Controllability of nonlinear fractional evolution systems in Banach spaces: A survey

  • Received: 01 June 2021 Revised: 01 August 2021 Published: 11 October 2021
  • Primary: 93B05, 47D06, 34K30; Secondary: 35R11

  • This paper presents a survey for some recent research on the controllability of nonlinear fractional evolution systems (FESs) in Banach spaces. The prime focus is exact controllability and approximate controllability of several types of FESs, which include the basic systems with classical initial and nonlocal conditions, FESs with time delay or impulsive effect. In addition, controllability results via resolvent operator are reviewed in detail. At last, the conclusions of this work and the research prospect are presented, which provides a reference for further study.

    Citation: Daliang Zhao, Yansheng Liu. Controllability of nonlinear fractional evolution systems in Banach spaces: A survey[J]. Electronic Research Archive, 2021, 29(5): 3551-3580. doi: 10.3934/era.2021083

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  • This paper presents a survey for some recent research on the controllability of nonlinear fractional evolution systems (FESs) in Banach spaces. The prime focus is exact controllability and approximate controllability of several types of FESs, which include the basic systems with classical initial and nonlocal conditions, FESs with time delay or impulsive effect. In addition, controllability results via resolvent operator are reviewed in detail. At last, the conclusions of this work and the research prospect are presented, which provides a reference for further study.



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    [1] A hybrid parametrization approach for a class of nonlinear optimal control problems. Numer. Algebra Control Optim. (2019) 9: 493-506.
    [2] Periodic solutions to functional-differential equations. Proc. Roy. Soc. Edinburgh Sect. A (1985) 101: 253-271.
    [3] Stabilization on input time-varying delay for linear switched systems with truncated predictor control. Numer. Algebra Control Optim. (2020) 10: 237-247.
    [4] Controllability of nonlinear systems via fixed-point theorems. J. Optim. Theory Appl. (1987) 53: 345-352.
    [5] Initial guess sensitivity in computational optimal control problems. Numer. Algebra Control Optim. (2020) 10: 39-43.
    [6] Local density of Caputo-stationary functions in the space of smooth functions. ESAIM Control Optim. Calc. Var. (2017) 23: 1361-1380.
    [7] Approximate controllability for fractional differential equations of Sobolev type via properties on resolvent operators. Fract. Calc. Appl. Anal. (2017) 20: 963-987.
    [8] J. Chen, X. Li and D. Wang, Asymptotic stability and exponential stability of impulsive delayed Hopfield neural networks, Abstr. Appl. Anal., 2013 (2013), 10pp. doi: 10.1155/2013/638496
    [9] Bearing rigidity and formation stabilization for multiple rigid bodies in $SE(3)$. Numer. Algebra Control Optim. (2019) 9: 257-267.
    [10] Stability of traveling wave fronts for nonlocal diffusion equation with delayed nonlocal response. Taiwanese J. Math. (2016) 20: 801-822.
    [11] The stability of the equilibria of the Allen-Cahn equation with fractional diffusion. Appl. Anal. (2019) 98: 600-610.
    [12] Optimal control of an HIV model with CTL cells and latently infected cells. Numer. Algebra Control Optim. (2020) 10: 207-225.
    [13] Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems. Comput. Math. Appl. (2011) 62: 1442-1450.
    [14] Stochastic stability and stabilization of $n$-person random evolutionary Boolean games. Appl. Math. Comput. (2017) 306: 1-12.
    [15] J. Du, W. Jiang, D. Pang and A. U. K. Niazi, Controllability for a new class of fractional neutral integro-differential evolution equations with infinite delay and nonlocal conditions, Adv. Difference Equ., 2017 (2017), 22pp. doi: 10.1186/s13662-017-1182-6
    [16] Some probability densities and fundamental solutions of fractional evolution equations. Chaos Solitons Fractals (2002) 14: 433-440.
    [17] Approximate controllability of semilinear evolution equations of fractional order with nonlocal and impulsive conditions via an approximating technique. Appl. Math. Comput. (2016) 275: 107-120.
