In the present paper, we investigate the controllability, and observability of a non-homogeneous continuous-time fractal dynamical system (NC-TFDS). We show that the controllability is equivalent to a controllability matrix that has a full rank. We also show a relationship between controllability and fractal differential Lyapunov equation (F-DLE). We give some theorems for the observability of a NC-TFDS. Finally, we offer two examples are provided to illustrate the effectiveness of our results and compare it with the two papers Al-Zhour (2022) and (Sadek et al., 2022).
AgrawalOP (2007). Analytical schemes for a new class of fractional differential equations. Journal of Physics A: Mathematical and Theoretical40(21): 5469–5477,
2.
Al-ZhourZ (2022) Controllability and observability behaviors of a non-homogeneous conformable fractional dynamical system compatible with some electrical applications. Alexandria Engineering Journal61(2): 1055–1067.
3.
AmerYAEl-SayedATAhmedEE (2022) Vibration reduction of a non-linear ship model using positive position feedback controllers. International Journal of Dynamics and Control10(2): 409–426.
BengocheaGOrtigueiraMD (2017). An operational approach to solve fractional continuoustime linear systems. International Journal of Dynamics and Control5(1): 61–71.
6.
BiswalSSSwainDRRoutPK (2022). Inter-area and intra-area oscillation damping for UPFC in a multi-machine power system based on tuned fractional PI controllers. International Journal of Dynamics and Control10: 1–19.
7.
BouazzaLMourllionBMakhloufA, et al. (2021). Controllability and observability of formal perturbed linear time invariant systems. International Journal of Dynamics and Control9(4): 1444–1455.
8.
BoudjeridaASebaD (2022). Controllability of nonlocal Hilfer fractional delay dynamic inclusions with non-instantaneous impulses and non-dense domain. International Journal of Dynamics and Control10: 1–13.
ChenWSunHZhangX, et al. (2010). Anomalous diffusion modeling by fractal and fractional derivatives. Computers & Mathematics with Applications59(5): 1754–1758.
12.
DasMSamantaGP (2021). Optimal control of a fractional order epidemic model with carriers. International Journal of Dynamics and Control10: 1–22.
13.
EzzahriLEl HarrakiIBoutouloutA (2018). Enlarged controllability for hyperbolic linear systems with zone actuators. International Journal of Dynamics and Control6(4): 1516–1522.
14.
FiuzyMShamaghdariS (2022). Stability analysis of fractional-order linear system with PID controller in the output feedback structure subject to input saturation. International Journal of Dynamics and Control10(2): 511–524.
15.
HuDLChenWSunHG (2020). Power-law stability of Hausdorff derivative nonlinear dynamical systems. International Journal of Systems Science51(4): 601–607.
16.
IbrahimMHameedI (2021). Disturbance rejection and reference tracking of time delayed systems using Gramian controllability. International Journal of Dynamics and Control10: 1–8.
17.
KhalilRAl HoraniMYousefA, et al. (2014). A new definition of fractional derivative. Journal of Computational and Applied Mathematics264: 65–70.
18.
KhanY (2021). Hausdorff fractal system arising in the field of plasma physics via variational technology. Chinese Journal of Physics77: 2288–2297,
19.
KhanYFarazNAlsulaimaniHA (2021). Two core optical fibers coupled nonlinear model in the framework of Hausdorff fractal derivative. Results in Physics24: 104103.
20.
KumarKAMALENDRAKumarRAKESH (2015). Controllability of Sobolev type nonlocal impulsive mixed functional integrodifferential evolution systems. Electronic Journal of Mathematical Analysis and Applications3(1): 122–132.
21.
KumarKKumarR (2017). Boundary controllability of delay differential systems of fractional order with nonlocal condition. Journal of Applied Nonlinear Dynamics6(4): 465–472.
22.
KumarKKumarR (2019). Boundary controllability of fractional order nonlocal semi-linear neutral evolution systems with impulsive condition. Discontinuity, Nonlinearity, and Complexity8(4): 419–428.
23.
LiouvilleJ (1835) Mmoire sur le changement de la variable indpendante, dans le calcul des diffrentielles a indices quelconques.
24.
LuQZuazuaE (2016) On the lack of controllability of fractional in time ODE and PDE. Mathematics of Control, Signals, and Systems28(2): 1–21.
MillerKSRossB (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley.
27.
MohtavipourSMMollajafariMNaseriA (2019). A guaranteed-comfort and safe adaptive cruise control by considering drivers acceptance level. International Journal of Dynamics and Control7(3): 966–980.
OldhamKSpanierJ (1974). The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order. Elsevier. http://worldcat.org/fr/title/1058445718.
30.
SadekLAbouzaidBSadekEM, et al. (2022). Controllability, observability and fractional linear-quadratic problem for fractional linear systems with conformable fractional derivatives and some applications. International Journal of Dynamics and Control75: 1–15.
31.
ShuklaAPatelR (2021a). Existence and optimal control results for second-order semilinear system in Hilbert spaces. Circuits, Systems, and Signal Processing40(9): 4246–4258.
32.
ShuklaAPatelR (2021b). Controllability results for fractional semilinear delay control systems. Journal of Applied Mathematics and Computing65(1): 861–875.
33.
ShuklaASukavanamNPandeyDN (2014). Controllability of semilinear stochastic system with multiple delays in control. IFAC Proceedings Volumes47(1): 306–312.
34.
ShuklaASukavanamNPandeyDN (2015a). Approximate controllability of semilinear stochastic control system with nonlocal conditions. Nonlinear Dynamics and Systems Theory15(3): 321–333.
35.
ShuklaASukavanamNPandeyDN (2015b). Complete controllability of semi-linear stochastic system with delay. Rendiconti del Circolo Matematico di Palermo (1952-)64(2): 209–220.
36.
ShuklaASukavanamNPandeyDN, et al. (2016). Approximate controllability of second-order semilinear control system. Circuits, Systems, and Signal Processing35(9): 3339–3354.
37.
SinghVPandeyDN (2019). Controllability of fractional impulsive quasilinear differential systems with state dependent delay. International Journal of Dynamics and Control7(1): 313–325.
38.
YangXJ (2019). New general calculi with respect to another functions applied to describe the Newton-like dashpot models in anomalous viscoelasticity. Thermal Science23(6 Part B): 3751–3757.
39.
YangXJ (2021a). A scaling law chaotic system. arXiv preprint arXiv:2111.09816.
40.
YangXJ (2021b). An insight on the fractal power law flow: from a Hausdorff vector calculus perspective. arXiv preprint arXiv:2111.04463.
41.
YangXJ (2021c). Theory and Applications of Special Functions for Scientists and Engineers. NY, USA: Springer Singapore.
42.
YangXJGaoFJuY (2022a). General fractional calculus with nonsingular kernels: new prospective on viscoelasticity. In: Methods of Mathematical Modelling and Computation for Complex Systems (pp. 135–157) Cham: Springer.
43.
YangXJLiuJGAbdel-AtyM (2022b). On the theory of the fractal scaling-law elasticity. Meccanica57(4): 943–955.
44.
ZhangYZhangXXiongFR, et al. (2022). Active vibration control of typical piping system of a nuclear power plant based on fractional PI controller. International Journal of Dynamics and Control102: 1–13.
