Approximate Controllability of Non-autonomous Evolution System with Nonlocal Conditions

  • Pengyu ChenEmail author
  • Xuping Zhang
  • Yongxiang Li


In this article, we are concerned with the existence of mild solutions as well as approximate controllability for a class of non-autonomous evolution system of parabolic type with nonlocal conditions in Banach spaces. Sufficient conditions of existence of mild solutions and approximate controllability for the desired problem are presented by introducing a new Green’s function and constructing a control function involving Gramian controllability operator. Some sufficient conditions of approximate controllability are formulated and proved here by using the resolvent operator condition. The discussions are based on Schauder’s fixed-point theorem as well as the theory of evolution family. An example is also given to illustrate the feasibility of our theoretical results.


Approximate controllability Non-autonomous evolution equation Nonlocal conditions Evolution family Resolvent operator 

Mathematics Subject Classification (2010)

34K30 34K35 93B05 


Funding Information

This research was financially supported by the National Natural Science Foundation of China (No. 11501455), the National Natural Science Foundation of China (No. 11661071), and the Key Project of Gansu Provincial National Science Foundation (No. 1606RJZA015).


  1. 1.
    Acquistapace P. Evolution operators and strong solution of abstract parabolic equations. Differential Integral Equations 1988;1:433–457.MathSciNetzbMATHGoogle Scholar
  2. 2.
    Acquistapace P, Terreni B. A unified approach to abstract linear parabolic equations. Rend Semin Mat Univ Padova 1987;78:47–107.MathSciNetzbMATHGoogle Scholar
  3. 3.
    Amann H. Parabolic evolution equations and nonlinear boundary conditions. J Differential Equations 1988;72:201–269.MathSciNetCrossRefGoogle Scholar
  4. 4.
    Boucherif A. Semilinear evolution inclutions with nonlocal conditions. Appl Math Lett 2009;22:1145–1149.MathSciNetCrossRefGoogle Scholar
  5. 5.
    Byszewski L. Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. J Math Anal Appl 1991;162:494–505.MathSciNetCrossRefGoogle Scholar
  6. 6.
    Byszewski L. Existence and uniqueness of a classical solutions to a functional-differential abstract nonlocal Cauchy problem. J Math Appl Stoch Anal 1999; 12:91–97.MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chang YK, Pereira A, Ponce R. Approximate controllability for fractional differential equations of Sobolev type via properties on resolvent operators. Fract Calc Appl Anal 2017;20:963–987.MathSciNetCrossRefGoogle Scholar
  8. 8.
    Chen P, Li Y. Monotone iterative technique for a class of semilinear evolution equations with nonlocal conditions. Results Math 2013;63:731–744.MathSciNetCrossRefGoogle Scholar
  9. 9.
    Chen P, Zhang X, Li Y. Study on fractional non-autonomous evolution equations with delay. Comput Math Appl 2017;73:794–803.MathSciNetCrossRefGoogle Scholar
  10. 10.
    Chen P, Zhang X, Li Y. 2017. Approximation technique for fractional evolution equations with nonlocal integral conditions. Mediterr J Math, Vol. 14. Art. 226.Google Scholar
  11. 11.
    Chen P, Zhang X, Li Y. A blowup alternative result for fractional nonautonomous evolution equation of Volterra type. Commun Pure Appl Anal 2018;17: 1975–1992.MathSciNetCrossRefGoogle Scholar
  12. 12.
    Deng K. Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions. J Math Anal Appl 1993;179:630–637.MathSciNetCrossRefGoogle Scholar
  13. 13.
    Ezzinbi K, Fu X, Hilal K. Existence and regularity in the α-norm for some neutral partial differential equations with nonlocal conditions. Nonlinear Anal 2007;67:1613–1622.MathSciNetCrossRefGoogle Scholar
  14. 14.
    Fan Z, Dong Q, Li G. Approximate controllability for semilinear composite fractional relaxation equations. Fract Calc Appl Anal 2016;19:267–284.MathSciNetCrossRefGoogle Scholar
  15. 15.
    Fitzgibbon WE. Semilinear functional equations in Banach space. J Differential Equations 1978;29:1–14.MathSciNetCrossRefGoogle Scholar
  16. 16.
    Fu X. Existence of solutions for non-autonomous functional evolution equations with nonlocal conditions. Electron J Differential Equations 2012;2012:1–15.MathSciNetGoogle Scholar
