## Topological Methods in Nonlinear Analysis

### Approximate controllability for abstract semilinear impulsive functional differential inclusions based on Hausdorff product measures

#### Abstract

A second order semilinear impulsive functional differential inclusion in a separable Hilbert space is considered. Without imposing hypotheses of the compactness on the cosine families of operators, some sufficient conditions of approximate controllability are formulated in the case where the multivalued nonlinearity of the inclusion is a completely continuous map dominated by a function. By the use of resolvents of controllability Gramian operators and developing appropriate computing techniques for the Hausdorff product measures of noncompactness, the results of approximate controllability for position and velocity are derived. An example is also given to illustrate the application of the obtained results.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 52, Number 2 (2018), 353-372.

Dates
First available in Project Euclid: 25 November 2018

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1543114845

Digital Object Identifier
doi:10.12775/TMNA.2018.030

Mathematical Reviews number (MathSciNet)
MR3915644

Zentralblatt MATH identifier
07051673

#### Citation

Xiao, Jian-Zhong; Zhu, Xing-Hua. Approximate controllability for abstract semilinear impulsive functional differential inclusions based on Hausdorff product measures. Topol. Methods Nonlinear Anal. 52 (2018), no. 2, 353--372. doi:10.12775/TMNA.2018.030. https://projecteuclid.org/euclid.tmna/1543114845

