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Approximate Controllability for a Class of Second-Order Stochastic Evolution Inclusions of Clarke’s Subdifferential Type

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Abstract

In this paper, we are dealing with the approximate controllability for a class of second-order stochastic evolution inclusions of Clarke’s subdifferential type. Initially, we show the existence of mild solutions for the stochastic evolution inclusions by using stochastic analysis, nonsmooth analysis and fixed point theorems of multivalued maps. Then we provide a set of sufficient conditions for the approximate controllability of the second-order stochastic evolution inclusions. Finally, an example is included for the illustration of the obtained theoretical results.

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References

  1. Arnold, L.: Stochastic Differential Equations: Theory and Applications. Wiley, Berlin (1974)

    MATH  Google Scholar 

  2. Arthi, G., Balachandran, K.: Controllability of second-order impulsive evolution systems with infinite delay. Nonlinear Anal. Hybrid Syst. 11, 139–153 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. Arthi, G., Park, J.H., Jung, H.Y.: Existence and controllability results for second-order impulsive stochastic evolution systems with state-dependent delay. Appl. Math. Comput. 248, 328–341 (2014)

    MathSciNet  MATH  Google Scholar 

  4. Balasubramaniam, P., Tamilalagan, P.: Approximate controllability of a class of fractional neutral stochastic integrodifferential inclusions with infinite delay by using Mainardi’s function. Appl. Math. Comput. 256, 232–246 (2015)

    MathSciNet  MATH  Google Scholar 

  5. Bashirov, A.E., Mahmudov, N.I.: On concepts of controllability for deterministic and stochastic systems. SIAM J. Control Optim. 37(6), 1808–1821 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  6. Carl, S.: Existence of extremal solutions of boundary hemivariational inequalities. J. Differ. Equ. 171, 370–396 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  7. Carl, S., Motreanu, D.: Extremal solutions of quasilinear parabolic inclusions with generalized Clarke’s gradient. J. Differ. Equ. 191, 206–233 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

    MATH  Google Scholar 

  9. Curtain, R.F., Zwart, H.: An Introduction to Infinite Dimensional Linear Systems Theorem. Springer, New York (1995)

    Book  MATH  Google Scholar 

  10. Das, S., Pandey, D.N., Sukavanam, N.: Approximate controllability of a second-order neutral differential equation with state-dependent delay. Differ. Equ. Dyn. Syst. 24(2), 201–214 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. Denkowski, Z., Migórski, S., Papageorgiou, N.S.: An Introduction to Nonlinear Analysis. Kluwer Academic, Boston (2003)

    Book  MATH  Google Scholar 

  12. Dhage, B.C.: Multi-valued mappings and fixed points II. Tamkang J. Math. 37(1), 27–46 (2006)

    MathSciNet  MATH  Google Scholar 

  13. Haslinger, J., Panagiotopoulos, P.D.: Optimal control of systems governed by hemivariational inequalities. Existence and approximation results. Nonlinear Anal. 24(1), 105–119 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  14. Henríquez, H.R.: Existence of solutions of non-autonomous second order functional differential equations with infinite delay. Nonlinear Anal. Theory Methods Appl. 74, 3333–3352 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Henríquez, H.R., Pierri, M., Rolnik, V.: Pseudo \(S\)-asymptotically periodic solutions of second-order abstract Cauchy problems. Appl. Math. Comput. 274, 590–603 (2016)

    MathSciNet  Google Scholar 

  16. Hu, S., Papageorgiou, N.S.: Handbook of Multivalued Analysis. Kluwer Academic Publishers, Dordrecht (1997)

    Book  MATH  Google Scholar 

  17. Huang, Y., Liu, Z.H., Migórski, S.: Elliptic hemivariational inequalities with nonhomogeneous Neumann boundary conditions and their applications to static frictional contact problems. Acta Appl. Math. 138, 153–170 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kisyński, J.: On cosine operator functions and one parameter group of operators. Stud. Math. 49, 93–105 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kozak, M.: A fundamental solution of a second order differential equation in a Banach space. Univ. Iagell. Acta Math. 32, 275–289 (1995)