  17. 17.
    Fu X. Approximate controllability of semilinear non-autonomous evolution systems with state-dependent delay. Evol Equ Control Theory 2017;6:517–534.MathSciNetCrossRefGoogle Scholar
  18. 18.
    Fu X, Huang R. Approximate controllability of semilinear non-autonomous evolutionary systems with nonlocal conditions. Autom Remote Control 2016;77:428–442.MathSciNetCrossRefGoogle Scholar
  19. 19.
    Fu X, Zhang Y. Exact null controllability of non-autonomous functional evolution systems with nonlocal conditions. Acta Math Sci Ser B Engl Ed 2013;33:747–757.MathSciNetCrossRefGoogle Scholar
  20. 20.
    Henry D, Vol. 840. Geometric theory of semilinear parabolic equations lecture notes in math. New York: Springer; 1981.CrossRefGoogle Scholar
  21. 21.
    George RK. Approximate controllability of non-autonomous semilinear systems. Nonlinear Anal 1995;24:1377–1393.MathSciNetCrossRefGoogle Scholar
  22. 22.
    Kalman RE. Controllablity of linear dynamical systems. Contrib Diff Equ 1963; 1:190–213.Google Scholar
  23. 23.
    Liang J, Liu JH, Xiao TJ. Nonlocal Cauchy problems for nonautonomous evolution equations. Commun Pure Appl Anal 2006;5:529–535.MathSciNetCrossRefGoogle Scholar
  24. 24.
    Liang J, Yang H. Controllability of fractional integro-differential evolution equations with nonlocal conditions. Appl Math Comput 2015;254:20–29.MathSciNetzbMATHGoogle Scholar
  25. 25.
    Liu Z, Li X. Approximate controllability of fractional evolution systems with Riemann-Liouville fractional derivatives. SIAM J Control Optim 2015;53:1920–1933.MathSciNetCrossRefGoogle Scholar
  26. 26.
    Mahmudov NI. Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces. SIAM J Control Optim 2003;42: 1604–1622.MathSciNetCrossRefGoogle Scholar
  27. 27.
    Mahmudov NI. Approximate controllability of evolution systems with nonlocal conditions. Nonlinear Anal 2008;68:536–546.MathSciNetCrossRefGoogle Scholar
  28. 28.
    Pazy A. Semigroups of linear operators and applications to partial differential equations. Berlin: Springer; 1983.CrossRefGoogle Scholar
  29. 29.
    Prüss J. Evolutionary integral equations and applications. Birkhäuser: Basel; 1993.CrossRefGoogle Scholar
  30. 30.
    Sakthivela R, Ren Y, Debbouchec A, Mahmudovd NI. Approximate controllability of fractional stochastic differential inclusions with nonlocal conditions. Appl Anal 2016;95:2361–2382.MathSciNetCrossRefGoogle Scholar
  31. 31.
    Tanabe H. Functional analytic methods for partial differential equations. New York: Marcel Dekker; 1997.zbMATHGoogle Scholar
  32. 32.
    Wang RN, Ezzinbi K, Zhu PX. Non-autonomous impulsive Cauchy problems of parabolic type involving nonlocal initial conditions. J Integral Equations Appl 2014; 26:275–299.MathSciNetCrossRefGoogle Scholar
  33. 33.
    Wang RN, Zhu PX. Non-autonomous evolution inclusions with nonlocal history conditions: Global integral solutions. Nonlinear Anal 2013;85:180–191.MathSciNetCrossRefGoogle Scholar
  34. 34.
    Wang J, Fec̆kan M, Zhou Y. Approximate controllability of Sobolev type fractional evolution systems with nonlocal conditions. Evol Equ Control Theory 2017;6: 471–486.MathSciNetCrossRefGoogle Scholar
  35. 35.
    Xiao TJ, Liang J. Existence of classical solutions to nonautonomous nonlocal parabolic problems. Nonlinear Anal 2005;63:225–232.MathSciNetCrossRefGoogle Scholar
  36. 36.
    Zhou HX. Approximate controllability for a class of semilinear abstract equations. SIAM J Control Optim 1983;21:551–565.MathSciNetCrossRefGoogle Scholar
  37. 37.
    Zhu B, Liu L, Wu Y. 2016. Existence and uniqueness of global mild solutions for a class of nonlinear fractional reactionCdiffusion equations with delay. Comput Math Appl.
  38. 38.
    Zhu B, Liu L, Wu Y. Local and global existence of mild solutions for a class of nonlinear fractional reaction-diffusion equations with delay. Appl Math Lett 2016;61: 73–79.MathSciNetCrossRefGoogle Scholar
  39. 39.
    Zhu B, Liu L, Wu Y. Local and global existence of mild solutions for a class of semilinear fractional integro-differential equations. Fract Calc Appl Anal 2017; 20:1338–1355.MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsNorthwest Normal UniversityLanzhouPeople’s Republic of China

Personalised recommendations