On the existence of optimal solutions to fractional optimal control problems

Kamocki, Rafał

doi:10.1016/j.amc.2014.02.086

Cited by 42 publications

(40 citation statements)

References 17 publications

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“…From the fact that u ∈ AC 0 (P, R m ) and from (7), (10) it follows that F u is well defined. Let us fix a positive integer k and consider a Bielecki type norm in AC 0 (P, R n 1 +n 2 )…”

Section: Case Of Compact Set Of Controlsmentioning

confidence: 95%

“…So, for sufficiently large k the operator F u is contractive and, consequently, possesses a unique fixed point. (7), (10) it follows that the solution z u = (z 1 u , z 2 u ) ∈ I α a+,x (AC 0 (P, R n 1 )) × I β c+,y (AC 0 (P, R n 2 )) of problem (14)-(2), corresponding to a control u ∈ AC 0 (P, R m ), belongs to the set AC 0 (P, R n 1 ) × AC 0 (P, R n 2 ) and (z 1 u (x, y), z 2 u (x, y)) = (I 1 a+,x I 1 c+,y I α a+,y μ 1 (x, y), I 1 a+,x I 1 c+,y I β c+,y μ 2 (x, y))…”

Section: Case Of Compact Set Of Controlsmentioning

confidence: 99%

“…The special case of Lipschitzian controls was studied by Lee and Marcus in [12] and case of piecewise constant controls with at most r points of discontinuity -by Strauss in [20]. In paper [10], a linear ordinary control system of fractional order involving the Riemann-Liouville derivative with the cost functional J is considered. Existence of optimal solutions to such problem under the assumption of convexity of g 0 with respect to u has been obtained.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Existence of optimal solutions to Lagrange problems for Roesser type systems of the first and fractional orders

Idczak

Kamocki

Majewski

et al. 2015

Applied Mathematics and Computation

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“…From the fact that u ∈ AC 0 (P, R m ) and from (7), (10) it follows that F u is well defined. Let us fix a positive integer k and consider a Bielecki type norm in AC 0 (P, R n 1 +n 2 )…”

Section: Case Of Compact Set Of Controlsmentioning

confidence: 95%

Section: Case Of Compact Set Of Controlsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Existence of optimal solutions to Lagrange problems for Roesser type systems of the first and fractional orders

Idczak

Kamocki

Majewski

et al. 2015

Applied Mathematics and Computation

Self Cite

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“…For example in [3,4], the authors have achieved the necessary conditions for optimization of FOCPs with the Caputo fractional derivative. The interested reader can refer to [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] for some recent works on FOCPs. In this paper, we propose a new efficient and accurate computational method based on HFs for solving the following FOCP [6]:…”

Section: Introductionmentioning

confidence: 99%

An efficient computational method based on the hat functions for solving fractional optimal control problems

et al. 2016

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In this paper, an efficient and accurate computational method based on the hat functions (HFs) is proposed for solving a class of fractional optimal control problems (FOCPs). In the proposed method, the fractional optimal control problem under consideration is reduced to a system of nonlinear algebraic equations which can be simply solved. To this end, the fractional state and control variables are expanded by the HFs with unknown coefficients. Then, the operational matrix of fractional integration of the HFs with some properties of these basis functions are employed to achieve a nonlinear algebraic equation, replacing the performance index and a nonlinear system of algebraic equations, replacing the dynamical system in terms of the unknown coefficients. Finally, the method of constrained extremum is applied, which consists of adjoining the constraint equations derived from the given dynamical system to the performance index by a set of undetermined Lagrange multipliers. As a result, the necessary conditions of optimality are derived as a system of algebraic equations in the unknown coefficients of the state variable, control variable and Lagrange multipliers. Furthermore, the efficiency of the proposed method is shown for some concrete examples. The results reveal that the proposed method is very accurate and efficient.

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“…Recently, many papers have been published on the control theory for fractional systems. For instance, optimal control problems were investigated in [12][13][14]. In [15], the author proves fractional version of Pontryagin principle.…”

Section: Introductionmentioning

confidence: 99%

On the Existence and Continuous Dependence on Parameter of Solutions to Some Fractional Dirichlet Problem with Application to Lagrange Optimal Control Problem

Kamocki

Majewski

2016

J Optim Theory Appl

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In the paper, a Lagrange optimal control problem governed by a fractional Dirichlet problem with the Riemann-Liouville derivative is considered. To begin with, based on some variational method, the existence and continuous dependence of solution to the aforementioned Dirichlet problem is investigated. Then, continuous dependence is applied to show the existence of optimal solution to the Lagrange problem. An important point is that the solution to Dirichlet problem does need to be unique; therefore, the above dependence should be understood as a continuity of some multifunction-the concept of the Kuratowski-Painlevé limit of the sequence of sets is used to formulate this property.

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On the existence of optimal solutions to fractional optimal control problems

Cited by 42 publications

References 17 publications

Existence of optimal solutions to Lagrange problems for Roesser type systems of the first and fractional orders

Existence of optimal solutions to Lagrange problems for Roesser type systems of the first and fractional orders

An efficient computational method based on the hat functions for solving fractional optimal control problems

On the Existence and Continuous Dependence on Parameter of Solutions to Some Fractional Dirichlet Problem with Application to Lagrange Optimal Control Problem

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