#### References

• N. Abada, M. Benchohra and H. Hammouche, Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions, J. Differential Equations 246 (2009), 3834–3863.
• U. Arora and N. Sukavanam, Approximate controllability of second order semilinear stochastic system with nonlocal conditions, Appl. Math. Comput. 258 (2015), 111–119.
• J.P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, 1990.
• K. Balachandran and J.H. Kim, Remarks on the paper “Controllability of second order differential inclusion in Banach spaces" [J. Math. Anal. Appl. 285 (2003), 537–550], J. Math. Anal. Appl. 324 (2006), 746–749.
• M. Benchohra, J.J. Nieto and A. Ouahab, Impulsive differential inclusions involving evolution operators in separable Banach spaces, Ukrainian Math. J. 64 (2012), no. 7, 991–1018.
• I. Benedetti, V. Obukhovskiĭ, and P. Zecca, Controllability for impulsive semilinear functional differential inclusions with a non-compact evolution operator, Discuss. Math. Differ. Incl. Control Optim. 31 (2011), 39–69.
• S. Das, D.N. Pandey and N. Sukavanam, Approximate controllability of a second order neutral differential equation with state dependent delay, Differ. Equ. Dyn. Syst. 24 (2016), 201–214.
• J.P. Dauer and N.I. Mahmudov, Approximate controllability of semilinear functional equations in Hilbert spaces, J. Math. Anal. Appl. 273 (2002), 310–327.
• J.P. Dauer, N.I. Mahmudov and M.M. Matar, Approximate controllability of backward stochastic evolution equations in Hilbert spaces, J. Math. Anal. Appl. 323 (2006), no. 1, 42–56.
• S. Djebali, L. Górniewicz and A. Ouahab, First-order periodic impulsive semilinear differential inclusions: Existence and structure of solution sets, Math. Comput. Model. 52 (2010), 683–714.
• K. Ezzinbi and S.L. Rhali, Existence and controllability for nondensely defined partial neutral functional differential inclusions, Appl. Math. 60 (2015), 321–340.
• H.O. Fattorini, Second Order Linear Differential Equations in Banach Spaces, North-Holland, Amsterdam, 1985.
• A. Grudzka and K. Rykaczewski, On approximate controllability of functional impulsive evolution inclusions in a Hilbert space, J. Optim. Theory Appl. 166 (2015), 414–439.
• L. Guedda, Some remarks in the study of impulsive differential equations and inclusions with delay, Fixed Point Theory 12 (2011), no. 2, 349–354.
• H.R. Henríquez and E. Hernández M, Approximate controllability of second-order distributed implicit functional systems, Nonlinear Anal. 70 (2009), 1023–1039.
• M. Kamenskiĭ, V. Obukhovskiĭ and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, de Gruyter Series in Nonlinear Analysis and Applications Vol. 7, Walter de Gruyter, Berlin, New York, 2001.
• J. Kisyński, On cosine operator functions and one parameter group of operators, Studia Math. 49 (1972), 93–105.
• J. Klamka, Controllability of dynamical systems –- a survey, Arch. Contol. Sci. 2 (1993), no. 3–4, 281–307.
• J. Klamka, Controllability of second order infinite-dimensional systems, IMA J. Math. Control Inform. 13 (1996), no. 1, 79–88.
• J. Klamka, Constrained approximate controllability, IEEE Trans. Automat. Control 45 (2000), no. 9, 1745–1749.
• J. Klamka, Constrained controllability of semilinear systems, Nonlinear Anal. 47 (2001), no. 5, 2939–2949.
• J. Klamka, Controllability of dynamical systems. A survey, Bull. Pol. Acad. Sci. Tech. Sci. 61 (2013), 335–241.
• J. Klamka and J. Wyrwal, Controllability of second order infinite-dimensional systems, Systems Control Lett. 57 (2008), 386–391.
• H. Leiva, Rothe's fixed point theorem and controllability of semilinear nonautonomous systems, Systems Control Lett. 67 (2014), 14–18.
• M. Martelli, A Rothe's type theorem for non-compact acyclic-valued map, Boll. Unione Mat. Ital. 4 (1975), 70–76.
• N.I. Mahmudov, Controllability of linear stochastic systems in Hilbert spaces, J. Math. Anal. Appl. 259 (2001), 64–82.
• N.I. Mahmudov, Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces, SIAM J. Control Optim. 42 (2003), no. 5, 1604–1622.
• F.Z. Mokkedem and X. Fu, Approximate controllability for a semilinear evolution system with infinite delay, J. Dyn. Control Syst. 22 (2016), 71–79.
• V. Obukhovskiĭ and P. Zecca, Controllability for systems governed by semilinear differential inclusions in a Banach space with a noncompact semigroup, Nonlinear Anal. 70 (2009), 3424–3436.
• S. Park, Generalized Leray–Schauder principles for condensing admissible multifunctions, Ann. Mat. Pura Appl. (IV) 172 (1997), 65–85.
• K. Rykaczewski, Approximate controllability of differential inclusions in Hilbert spaces, Nonlinear Anal. 75 (2012), 2701–2702.
• R. Sakthivel, Approximate controllability of impulsive stochastic evolution equations, Funkcial. Ekvac. 52 (2009), 381–393.
• R. Sakthivel, N.I. Mahmudov and J.H. Kim, Approximate controllability of nonlinear impulsive differential systems, Rep. Math. Phys. 60 (2007),no. 1, 85–96.
• E. Schüler, On the spectrum of cosine functions, J. Math. Anal. Appl. 229 (1999), 376–398.
• C.C. Travis and G.F. Webb, Compactness, regularity and uniform continuity properties of strongly continuous cosine families, Houston J. Math. 3 (1977), 555–567.
• C.C. Travis and G.F. Webb, Cosine families and abstract nonlinear second order differential equations, Acta Math. Acad. Sci. Hungaricae 32 (1978), 75–96.
• R. Triggiani, A note on the lack of exact controllability for mild solutions in Banach spaces, SIAM J. Control Optim. 15 (1977), 407–411.
• X. Wan and J. Sun, Approximate controllability for abstract measure differential systems, Systems Control Lett. 61 (2012), 50–54.
• J.Z. Xiao, Y.H. Cang and Q.F. Liu, Existence of solutions for a class of boundary value problems of semilinear differential inclusions, Math. Comput. Modelling 57 (2013), 671–683.
• J.Z. Xiao, Z.Y. Wang and J. Liu, Hausdorff product measures and $C^1$-solution sets of abstract semilinear functional differential inclusions, Topol. Methods Nonlinear Anal. 49 (2017), no. 1, 273–298.
• J.Z. Xiao, X.H. Zhu and Z.H. Yao, Controllability results for second order semilinear functional differential inclusions based on Kuratowski product measures, IMA J. Math. Control Inform. 35 (2018), i31–i50.