    [18] Numerical solution of bilateral obstacle optimal control problem, where the controls and the obstacles coincide. Numer. Algebra Control Optim. (2020) 10: 275-300.
    [19] Controllability of Volterra-Fredholm type systems in Banach spaces. J. Franklin. Inst. (2009) 346: 95-101.
    [20] On recent developments in the theory of abstract differential equations with fractional derivatives. Nonlinear Anal. (2010) 73: 3462-3471.
    [21] Y. Hino, S. Murakami and T. Naito, Functional-Differential Equations with Infinite Delay, Lecture Notes in Mathematics, 1473, Springer-Verlag, Berlin, 1991. doi: 10.1007/BFb0084432
    [22] Approximate controllability for semilinear retarded systems. J. Math. Anal. Appl. (2006) 321: 961-975.
    [23] Approximate controllability of semilinear nonlocal fractional differential systems via an approximating method. Appl. Math. Comput. (2014) 236: 43-53.
    [24] A fast finite volume method for conservative space-time fractional diffusion equations discretized on space-time locally refined meshes. Comput. Math. Appl. (2019) 78: 1345-1356.
    [25] A preconditioned fast finite element approximation to variable-order time-fractional diffusion equations in multiple space dimensions. Appl. Numer. Math. (2021) 163: 15-29.
    [26] The controllability of nonlinear implicit fractional delay dynamical systems. Int. J. Appl. Math. Comput. Sci. (2017) 27: 501-513.
    [27] Controllability of nonlinear fractional delay dynamical systems. Rep. Math. Phys. (2016) 77: 87-104.
    [28] K. Kavitha, V. Vijayakumar and R. Udhayakumar, Results on controllability of Hilfer fractional neutral differential equations with infinite delay via measures of noncompactness, Chaos Solitons Fractals, 139 (2020), 12pp. doi: 10.1016/j.chaos.2020.110035
    [29] Linear optimal control of time delay systems via Hermite wavelet. Numer. Algebra Control Optim. (2020) 10: 143-156.
    [30] Solving optimal control problem using Hermite wavelet. Numer. Algebra Control Optim. (2019) 9: 101-112.
    [31] Controllability results for non densely defined impulsive fractional differential equations in abstract space. Differ. Equ. Dyn. Syst. (2021) 29: 227-237.
    [32] Approximate controllability of fractional order semilinear systems with bounded delay. J. Differential Equations (2012) 252: 6163-6174.
    [33] A control Lyapunov function approach to feedback stabilization of logical control networks. SIAM J. Control Optim. (2019) 57: 810-831.
    [34] Controllability analysis and control design for switched Boolean networks with state and input constraints. SIAM J. Control Optim. (2015) 53: 2955-2979.
    [35] Numerical solution of the time-fractional sub-diffusion equation on an unbounded domain in two-dimensional space. East Asian J. Appl. Math. (2017) 7: 439-454.
    [36] H. Li and Y. Wu, Artificial boundary conditions for nonlinear time fractional Burgers' equation on unbounded domains, Appl. Math. Lett., 120 (2021), 8pp. doi: 10.1016/j.aml.2021.107277
    [37] Finite-time stability analysis of stochastic switched Boolean networks with impulsive effect. Appl. Math. Comput. (2019) 347: 557-565.
    [38] Algebraic formulation and topological structure of Boolean networks with state-dependent delay. J. Comput. Appl. Math. (2019) 350: 87-97.
    [39] Controllability of nonlocal fractional differential systems of order $\alpha\in (1, 2]$ in Banach spaces. Rep. Math. Phys. (2013) 71: 33-43.
    [40] L. Li, Z. Jiang and Z. Yin, Compact finite-difference method for 2D time-fractional convection-diffusion equation of groundwater pollution problems, Comput. Appl. Math., 39 (2020), 34pp. doi: 10.1007/s40314-020-01169-9
    [41] P. Li, X. Li and J. Cao, Input-to-state stability of nonlinear switched systems via Lyapunov method involving indefinite derivative, Complexity, 2018 (2018). doi: 10.1155/2018/8701219
    [42] Further analysis on uniform stability of impulsive infinite delay differential equations. Appl. Math. Lett. (2012) 25: 133-137.