    MathSciNet  MATH  Google Scholar 

  20. Li, Y.X., Lu, L.: Existence and controllability for stochastic evolution inclusions of Clarke’s subdifferential type. Electron. J. Qual. Theory Differ. Equ. 59, 1–16 (2015)

    MathSciNet  MATH  Google Scholar 

  21. Li, X., Liu, Z.H., Migórski, S.: Approximate controllability for second order nonlinear evolution hemivariational inequalities. Electron. J. Qual. Theory Differ. Equ. 100, 1–16 (2015)

    MathSciNet  MATH  Google Scholar 

  22. Liu, Z.H., Li, X., Motreanu, D.: Approximate controllability for nonlinear evolution hemivariational inequalities in Hilbert spaces. SIAM J. Control Optim. 53(5), 3228–3244 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  23. Liu, Z.H., Li, X.: Approximate controllability for a class of hemivariational inequalities. Nonlinear Anal. Real World Appl. 22, 581–591 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  24. Liu, Z.H.: Existence results for quasilinear parabolic hemivariational inequalities. J. Differ. Equ. 244, 1395–1409 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Liu, Z.H., Zeng, B.: Existence and controllability for fractional evolution inclusions of Clarke’s subdifferential type. Appl. Math. Comput. 257, 178–189 (2015)

    MathSciNet  MATH  Google Scholar 

  26. Lu, L., Liu, Z.H.: Existence and controllability results for stochastic fractional evolution hemivariational inequalities. Appl. Math. Comput. 268, 1164–1176 (2015)

    MathSciNet  Google Scholar 

  27. Lu, L., Liu, Z.H., Bin, M.: Approximate controllability for stochastic evolution inclusions of Clarke’s subdifferential type. Appl. Math. Comput. 286, 201–212 (2016)

    MathSciNet  Google Scholar 

  28. Ma, T.W.: Topological degrees for set-valued compact vector fields in locally convex spaces. Diss. Math. 92, 1–43 (1972)

    Google Scholar 

  29. Migórski, S.: On existence of solutions for parabolic hemivariational inequalities. J. Comput. Appl. Math. 129, 77–87 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  30. Migórski, S., Ochal, A., Sofonea, M.: Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics, vol. 26. Springer, New York (2013)

    MATH  Google Scholar 

  31. Migórski, S., Ochal, A.: Quasi-static hemivariational inequality via vanishing acceleration approach. SIAM J. Math. Anal. 41, 1415–1435 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  32. Mahmudov, N.I., Denker, A.: On controllability of linear stochastic systems. Int. J. Control 73, 144–151 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  33. Mahmudov, N.I.: Existence and approximate controllability of Sobolev type fractional stochastic evolution equations. Bull. Pol. Acad. Sci. Tech. Sci. 62(2), 205–215 (2014)

    Google Scholar 

  34. Mahmudov, N.I., Vijayakumar, V., Murugesu, R.: Approximate controllability of second-order evolution differential inclusions in Hilbert spaces. Mediter. J. Math. 13(5), 3433–3454 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  35. Mahmudov, N.I., Murugesu, R., Ravichandran, C., Vijayakumar, V.: Approximate controllability results for fractional semilinear integro-differential inclusions in Hilbert spaces. Results Math. 71(1), 45–61 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  36. Naniewicz, Z., Panagiotopoulos, P.D.: Mathematical Theory of Hemivariational Inequalities and Applications. Marcel Dekker, New York (1995)

    MATH  Google Scholar 

  37. Oksendal, B.: Stochastic Differential Equations. An Introduction with Applications. Springer, Berlin (2000)

    MATH  Google Scholar 

  38. Panagiotopoulos, P.D.: Hemivariational Inequalities, Applications in Mechanics and Engineering. Springer, Berlin (1993)

