Fractional differential repetitive processes with Riemann–Liouville and Caputo derivatives

Idczak, Dariusz; Kamocki, Rafał

doi:10.1007/s11045-013-0249-0

Cited by 7 publications

(5 citation statements)

References 36 publications

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“…Because

{(b - t)}^{(k + 1) α} = - ((k + 1) α) \int_{t}^{b} - {(b - τ)}^{(k + 1) α - 1} dτ

and an integrand is summable on [ a , b ], therefore, the right‐side hand of equality is an absolutely continuous function on [ a , b ], which equals zero at the point b . Moreover, similarly as in , proof of Theorem 4], we show that

R_{m} \in AC ((), [a, b], {double-struckR}^{n})

and R m ( b ) = 0 for sufficiently large

m \in double-struckN

. So, we proved that a solution to system is an absolutely continuous function, which equals zero at the point b .…”

Section: Optimal Control Problem With Caputo Derivativesupporting

confidence: 77%

“…Consequently, in order to obtain an existence and uniqueness of a solution to problem with x 0 =0, it suffices to show that there exists a unique solution

x \in I_{a +}^{α} (L^{p})

to problem

({, \begin{matrix} (D_{a +}^{α} x) (t) = Ax (t) + v (t), t \in [a, b] a.e. \\ (I_{a +}^{1 - α} x) (a) = 0 \end{matrix})

such that

\{\begin{matrix} x \in A C^{p} ((), [a, b], {double-struckR}^{n}) \\ x (a) = 0 \end{matrix} .

So, we have the following. Theorem If

v \in I_{a +}^{1 - α} ((), L^{p} ((), [a, b], {double-struckR}^{n}))

, then problem possesses a unique solution in

A C^{p} ((), [a, b], {double-struckR}^{n})

. Proof In order to prove this result, one can use analogous arguments as in , proof of theorem 4]. So, we give only a brief overview of the proof.…”

Section: The Linear Cauchy Problem With Caputo Derivativementioning

confidence: 97%

“…In paper , Theorem 4], a theorem on the existence of a unique solution to problem in

AC ((), [a, b], {double-struckR}^{n})

has been proved. In this section, we shall prove a theorem of a such type for the space

A C^{p} ((), [a, b], {double-struckR}^{n})

(in the next section we shall work on the optimization problems, which will be considered in such space).…”

Section: The Linear Cauchy Problem With Caputo Derivativementioning

confidence: 99%

“…In Section 3, we consider the following Cauchy problem:

({, \begin{matrix} (^{C} D_{a +}^{α} x) (t) = Ax (t) + v (t), t \in [a, b] a . e . \\ x (a) = x_{0}, \end{matrix})

where 0 < α < 1,

x_{0} \in {double-struckR}^{n}

, and

A \in {double-struckR}^{n \times n}

. In paper , the existence and uniqueness of a solution in the space of absolutely continuous functions to such problem is given. In this paper, using methods from the mentioned paper, we shall prove a theorem on the existence and uniqueness of a solution to problem in the space of absolutely continuous functions, in which the first‐order derivative belongs to L p with p > 1 (the derivative is a summable function with p > 1).…”

Section: Introductionmentioning

confidence: 99%

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Fractional linear control systems with Caputo derivative and their optimization

Kamocki

Majewski

2014

Optim. Control Appl. Meth.

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SUMMARYIn this paper, a fractional linear control system, containing Caputo derivative, with an integral performance index is studied. First, the existence and uniqueness of a solution to the mentioned control system is obtained. The main result is a theorem on the existence of optimal solutions to considered optimal control problems. Moreover, in order to find these solutions, the necessary optimality conditions (Pontryagin maximum principle) are derived. Our considerations consist of two parts: first, we consider starting a problem with zero initial condition and, next, with nonzero initial condition. All results are obtained by using results of such a type for equivalent fractional optimal control problem containing a Riemann-Liouville derivative. Copyright

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“…Because

{(b - t)}^{(k + 1) α} = - ((k + 1) α) \int_{t}^{b} - {(b - τ)}^{(k + 1) α - 1} dτ

R_{m} \in AC ((), [a, b], {double-struckR}^{n})

and R m ( b ) = 0 for sufficiently large

m \in double-struckN

. So, we proved that a solution to system is an absolutely continuous function, which equals zero at the point b .…”

Section: Optimal Control Problem With Caputo Derivativesupporting

confidence: 77%

“…Consequently, in order to obtain an existence and uniqueness of a solution to problem with x 0 =0, it suffices to show that there exists a unique solution

x \in I_{a +}^{α} (L^{p})

to problem

({, \begin{matrix} (D_{a +}^{α} x) (t) = Ax (t) + v (t), t \in [a, b] a.e. \\ (I_{a +}^{1 - α} x) (a) = 0 \end{matrix})

such that

\{\begin{matrix} x \in A C^{p} ((), [a, b], {double-struckR}^{n}) \\ x (a) = 0 \end{matrix} .

So, we have the following. Theorem If

v \in I_{a +}^{1 - α} ((), L^{p} ((), [a, b], {double-struckR}^{n}))

, then problem possesses a unique solution in

A C^{p} ((), [a, b], {double-struckR}^{n})

. Proof In order to prove this result, one can use analogous arguments as in , proof of theorem 4]. So, we give only a brief overview of the proof.…”

Section: The Linear Cauchy Problem With Caputo Derivativementioning

confidence: 97%

“…In paper , Theorem 4], a theorem on the existence of a unique solution to problem in

AC ((), [a, b], {double-struckR}^{n})

has been proved. In this section, we shall prove a theorem of a such type for the space

A C^{p} ((), [a, b], {double-struckR}^{n})

(in the next section we shall work on the optimization problems, which will be considered in such space).…”

Section: The Linear Cauchy Problem With Caputo Derivativementioning

confidence: 99%

“…In Section 3, we consider the following Cauchy problem:

({, \begin{matrix} (^{C} D_{a +}^{α} x) (t) = Ax (t) + v (t), t \in [a, b] a . e . \\ x (a) = x_{0}, \end{matrix})

where 0 < α < 1,

x_{0} \in {double-struckR}^{n}

, and

A \in {double-struckR}^{n \times n}

Section: Introductionmentioning

confidence: 99%

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Fractional linear control systems with Caputo derivative and their optimization

Kamocki

Majewski

2014

Optim. Control Appl. Meth.

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“…a+ is equivalent to the problem 4) in I α a+ (L 2 ), i.e. the set of solutions to (1.1) in AC α,2 a+ is the same as the set of solutions to (2.4) …”

Section: Basics Of Fractional Calculusmentioning

confidence: 99%

Optimization of a fractional Mayer problem - existence of solutions, maximum principle, gradient methods

Idczak¹,

Walczak²

2014

OpMath

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Abstract. In the paper, we study a linear-quadratic optimal control problem of Mayer type given by a fractional control system. First, we prove a theorem on the existence of a solution to such a problem. Next, using the local implicit function theorem, we derive a formula for the gradient of a cost functional under constraints given by a control system and prove a maximum principle in the case of a control constraint set. The formula for the gradient is used to implement the gradient methods for the problem under consideration.

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Wavelet inpainting by fractional order total variation

Jiang

Yin

2016

Multidim Syst Sign Process

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Fractional differential repetitive processes with Riemann–Liouville and Caputo derivatives

Cited by 7 publications

References 36 publications

Fractional linear control systems with Caputo derivative and their optimization

Fractional linear control systems with Caputo derivative and their optimization

Optimization of a fractional Mayer problem - existence of solutions, maximum principle, gradient methods

Wavelet inpainting by fractional order total variation

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