EDP Sciences] [MathSciNet] [Google Scholar]
  2. A. Cabot, The steepest descent dynamical system with control. Applications to constrained minimization. ESAIM: Control Optim. Calc. Var. 10 (2004) 243–258. [CrossRef] [EDP Sciences] [MathSciNet] [Google Scholar]
  3. P. Nistri and M. Quincampoix, On the dynamics of a differential inclusion built upon a nonconvex constrained minimization problem. J. Optim. Theory Appl. 124 (2005) 659–672. [CrossRef] [MathSciNet] [Google Scholar]
  4. P. Saint-Pierre, Newton and other continuation methods for multivalued inclusions. Set-Valued Anal. 3 (1995) 143–156. [CrossRef] [MathSciNet] [Google Scholar]
  5. A. Opial and Z. Lasota, An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations. Bull. Acad. Polon. Sci., Ser. Sci. Math. Astron. phys. 13 (1965) 781–786. [Google Scholar]
  6. N. Kikuchi, Control problems of contingent equation. Publ. Res. Inst. Math. Sci. 3 (1967) 85–99. [CrossRef] [Google Scholar]
  7. J.-P. Aubin and A. Cellina, Differential Inclusions. Springer-Verlag Publ., Berlin (1984) 344. [Google Scholar]
  8. G.V. Smirnov, Introduction to the Theory of Differential Inclusions. American Mathematical Society, Providence (2000) 227. [Google Scholar]
  9. F.H. Clarke and P.R. Wolenski, Control of systems to sets and their interiors. J. Optim. Theory Appl. 88 (1996) 3–23. [CrossRef] [MathSciNet] [Google Scholar]
  10. M. Krastanov and M. Quincampoix, Local small time controllability and attainability of a set for nonlinear control system. ESAIM: Control Optim. Calc. Var. 6 (2001) 499–516. [CrossRef] [EDP Sciences] [MathSciNet] [Google Scholar]
  11. A. Cernea an C. Georgescu, Necessary optimality conditions for differential-difference inclusions with state constraints. J. Math. Anal. Appl. 344 (2007) 43–53. [Google Scholar]
  12. G.S. Pappas, Optimal solutions to differential inclusions in presence of state constraints. J. Optim. Theory Appl. 44 (1984) 657–679. [CrossRef] [MathSciNet] [Google Scholar]
  13. Q.J. Zhu, Necessary optimality conditions for nonconvex differential inclusions with endpoint constraints. J. Diff. Equ. 124 (1996) 186–204. [CrossRef] [Google Scholar]
  14. A.V. Arutyunov, S.M. Aseev and V.I. Blagodatskikh, Necessary conditions of the first order in the problem of optimal control of a differential inclusion with phase constraints. Russian Acad. Sci. Sb. Math. 79 (1994) 117–139. [MathSciNet] [Google Scholar]
  15. M. Sandberg, Convergence of the forward Euler method for nonconvex differential inclusions. SIAM J. Numer. Anal. 47 (2008) 308–320. [Google Scholar]
  16. J. Bastien, Convergence order of implicit Euler numerical scheme for maximal monotone differential Inclusions. Z. Angew. Math. Phys. 64 (2013) 955–966. [CrossRef] [MathSciNet] [Google Scholar]
  17. W.-J. Beyn and J. Rieger, The implicit Euler scheme for one-sided Lipschitz differential inclusions. Discr. Cont. Dynam. Syst. Ser. B 14 (2010) 409–428. [Google Scholar]
  18. F. Lempio, Modified Euler methods for differential inclusions. Proc. of the Worksh. on Set-Val. Anal. and Diff. Incl., Pamporovo, Bulgaria (1990). [Google Scholar]
  19. V. Veliov, Second order discrete approximations to strongly convex differential inclusions. Syst. Contr. Lett. 13 (1989) 263–269. [CrossRef] [Google Scholar]