    Book  MATH  Google Scholar 

  39. Panagiotopoulos, P.D.: Hemivariational inequality and fan-variational inequality, new applications and results. Atti del Seminario Matematico e Fisico dell’ Universita di Modena XLIII, 159–191 (1995)

    MATH  Google Scholar 

  40. Panagiotopoulos, P.D., Pop, G.: On a type of hyperbolic variational-hemivariational inequalities. J. Appl. Anal. 5(1), 95–112 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  41. Park, J.Y., Balasubramaniam, P., Kumaresan, N.: Controllability for neutral stochastic functional integrodifferential infinite delay systems in abstract space. Numer. Funct. Anal. Optim. 28, 1369–1386 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  42. Prato, G.D., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992)

    Book  MATH  Google Scholar 

  43. Rykaczewski, K.: Approximate controllability of differential inclusions in Hilbert spaces. Nonlinear Anal. 75, 2701–2712 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  44. Serizawa, H., Watanabe, M.: Time-dependent perturbation for cosine families in Banach spaces. Houst. J. Math. 12, 579–586 (1986)

    MathSciNet  MATH  Google Scholar 

  45. Shen, L.J., Shi, J.P., Sun, J.T.: Complete controllability of impulsive stochastic integro-differential systems. Automatica 46, 1068–1073 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  46. Shen, L.J., Sun, J.T.: Approximate controllability of stochastic impulsive functional systems with infinite delay. Automatica 48, 2705–2709 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  47. Sobczyk, K.: Stochastic Differential Equations with Applications to Physics and Engineering. Kluwer Academic Publishers, London (1991)

    MATH  Google Scholar 

  48. Travis, C.C., Webb, G.F.: Compactness, regularity, and uniform continuity properties of strongly continuous cosine families. Houst. J. Math. 3(4), 555–567 (1977)

    MathSciNet  MATH  Google Scholar 

  49. Travis, C.C., Webb, G.F.: Cosine families and abstract nonlinear second order differential equations. Acta Math. Acad. Sci. Hung. 32, 76–96 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  50. Vijayakumar, V.: Approximate controllability results for abstract neutral integro-differential inclusions with infinite delay in Hilbert spaces. IMA J. Math. Control Inf. (2016). https://doi.org/10.1093/imamci/dnw049

    Google Scholar 

  51. Vijayakumar, V.: Approximate controllability results for analytic resolvent integro-differential inclusions in Hilbert spaces. Int. J. Control (2017). https://doi.org/10.1080/00207179.2016.1276633

    Google Scholar 

  52. Vijayakumar, V.: Approximate controllability results for impulsive neutral differential inclusions of Sobolev-type with infinite delay. Int. J. Control (2017). https://doi.org/10.1080/00207179.2017.1346300

    Google Scholar 

  53. Vijayakumar, V., Murugesu, R., Poongodi, R., Dhanalakshmi, S.: Controllability of second order impulsive nonlocal Cauchy problem via measure of noncompactness. Mediter. J. Math. 14(1), 29–51 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  54. Vijayakumar, V., Murugesu, R., Tamil Selvan, M.: Controllability for a class of second order functional evolution differential equations without uniqueness. IMA J. Math. Control Inf. (2017). https://doi.org/10.1093/imamci/dnx048

    Google Scholar 

  55. Vijayakumar, V., Murugesu, R.: Controllability for a class of second-order evolution differential inclusions without compactness. Appl. Anal. (2018). https://doi.org/10.1080/00036811.2017.1422727

    Google Scholar 

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  1. Department of Mathematics, Info Institute of Engineering, Kovilpalayam, Coimbatore, Tamil Nadu, 641 107, India

    V. Vijayakumar

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Vijayakumar, V. Approximate Controllability for a Class of Second-Order Stochastic Evolution Inclusions of Clarke’s Subdifferential Type. Results Math 73, 42 (2018). https://doi.org/10.1007/s00025-018-0807-8

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