    [43] Function perturbation impact on stability in distribution of probabilistic Boolean networks. Math. Comput. Simulation (2020) 177: 1-12.
    [44] X. Li, Z. Liu and C. C. Tisdell, Approximate controllability of fractional control systems with time delay using the sequence method, Electron. J. Differential Equations, 2017 (2017), 11pp.
    [45] Global exponential stabilization of impulsive neural networks with unbounded continuously distributed delays. IMA J. Appl. Math. (2015) 80: 85-99.
    [46] LMI-based stability for singularly perturbed nonlinear impulsive differential systems with delays of small parameter. Appl. Math. Comput. (2015) 250: 798-804.
    [47] Sliding mode control for uncertain T-S fuzzy systems with input and state delays. Numer. Algebra Control Optim. (2020) 10: 345-354.
    [48] Controllability of fractional integro-differential evolution equations with nonlocal conditions. Appl. Math. Comput. (2015) 254: 20-29.
    [49] Structural stability analysis of gene regulatory networks modeled by Boolean networks. Math. Methods Appl. Sci. (2019) 42: 2221-2230.
    [50] L. Lin, Y. Liu and D. Zhao, Study on implicit-type fractional coupled system with integral boundary conditions, Mathematics, 9 (2021). doi: 10.3390/math9040300
    [51] B. Liu and Y. Liu, Positive solutions of a two-point boundary value problem for singular fractional differential equations in Banach space, J. Funct. Spaces Appl., 2013 (2013), 9pp. doi: 10.1155/2013/585639
    [52] Fault estimation and optimization for uncertain disturbed singularly perturbed systems with time-delay. Numer. Algebra Control Optim. (2020) 10: 367-379.
    [53] M. Liu, S. Li, X. Li, L. Jin, C. Yi and Z. Huang, Intelligent controllers for multirobot competitive and dynamic tracking, Complexity, 2018 (2018). doi: 10.1155/2018/4573631
    [54] Approximate controllability of impulsive fractional neutral evolution equations with Riemann-Liouville fractional derivatives. J. Comput. Anal. Appl. (2014) 17: 468-485.
    [55] Bifurcation techniques for a class of boundary value problems of fractional impulsive differential equations. J. Nonlinear Sci. Appl. (2015) 8: 340-353.
    [56] Y. Liu, Positive solutions using bifurcation techniques for boundary value problems of fractional differential equations, Abstr. Appl. Anal., 2013 (2013), 7pp. doi: 10.1155/2013/162418
    [57] Y. Liu and D. O'Regan, Controllability of impulsive functional differential systems with nonlocal conditions, Electron. J. Differential Equations, 2013 (2013), 10pp.
    [58] Y. Liu and H. Yu, Bifurcation of positive solutions for a class of boundary value problems of fractional differential inclusions, Abstr. Appl. Anal., 2013 (2013), 8pp. doi: 10.1155/2013/942831
    [59] Control design for output tracking of delayed Boolean control networks. J. Comput. Appl. Math. (2018) 327: 188-195.
    [60] Approximate controllability of impulsive Riemann-Liouville fractional equations in Banach spaces. J. Integral Equations Appl. (2014) 26: 527-551.
    [61] Controllability of nonlinear fractional impulsive evolution systems. J. Integral Equations Appl. (2013) 25: 395-406.
    [62] X. Lv, X. Li, J. Cao and P. Duan, Exponential synchronization of neural networks via feedback control in complex environment, Complexity, 2018 (2018). doi: 10.1155/2018/4352714
    [63] The multiplicity solutions for nonlinear fractional differential equations of Riemann-Liouville type. Fract. Calc. Appl. Anal. (2018) 21: 801-818.
    [64] Partial-approximate controllability of nonlocal fractional evolution equations via approximating method. Appl. Math. Comput. (2018) 334: 227-238.
