Numerical controllability of fractional dynamical systems

Balachandran, K.; Govindaraj, V.

doi:10.1080/02331934.2014.906416

Cited by 31 publications

(15 citation statements)

References 24 publications

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“…According to Theorem 2.1 of , the controllability Grammian for this system is

\begin{matrix} W (0, 2) & = \int_{0}^{2} {(2 - s)}^{\frac{16}{9}} {((), E_{\frac{17}{9}, \frac{17}{9}} (- 3 {(2 - s)}^{\frac{17}{9}}))}^{2} e^{2 s} normald s \\ = 3.4035 > 0 . \end{matrix}

Therefore the linear system of is controllable on [0,2]. Further, the nonlinear functions

f ((), t, w (t), \int_{0}^{t} g (t, s, w (s))) = \cos w (t) + \int_{0}^{t} \sin w (s) normald s

and

g (t, s, w (s)) = \sin w (s)

satisfy the conditions ( H 1) and ( H 2) of with constants k 1 =1 and k 2 =1 respectively. Hence, it follows, from Theorem 4.4 of , that the nonlinear system is controllable on [0,2].…”

Section: T‐controllability Of Finite‐dimensional Systemsmentioning

confidence: 98%

“…Further, the nonlinear functions

f ((), t, w (t), \int_{0}^{t} g (t, s, w (s))) = \cos w (t) + \int_{0}^{t} \sin w (s) normald s

and

g (t, s, w (s)) = \sin w (s)

satisfy the conditions ( H 1) and ( H 2) of with constants k 1 =1 and k 2 =1 respectively. Hence, it follows, from Theorem 4.4 of , that the nonlinear system is controllable on [0,2]. The controlled trajectories of the system steering from the initial state w (0)=2 to a desired final state w (2)=6 during the interval [0,2] can be approximated from the following algorithm

\begin{matrix} u_{n} (t) & = 0.2938 {(2 - t)}^{8 / 9} e^{t} E_{17 / 9, 17 / 9} (- 3 {(2 - t)}^{17 / 9}) \\ \times ([, 6 - 2 E_{17 / 9} (- 3 {(2)}^{17 / 9}) - \int_{0}^{2} {(2 - s)}^{8 / 9}) \\ \times E_{17 / 9, 17 / 9} (- 3 {(2 - s)}^{17 / 9}) ((, \cos w_{n} (s)) \\ + (], (), \int_{0}^{s} \sin w_{n} (τ) normald τ) normald s), \\ w_{n + 1} (t) & = 2 E_{17 / 9} ( \end{matrix}

…”

Section: T‐controllability Of Finite‐dimensional Systemsmentioning

confidence: 99%

“…= cos w(t) + ∫ t 0 sin w(s)ds and g(t, s, w(s)) = sin w(s) satisfy the conditions (H1) and (H2) of [18] with constants k 1 = 1 and k 2 = 1 respectively. Hence, it follows, from Theorem 4.4 of [18], that the nonlinear system (4) is controllable on [0, 2].…”

mentioning

confidence: 99%

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Trajectory Controllability of Fractional Integro‐Differential Systems in Hilbert Spaces

Govindaraj¹,

George

2017

Asian Journal of Control

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In this paper, sufficient conditions for trajectory controllability of nonlinear fractional integro‐differential systems involving Caputo fractional derivative of order α∈(1,2] in finite and as well as in infinite dimensional Hilbert spaces are obtained. Our tools of study include set‐valued functions, theory of monotone operators and α‐order cosine family of operators. The main results are well illustrated with the aid of examples.

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“…According to Theorem 2.1 of , the controllability Grammian for this system is

\begin{matrix} W (0, 2) & = \int_{0}^{2} {(2 - s)}^{\frac{16}{9}} {((), E_{\frac{17}{9}, \frac{17}{9}} (- 3 {(2 - s)}^{\frac{17}{9}}))}^{2} e^{2 s} normald s \\ = 3.4035 > 0 . \end{matrix}

Therefore the linear system of is controllable on [0,2]. Further, the nonlinear functions

f ((), t, w (t), \int_{0}^{t} g (t, s, w (s))) = \cos w (t) + \int_{0}^{t} \sin w (s) normald s

and

g (t, s, w (s)) = \sin w (s)

satisfy the conditions ( H 1) and ( H 2) of with constants k 1 =1 and k 2 =1 respectively. Hence, it follows, from Theorem 4.4 of , that the nonlinear system is controllable on [0,2].…”

Section: T‐controllability Of Finite‐dimensional Systemsmentioning

confidence: 98%

“…Further, the nonlinear functions

f ((), t, w (t), \int_{0}^{t} g (t, s, w (s))) = \cos w (t) + \int_{0}^{t} \sin w (s) normald s

and

g (t, s, w (s)) = \sin w (s)

\begin{matrix} u_{n} (t) & = 0.2938 {(2 - t)}^{8 / 9} e^{t} E_{17 / 9, 17 / 9} (- 3 {(2 - t)}^{17 / 9}) \\ \times ([, 6 - 2 E_{17 / 9} (- 3 {(2)}^{17 / 9}) - \int_{0}^{2} {(2 - s)}^{8 / 9}) \\ \times E_{17 / 9, 17 / 9} (- 3 {(2 - s)}^{17 / 9}) ((, \cos w_{n} (s)) \\ + (], (), \int_{0}^{s} \sin w_{n} (τ) normald τ) normald s), \\ w_{n + 1} (t) & = 2 E_{17 / 9} ( \end{matrix}

…”

Section: T‐controllability Of Finite‐dimensional Systemsmentioning

confidence: 99%

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Trajectory Controllability of Fractional Integro‐Differential Systems in Hilbert Spaces

Govindaraj¹,

George

2017

Asian Journal of Control

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“…On the other hand, control system is an interconnection of components forming a system configuration that will provide a desired system response. Recently Balachandran et al [25][26][27] established the controllability results for fractional oscillator type dynamical systems by using fixed point theorems and iterative technique.…”

Section: Introductionmentioning

confidence: 99%

Controllability of fractional damped dynamical systems

Balachandran

Govindaraj

Rivero

et al. 2015

Applied Mathematics and Computation

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“…Moreover, controllability analysis of nonlinear fractional order systems is considered, too (Balachandran et al 2012. Balachandran and Govindaraj (2014) presented a computational approach for controllability analysis of a special case of fractional order systems. Some papers was dedicated to constrained controllability of fractional order systems (Balachandran and Kokila 2013;Krishnan and Jayakumar 2013).…”

Section: Introductionmentioning

confidence: 99%

Controllability and observability analysis of continuous-time multi-order fractional systems

Tavakoli

2015

Multidim Syst Sign Process

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This paper proposes some analytical criteria for controllability and observability analysis of fractional order systems describing by multi-order state space equations. The controllability and observability gramians are presented in which their non-singularity is equivalent to controllability and observability of the mentioned systems. Moreover, to overcome the gramian calculation complexities, the controllability and observability matrices are introduced. It is proved that the sufficient condition for controllability or observability of a multi-order fractional system is that its controllability or observability matrix is nonsingular.Some illustrative examples are given to show the efficiency of the proposed method.

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Numerical controllability of fractional dynamical systems

Cited by 31 publications

References 24 publications

Trajectory Controllability of Fractional Integro‐Differential Systems in Hilbert Spaces

Trajectory Controllability of Fractional Integro‐Differential Systems in Hilbert Spaces

Controllability of fractional damped dynamical systems

Controllability and observability analysis of continuous-time multi-order fractional systems

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