Controllability analysis of damped elastic systems using sequence method
DOI:
https://doi.org/10.12775/TMNA.2025.015Keywords
Approximate controllability, damped elastic systems, mild solutions, sequence methodAbstract
In this paper, we are focused on the existence of mild solutions and approximate controllability for a class of Volterra-Fredholm type integro-differential damped elastic systems in Banach spaces. Firstly, the existence of mild solution is obtained by means of Banach fixed point theorem as well as operator semigroup theory. Secondly, the sufficient conditions of approximate controllability for our desired problem are formulated and proved, in order to test the approximate controllability results, the sequence method is used without assuming that the corresponding linear system is approximately controllable. Finally, an example is given to illustrate the theory.References
H. Amann, Periodic solutions of semilinear parabolic equations, Nonlinear Anal., A Collection of Papers in Honor of Erich H. Rothe (L. Cesari, R. Kannan, R. Weinberger, eds.), New York, Academic Press, 1978, pp. 1–29.
L. Byszewski, Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J. Math. Anal. Appl. 162 (1991), no. 2, 497–505.
Y. Cao and J. Sun, Approximate controllability of semilinear measure driven systems, Math. Nachr. 291(2018), 1979–1988.
G. Chen and D.L. Russell, A mathematical model for linear elastic systems with structural damping, Quart. Appl. Math. 39(1982), 433-454.
P. Chen, X. Zhang and Y. Li, Existence and approximate controllability of fractional evolution equations with nonlocal conditions via resolvent operators, Fract. Calc. Appl. Anal. 23 (2020), no. 1, 268–291.
S. Chen and R. Triggiani, Proof of extensions of two conjectures on structural damping for elastic systems: the systems: the case 1/2 ≤ α ≤ 1, Pacific J. Math. 136 (1989), no. 1, 15–55.
S. Chen and R. Triggiani, Gevrey class semigroups arising from elastic systems with gentle dissipation: the case 0 < α < 1/2, Proc. Amer. Math. Soc. 110 (1990), no. 2, 401–415.
K. Deng, Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions, J. Math. Anal. Appl. 179 (1993), no. 2, 630–637.
T. Diagana, Semilinear Evolution Eqautions and Their Applications, Springer Nature Switzerland AG., 2018.
T. Diagana, Well-posedness for some damped elastic systems in Banach spaces, Appl. Math. Lett. 71 (2017), 74–80.
H. Fan and F. Gao, Asymptotic stability of solutions to elastis systems with structural damping, Electron. J. Differential Equations 245 (2014), 1–9.
H. Fan and Y. Li, Monotone iterative technique for the elastic systems with structural damping in Banach spaces, Comput. Math. Appl. 68 (2014), 384–391.
H. Fan and Y. Li, Analyticity and exponential stability of semigroups for the elastic systems with structural damping in Banach spaces, J. Math. Anal. Appl. 410 (2014), 316–322.
H. Fan, Y. Li and P. Chen, Existence of mild solutions for the elastic systems with structural damping in Banach spaces, Abstract and Applied Analysis (2013), Article ID 746893, 1–6.
X. Fu, Approximate controllability of semilinear non-autonomous evolution systems with state-dependent delay, Evol. Equ. Control Theory 6 (2017), 517–534.
X. Fu and R. Huang, Approximate controllability of semilinear non-autonomous evolutionary systems with nonlocal conditions, Autom. Remote Control 77 (2016), 428–442.
X.T. Fu and Y. Zhang, Exact null controllability of non-autonomous functional evolution systems with nonlocal conditions, Acta Math. Sci. Ser. B (Engl. Ed.) 33 (2013), no. 840, 747–757.
P. Gautam, A. Shukla, M. Johnson and V. Vijayakumar, Approximate controllability of third order dispersion systems, Bull. Sci. Math. 191 (2024), 10339.
F.D. Ge, H.C. Zhou and C.H. Kou, Approximate controllability of semilinear evolution equations of fractional order with nonlocal and impulsive conditions via an approximating technique, Appl. Math. Comput. 275 (2016), 107–120.
R.K. George, Approximate controllability of non-autonomous semilinear systems, Nonlinear Anal. 24 (1995), 1377–1393.
H. Gou and Y. Li, Mixed monotone iterative technique for damped elastic systems in Banach spaces, J. Pseudo-Differ. Oper. Appl. 11 (2020), 917–933.
H. Gou and Y. Li, A study on damped elastic systems in Banach spaces, Numer. Func. Anal. Opt. 41 (2020), 542–570.
P.J. Graber and I. Lasiecka, Analyticity and Gevrey class regularity for a strongly damped wave equation with hyperbolic dynamic boundary conditions, Semigroup Forum 88 (2014), no. 2, 333–365.
A. Haq and B. Ahmad, Mild solution and finite-approximate controllability of higherorder fractional integro-differential equations with nonlocal conditions, Evol. Equ. Control Theory 13 (2024), no. 5, 1386–1400.