    [65] J. Mao and D. Zhao, Multiple positive solutions for nonlinear fractional differential equations with integral boundary value conditions and a parameter, J. Funct. Spaces, 2019 (2019), 11pp. doi: 10.1155/2019/2787569
    [66] Bifurcation analysis of a singular nutrient-plankton-fish model with taxation, protected zone and multiple delays. Numer. Algebra Control Optim. (2020) 10: 391-423.
    [67] Existence and uniqueness of mild solution to impulsive fractional differential equations. Nonlinear Anal. (2010) 72: 1604-1615.
    [68] Existence of solutions and approximate controllability of fractional nonlocal neutral impulsive stochastic differential equations of order $1<q<2$ with infinite delay and Poisson jumps. J. Dyn. Control Syst. (2017) 23: 213-235.
    [69] Controllability of semilinear control systems dominated by the linear part. SIAM J. Control Optim. (1987) 25: 715-722.
    [70] Stability analysis of stagnation point flow in nanofluid over stretching/shrinking sheet with slip effect using Buongiorno's model. Numer. Algebra Control Optim. (2019) 9: 423-431.
    [71] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1
    [72] D. Peng, X. Li, R. Rakkiyappan and Y. Ding, Stabilization of stochastic delayed systems: Event-triggered impulsive control, Appl. Math. Comput., 401 (2021), 12pp. doi: 10.1016/j.amc.2021.126054
    [73] (1999) Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Inc., San Diego, CA: Mathematics in Science and Engineering, 198, Academic Press.
    [74] J. Prüss, Evolutionary Integral Equations and Applications, Monographs in Mathematics, 87, Birkhäuser Verlag, Basel, 1993. doi: 10.1007/978-3-0348-8570-6
    [75] T. Qi, Y. Liu and Y. Cui, Existence of solutions for a class of coupled fractional differential systems with nonlocal boundary conditions, J. Funct. Spaces, 2017 (2017), 9pp. doi: 10.1155/2017/6703860
    [76] Existence result for a class of coupled fractional differential systems with integral boundary value conditions. J. Nonlinear Sci. Appl. (2017) 10: 4034-4045.
    [77] Local smooth representation of solution sets in parametric linear fractional programming problems. Numer. Algebra Control Optim. (2019) 9: 45-52.
    [78] Passive control for a class of nonlinear systems by using the technique of adding a power integrator. Numer. Algebra Control Optim. (2020) 10: 381-389.
    [79] H. Qin, Z. Gu, Y. Fu and T. Li, Existence of mild solutions and controllability of fractional impulsive integrodifferential systems with nonlocal conditions, J. Funct. Spaces, 2017 (2017), 11pp. doi: 10.1155/2017/6979571
    [80] H. Qin, J. Liu and X. Zuo, Controllability problem for fractional integrodifferential evolution systems of mixed type with the measure of noncompactness, J. Inequal. Appl., 2014 (2014), 15pp. doi: 10.1186/1029-242X-2014-292
    [81] H. Qin, X. Zuo and J. Liu, Existence and controllability results for fractional impulsive integrodifferential systems in Banach spaces, Abstr. Appl. Anal., 2013 (2013), 12pp. doi: 10.1155/2013/295837
    [82] M. M. Raja, V. Vijayakumar, R. Udhayakumar and Y. Zhou, A new approach on the approximate controllability of fractional differential evolution equations of order $1<r<2$ in Hilbert spaces, Chaos Solitons Fractals, 141 (2020), 10pp. doi: 10.1016/j.chaos.2020.110310
    [83] Approximate controllability of nonlinear fractional dynamical systems. Commun. Nonlinear Sci. Numer. Simul. (2013) 18: 3498-3508.
    [84] On the approximate controllability of semilinear fractional differential systems. Comput. Math. Appl. (2011) 62: 1451-1459.
    [85] Approximate controllability of fractional stochastic evolution equations. Comput. Math. Appl. (2012) 63: 660-668.