  20. A. Dontchev and F. Lempio, Difference methods for differential inclusions: a survey. SIAM Rev. 34 (1992) 263–294. [CrossRef] [MathSciNet] [Google Scholar]
  21. K. Schilling, An algorithm to solve boundary value problems for differential inclusions and applications in optimal control. Numer. Funct. Anal. Optim. 10 (1989) 733–764. [Google Scholar]
  22. E.K. Kostousova and A.B. Kurzhanski, Guaranteed estimates of accuracy of computations in problems of control and estimation. Computat. Technol. 2 (1977) 19–27. [Google Scholar]
  23. F.L. Chernousko, State estimation for dynamical systems by means of ellipsoids. Dyn. Syst. (1987) 95–105. [Google Scholar]
  24. A.I. Ovseevich, On equations of ellipsoids approximating attainable sets. J. Optim. Theory Appl. 95 (1997) 659–676. [CrossRef] [MathSciNet] [Google Scholar]
  25. V.A. Komarov, Estimates for the attainable set for differential inclusions. Math. Notes Acad. Sci. USSR 37 (1985) 501–506. [Google Scholar]
  26. Z. Artstein, First-order approximations for differential inclusions. Set-Valued Anal. 2 (1994) 7–17. [CrossRef] [MathSciNet] [Google Scholar]
  27. S. Otakulov, Approximation of the optimal-time problem for controlled differential inclusions. Cybernet. Syst. Anal. 30 (1994) 458–462. [CrossRef] [Google Scholar]
  28. B.S. Mordukhovich, Optimization and finite difference approximations of nonconvex differential inclusions with free time, in Nonsmooth Analysis and Geometric Methods in Deterministic Optimal Control, edited by B.S. Mordukhovich and H.J. Sussmann. (1996) 153–202 [CrossRef] [Google Scholar]
  29. A.I. Panasyuk, Equations of attainable set dynamics, part 1: integral funnel equations. J. Optim. Theory Appl. 64 (1990) 349–366. [CrossRef] [MathSciNet] [Google Scholar]
  30. P. Saint-Pierre, Approximation of the viability kernel. Appl. Math. Optim. 29 (1994) 187–209. [CrossRef] [MathSciNet] [Google Scholar]
  31. J. Rieger, Shadowing and the viability kernel algorithm. Appl. Math. Optim. 60 (2009) 429–441. [CrossRef] [MathSciNet] [Google Scholar]
  32. V.I. Gurman and E.A. Trushkova, Estimates for attainability sets of control systems. Diff. Equ. 45 (2009) 1636–1644. [CrossRef] [Google Scholar]
  33. A.R. Matviychuk and V.N. Ushakov, On the construction of resolving controls in control problems with phase constraints. J. Comput. Syst. Sci. Int. 45 (2006) 1–16. [CrossRef] [MathSciNet] [Google Scholar]
  34. A.V. Fominyh, The subdifferential descent method in a nonsmooth variational problem. Optim. Lett. 17 (2023) 675–698. [CrossRef] [MathSciNet] [Google Scholar]
  35. A.V. Fominyh, A numerical method for finding the optimal solution of a differential inclusion. Vestnik St. Petersburg Univers. Math. 51 (2018) 397–406. [CrossRef] [Google Scholar]
  36. A.V. Fominyh, A method for solving differential inclusions with fixed right end. Vestnik Saint Petersburg Univ. Appl. Math. Comput. Sci. Control Processes 14 (2018) 302–315. [Google Scholar]
  37. A.V. Fominyh, Method for finding a solution to a linear nonstationary interval ODE system. Vestnik Saint Petersburg Univ. Appl. Math. Comput. Sci. Control Processes 17 (2021) 148–165. [Google Scholar]
  38. M.V. Dolgopolik and A.V. Fominyh, Exact penalty functions for optimal control problems. I. Main theorem and free-endpoint problems. Optimal Control Appl. Methods 40 (2019) 1018–1044. [CrossRef] [MathSciNet] [Google Scholar]