A. Haq and N. Sukavanam, Mild solution and approximate controllability of retarded semilinear systems with control delays and nonlocal conditions, Numer. Funct. Anal. Optim. 42 (2021), no. 6, 721–737.
A. Haq and N. Sukavanam, Mild solution and approximate controllability of secondorder retarded systems with control delays and nonlocal conditions, Bull. Iranian Math. Soc. 48 (2022), 447–464.
A. Haq and N. Sukavanam, Existence and controllability of higher-order nonlinear fractional integro-differential systems via fractional resolvent, Math. Methods Appl. Sci. 45 (2022), no. 16, 9034–9048.
E. Hernández and D.O. Regan, Controllability of Volterra–Fredholm type systems in Banach spaces, J. Franklin Inst. 346 (2009), 95–101.
F. Huang, On the holomorphic property of the semigroup associated with linear elastic systems with structural damping, Acta Math. Sci. Ser. A (Chin. Ed.) 5 (1985), 271–277.
F. Huang and K. Liu, Holomorphic property and exponential stability of the semigroup associated with linear elastic systems with damping, Ann. Diff. Eqs. 4 (1988), no. 4, 411–424.
J.M. Jeong, E.Y. Ju and S.H. Cho, Control problems for semilinear second order equations with cosine families, Adv. Difference Equ. 125 (2016), 13 pp. [32] R.E. Kalman, Controllablity of linear dynamical systems, Contrib. Differential Equations 1 (1963), 190–213.
K. Liu and Z. Liu, Analyticity and differentiability of semigroups associated with elastic systems with damping and gyroscopic forces, J. Differential Equations 141 (1997), 340–355.
Z. Liu, J. Lv and R. Sakthivel, Approximate controllability of fractional functional evolution inclusions with delay in Hilbert spaces, IMA J. Math. Control Inform. 31 (2014), no. 3, 363–383.
V.T. Luong and N.T. Tung, Decay mild solutions for elastic systems with structural damping involving nonlocal conditions, Vest. St.-Petersburg Univ. Math. 50 (2017), no. 1, 55–67.
V.T. Luong and N.T. Tung, Exponential decay for elastic systems with structural damping and infinite delay, Appl. Anal. 99 (2020), 13–28.
N.I. Mahmudov, Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces, SIAM J Control Optim. 42 (2003), 1604–1622.
N.I. Mahmudov, Approximate controllability of evolution systems with nonlocal conditions, Nonlinear Anal. 68 (2008), 536–546.
N.I. Mahmudov and S. Zorlu, On the approximate controllability of fractional evolution equations with compact analytic semigroup, J. Comput. Appl. Math. 259 (2014), 194–204.
F.Z. Mokkedem and X. Fu, Approximate controllability for a semilinear evolution system with infinite delay, J. Dyn. Control. Syst. 22 (2016), 71–89.
X. Pang, X. Li and Z. Liu, Decay mild solutions of Hilfer fractional differential variational-hemivariational inequalities, Nonlinear Anal. 71 (2023), 103834.
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer–Verlag, New York, 1983.
L. Sahijwani, N. Sukavana, and A. Haq, Non-instantaneous impulsive Riemann–Liouville fractional differential systems: Existence and controllability analysis, Math. Methods Appl. Sci. 46 (2023), no. 13, 14509–14526.
R. Sakthivel and E. Anandhi, Approximate controllability of impulsive differential equations with state-dependent delay, Internat. J. Control. 83 (2010), 387–493.
A. Shukla, N. Sukavanam and D.N. Pandey, Approximate controllability of semilinear system with state delay using sequence method, J. Franklin Inst. 352 (2015), 5380–5392.
S. Wei, Global existence of mild solutions for the elastic system with structural damping, Ann. Appl. Math. 35 (2019), 180–188.
M. Wei and Y. Li, Existence and global asymptotic behavior of mild solutions for damped elastic systems with delay and nonlocal conditions, J. Anal. Appl. Comput. 13 (2023), no. 2, 874–892.
M. Wei, Y. Li and Q. Li, Positive mild solutions for damped elastic systems with delay and nonlocal conditions in ordered Banach space, Qual. Theory Dyn. Syst. 21 (2022), 128.
Z. Yan and F. Lu, On approximate controllability of fractional stochastic neutral integrodifferential inclusions with infinite delay, Appl. Anal. 94 (2015), 1235–1258.
H.X. Zhou, Approximate controllability for a class of semilinear abstract equations, SIAM J. Control Optim. 21 (1983),no. 4, 551–565.
Y. Zhou, V. Vijayakumar, C. Ravichandran and R. Murugesu, Controllability results for fractioanl order neutral functional differential inclusions with infinite delay, Fixed Point Theory 18 (2017), no. 2, 773–798.
Published
How to Cite
Issue
Section
License
Copyright (c) 2025 Haide Gou
This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.
Stats
Number of views and downloads: 0
Number of citations: 0