    [86] T. Sathiyaraj, M. Fečkan and J. Wang, Null controllability results for stochastic delay systems with delayed perturbation of matrices, Chaos Solitons Fractals, 138 (2020), 11pp. doi: 10.1016/j.chaos.2020.109927
    [87] A study on the mild solution of impulsive fractional evolution equations. Appl. Math. Comput. (2016) 273: 465-476.
    [88] Controllability results for fractional semilinear delay control systems. J. Appl. Math. Comput. (2021) 65: 861-875.
    [89] Approximate controllability of semilinear fractional control systems of order $\alpha\in(1, 2]$ with infinite delay. Mediterr. J. Math. (2016) 13: 2539-2550.
    [90] Approximate controllability of semilinear system with state delay using sequence method. J. Franklin Inst. (2015) 352: 5380-5392.
    [91] N. Sukavanam, Approximate controllability of semilinear control system with growing nonlinearity, in Mathematical Theory of Control, Lecture Notes in Pure and Appl. Math., 142, Dekker, New York, 1993,353–357.
    [92] Quasilinear iterative method for the boundary value problem of nonlinear fractional differential equation. Numer. Algebra Control Optim. (2020) 10: 157-164.
    [93] Identification of Hessian matrix in distributed gradient-based multi-agent coordination control systems. Numer. Algebra Control Optim. (2019) 9: 297-318.
    [94] I. Tarnove, A controllability problem for nonlinear systems, in Mathematical Theory of Control, Academic Press, New York, 1967, 170-179.
    [95] Y. Tian, Some results on the eigenvalue problem for a fractional elliptic equation, Bound. Value. Probl., 2019 (2019), 11pp. doi: 10.1186/s13661-019-1127-y
    [96] Y. Tian and S. Zhao, Existence of solutions for perturbed fractional equations with two competing weighted nonlinear terms, Bound. Value. Probl., 2018 (2018), 16pp. doi: 10.1186/s13661-018-1074-z
    [97] Cosine families and abstract nonlinear second order differential equations. Acta Math. Acad. Sci. Hungar. (1978) 32: 75-96.
    [98] On the lack of exact controllability for mild solutions in Banach spaces. J. Math. Anal. Appl. (1975) 50: 438-446.
    [99] F. Wang, Z. Zhang and Z. Zhou, A spectral Galerkin approximation of optimal control problem governed by fractional advection-diffusion-reaction equations, J. Comput. Appl. Math., 386 (2021), 16pp. doi: 10.1016/j.cam.2020.113233
    [100] Nonlocal controllability of semilinear dynamic systems with fractional derivative in Banach spaces. J. Optim. Theory Appl. (2012) 154: 292-302.
    [101] Complete controllability of fractional evolution systems. Commun. Nonlinear Sci. Numer. Simul. (2012) 17: 4346-4355.
    [102] Y. Wang, Y. Liu and Y. Cui, Infinitely many solutions for impulsive fractional boundary value problem with $p$-Laplacian, Bound. Value Probl., 2018 (2018), 16pp. doi: 10.1186/s13661-018-1012-0
    [103] Y. Wang, Y. Liu and Y. Cui, Multiple sign-changing solutions for nonlinear fractional Kirchhoff equations, Bound. Value Probl., 2018 (2018), 21pp. doi: 10.1186/s13661-018-1114-8
    [104] Y. Wang, Y. Liu and Y. Cui, Multiple solutions for a nonlinear fractional boundary value problem via critical point theory, J. Funct. Spaces, 2017 (2017), 8pp. doi: 10.1155/2017/8548975
    [105] Unified vector quasiequilibrium problems via improvement sets and nonlinear scalarization with stability analysis. Numer. Algebra Control Optim. (2020) 10: 107-125.
    [106] Input-to-state stability of delayed reaction-diffusion neural networks with multiple impulses. AIMS Math. (2021) 6: 5786-5800.
    [107] Output tracking control of Boolean control networks with impulsive effects. Math. Methods Appl. Sci. (2018) 41: 1554-1564.
    [108] Robust set stabilization of Boolean control networks with impulsive effects. Nonlinear Anal. Model. Control (2018) 23: 553-567.