  39. A.V. Fominyh, Open-loop control of a plant described by a system with nonsmooth right-hand side. Computat. Math. Math. Phys. 59 (2019) 1639–1648. [CrossRef] [Google Scholar]
  40. V.F. Demyanov and V.N. Malozemov, Introduction to minimax. Dover Publications Inc., New York (1990) 320 p. [Google Scholar]
  41. M. Dolgopolik, Nonsmooth problems of calculus of variations via codifferentiation. ESAIM: Control Optim. Calc. Var. 20 (2014) 1153–1180. [CrossRef] [EDP Sciences] [MathSciNet] [Google Scholar]
  42. S. Amat and P. Pedregal, A variational approach to implicit ODEs and differential Inclusions. ESAIM: Control Optim. Calc. Var. 15 (2009) 139–148. [CrossRef] [EDP Sciences] [MathSciNet] [Google Scholar]
  43. A.F. Filippov, Differential Equations with Discontinuous Righthand Sides. Springer Science+Business Media B.V. Dordrecht (1988) 304. [Google Scholar]
  44. D.E. Stewart and J.C. Trinkle, An implicit time-stepping scheme for rigid body dynamics with inelastic collisions and Coulomb friction. Int. J. Numer. Methods Eng. 39 (1996) 2673–2691. [CrossRef] [Google Scholar]
  45. A. Levant, Higher-order sliding modes, differentiation and output-feedback control. Int. J. Control 76 (2003) 924–941. [CrossRef] [Google Scholar]
  46. V.I. Utkin, Sliding Modes in Control and Optimization. Springer-Verlag, Berlin (1992) 286. [Google Scholar]
  47. V.F. Demyanov, G. Stavroulakis, L.N. Polyakova and P.D. Panagiotopoulos, Quasidifferentiability and Nonsmooth Modelling in Mechanics, Engineering and Economics. Kluwer Academic Publishers, Dordrecht, London (1996) 348. [Google Scholar]
  48. M. Makela and P. Neittaanmaki, Nonsmooth Optimization: Analysis and Algorithms with Applications to Optimal Control. World Scientific Publishing, London (1992) 254. [Google Scholar]
  49. M. Dolgopolik, Constrained nonsmooth problems of the calculus of variations. ESAIM: Control Optim. Calc. Var. 27 (2021) 1–35. [Google Scholar]
  50. M.A. Aizerman and E.S. Pyatniskii, Fundamentals of the theory of discontinuous systems I. Autom. Remote Control 7 (1974) 33–47. [Google Scholar]
  51. V.I. Blagodatskikh and A.F. Filippov, Differential inclusions and optimal control. Proc. Steklov Inst. Math. 169 (1986) 199–259. [Google Scholar]
  52. A.V. Fominyh, A method for solving discontinuous systems with sliding modes. (2022) 1–17. arXiv:2204.11266. [Google Scholar]
  53. J.F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems. Springer Science+Business Media Publ., New York (2000) 601 p. [Google Scholar]
  54. V.F. Demyanov and L.V. Vasil’ev, Nondifferentiable Optimization. Springer-Optimization Software, New York (1986) 452. [Google Scholar]
  55. V.F. Demyanov and A.M. Rubinov, Basics of Nonsmooth Analysis and Quasidifferential Calculus. Nauka, Moscow (1990) 432. [Google Scholar]
  56. J.-P. Aubin and H. Frankowska, Set-valued Analysis. Birkhauser Basel, Boston (1990) 482. [Google Scholar]
  57. A.F. Filippov, On certain questions in the theory of optimal control. J. Soc. Ind. Appl. Math. Ser. A: Control 1 (1959) 76–84. [Google Scholar]
  58. V.F. Demyanov, Extremum Conditions and Variation Calculus. Vysshaya shkola, Moscow (2005) 335 p. [Google Scholar]
  59. L.V. Kantorovich and G.P. Akilov, Functional Analysis. Nauka Publ., Moscow (1977) 752 p. [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.