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Research Papers

Fractional Order Version of the Hamilton–Jacobi–Bellman Equation

[+] Author and Article Information
Abolhassan Razminia

Electrical Engineering Department,
School of Engineering,
Persian Gulf University,
P.O. Box 75169,
Bushehr 75169-13817, Iran
e-mail: razminia@pgu.ac.ir

Mehdi Asadizadehshiraz

Electronic and Electrical Engineering Department,
Shiraz University of Technology,
P.O. Box 71555-313
Shiraz 71557-13876, Iran
e-mail: M.Asadizadehshiraz@sutech.ac.ir

Delfim F. M. Torres

R&D Unit CIDMA,
Department of Mathematics,
Center for Research and Development in
Mathematics and Applications,
University of Aveiro,
Aveiro 3810-193, Portugal
e-mail: delfim@ua.pt

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received June 29, 2018; final manuscript received October 30, 2018; published online November 28, 2018. Assoc. Editor: Dumitru Baleanu.

J. Comput. Nonlinear Dynam 14(1), 011005 (Nov 28, 2018) (6 pages) Paper No: CND-18-1287; doi: 10.1115/1.4041912 History: Received June 29, 2018; Revised October 30, 2018

We consider an extension of the well-known Hamilton–Jacobi–Bellman (HJB) equation for fractional order dynamical systems in which a generalized performance index is considered for the related optimal control problem. Owing to the nonlocality of the fractional order operators, the classical HJB equation, in the usual form, does not hold true for fractional problems. Effectiveness of the proposed technique is illustrated through a numerical example.

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Figures

Grahic Jump Location
Fig. 1

Optimal system states of problem (32)(34), which converge to the equilibrium points

Grahic Jump Location
Fig. 2

The optimal controller of the system (32)(33) that minimizes the performance index (34)

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