    [109] Output tracking control of delayed switched systems via state-dependent switching and dynamic output feedback. Nonlinear Anal. Hybrid Syst. (2019) 32: 294-305.
    [110] State-dependent switching control of delayed switched systems with stable and unstable modes. Math. Methods Appl. Sci. (2018) 41: 6968-6983.
    [111] H. Yang and Y. Zhao, Controllability of fractional evolution systems of Sobolev type via resolvent operators, Bound. Value Probl., 2020 (2020), 13pp. doi: 10.1186/s13661-020-01417-1
    [112] Q. Yang, H. Li and Y. Liu, Pinning control design for feedback stabilization of constrained Boolean control networks, Adv. Difference Equ., 2016 (2016), 16pp. doi: 10.1186/s13662-016-0909-0
    [113] Review of stability and stabilization for impulsive delayed systems. Math. Biosci. Eng. (2018) 15: 1495-1515.
    [114] On fractional quadratic optimization problem with two quadratic constraints. Numer. Algebra Control Optim. (2020) 10: 301-315.
    [115] D. Zhang and Y. Liang, Existence and controllability of fractional evolution equation with sectorial operator and impulse, Adv. Difference Equ., 2018 (2018), 12pp. doi: 10.1186/s13662-018-1664-1
    [116] Spectral Galerkin approximation of optimal control problem governed by Riesz fractional differential equation. Appl. Numer. Math. (2019) 143: 247-262.
    [117] X. Zhang, C. Zhu and C. Yuan, Approximate controllability of impulsive fractional stochastic differential equations with state-dependent delay, Adv. Difference Equ., 2015 (2015), 12pp. doi: 10.1186/s13662-015-0412-z
    [118] New results on controllability for a class of fractional integrodifferential dynamical systems with delay in Banach spaces. Fractal Fract. (2021) 5: 1-18.
    [119] D. Zhao and Y. Liu, Eigenvalues of a class of singular boundary value problems of impulsive differential equations in Banach spaces, J. Funct. Spaces, 2014 (2014), 12pp. doi: 10.1155/2014/720494
    [120] D. Zhao and Y. Liu, Multiple positive solutions for nonlinear fractional boundary value problems, The Scientific World Journal, 2013 (2013). doi: 10.1155/2013/473828
    [121] Positive solutions for a class of fractional differential coupled system with integral boundary value conditions. J. Nonlinear Sci. Appl. (2016) 9: 2922-2942.
    [122] Twin solutions to semipositone boundary value problems for fractional differential equations with coupled integral boundary conditions. J. Nonlinear Sci. Appl. (2017) 10: 3544-3565.
    [123] Controllability for a class of semilinear fractional evolution systems via resolvent operators. Commun. Pur. Appl. Anal. (2019) 18: 455-478.
    [124] D. Zhao and J. Mao, New controllability results of fractional nonlocal semilinear evolution systems with finite delay, Complexity, 2020 (2020). doi: 10.1155/2020/7652648
    [125] Positive solutions for a class of nonlinear singular fractional differential systems with Riemann-Stieltjes coupled integral boundary value conditions. Symmetry (2021) 13: 1-19.
    [126] Stability analysis of activation-inhibition Boolean networks with stochastic function structures. Math. Methods Appl. Sci. (2020) 43: 8694-8705.
    [127] Y. Zhao, X. Li and J. Cao, Global exponential stability for impulsive systems with infinite distributed delay based on flexible impulse frequency, Appl. Math. Comput., 386 (2020), 10pp. doi: 10.1016/j.amc.2020.125467
    [128] Approximate controllability for a class of semilinear abstract equations. SIAM J. Control Optim. (1983) 22: 551-565.
    [129] New results on controllability of fractional evolution systems with order $\alpha\in (1, 2)$. Evol. Equ. Control Theory (2021) 10: 491-509.
    [130] Finite element approximation of optimal control problems governed by time fractional diffusion equation. Comput. Math. Appl. (2016) 71: 301-318.
    [131] A survey of numerical methods for convection-diffusion optimal control problems. J. Numer. Math. (2014) 22: 61-85.
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