Constrained Controllability of Fractional Dynamical Systems

Balachandran, K.; Kokila, J.

doi:10.1080/01630563.2013.778868

Cited by 12 publications

(14 citation statements)

References 18 publications

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“…First, consider the following linear fractional integro‐differential equation:

\begin{matrix} ^{C} D^{α} x (t) & = Ax (t) + \int_{0}^{t} H (t - s) x (s) normald s + f (t), 0 < α < 1, t \in [0, T] : = J, and \\ x (0) & = x_{0}, \end{matrix}

where the state vector

x (t) \in {double-struckR}^{n}

, A is an n × n matrix, H is an n × n continuous matrix, and f ( t ) is a continuous function. By Laplace transformation approach, we have the following solution :

x (t) = R_{α} (t) + \int_{0}^{t} {(t - s)}^{α - 1} R_{α, α} (t - s) f (s) d s,

where R α ( t ) is an n × n matrix satisfying the condition stated in , and

R_{α, α} (θ) = θ^{1 - α} D I^{α} R_{α} (θ) .

Next, consider the linear neutral fractional Volterra integro‐differential equation of the following form:

\begin{matrix} ^{C} D^{α} ([], x (t) - \int_{0}^{t} C (t - s) x (s) normald s) & = Ax (t) + \int_{0}^{} \end{matrix}

…”

Section: Preliminariesmentioning

confidence: 99%

“…Balachandran and Dauer established sufficient conditions for the relative controllability of nonlinear neutral Volterra integro‐differential systems. Recently, Balachandran et al investigated the relative controllability of fractional dynamical systems with distributed delays in control. However, for the nonlinear neutral Volterra integro‐differential systems with fractional order, the problem of controllability with distributed delays in control remains to be studied.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Relative controllability of nonlinear neutral fractional integro‐differential systems with distributed delays in control

Balachandran

Divya

Rodríguez-Germá

et al. 2015

Math Methods in App Sciences

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“…First, consider the following linear fractional integro‐differential equation:

\begin{matrix} ^{C} D^{α} x (t) & = Ax (t) + \int_{0}^{t} H (t - s) x (s) normald s + f (t), 0 < α < 1, t \in [0, T] : = J, and \\ x (0) & = x_{0}, \end{matrix}

where the state vector

x (t) \in {double-struckR}^{n}

, A is an n × n matrix, H is an n × n continuous matrix, and f ( t ) is a continuous function. By Laplace transformation approach, we have the following solution :

x (t) = R_{α} (t) + \int_{0}^{t} {(t - s)}^{α - 1} R_{α, α} (t - s) f (s) d s,

where R α ( t ) is an n × n matrix satisfying the condition stated in , and

R_{α, α} (θ) = θ^{1 - α} D I^{α} R_{α} (θ) .

Next, consider the linear neutral fractional Volterra integro‐differential equation of the following form:

\begin{matrix} ^{C} D^{α} ([], x (t) - \int_{0}^{t} C (t - s) x (s) normald s) & = Ax (t) + \int_{0}^{} \end{matrix}

…”

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Relative controllability of nonlinear neutral fractional integro‐differential systems with distributed delays in control

Balachandran

Divya

Rodríguez-Germá

et al. 2015

Math Methods in App Sciences

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show abstract

“…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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“…Now, taking Laplace transform on (19) and using the property (iii) as well as a simple partial fraction method, we obtain the solution of (19) as (Balachandran and Kokila, 2013a) x(t)…”

Section: Examplesmentioning

confidence: 99%

“…Klamka (2010) discussed the minimum energy control problem of infinite-dimensional fractional-discrete time linear systems and established necessary and sufficient conditions for exact controllability of such systems. Recently, Balachandran et al (2012a;2013a;2013b;2012b;2012c;2012d) studied the controllability problem for various types of nonlinear fractional dynamical systems by using fixed point theorems.…”

Section: Introductionmentioning

confidence: 99%