A necessary condition of Pontryagin type for fuzzy control systems
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A necessary condition of Pontryagin type for fuzzy control systems J. Soolaki1 · O. S. Fard1 · Akbar Hashemi Borzabadi1
Received: 11 June 2016 / Revised: 27 October 2016 / Accepted: 2 November 2016 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2016
Abstract In the present paper, we prove necessary optimality conditions of Pontryagin type for a class of fuzzy optimal control problems. The new results are illustrated by computing the extremals of two fuzzy optimal control systems, which improve recent results of Najariyan and Farahi. Keywords Fuzzy Pontryagin maximum principle · Fuzzy variational problems · Fuzzy Hamiltonian function Mathematics Subject Classification Primary 93C42; Secondary 34N05 · 93D05
1 Introduction Optimal control problems are usually solved with the help of the famous pontryagin maximum principle (PMP), which is a generalization of the classic Euler–Lagrange and Weierstrass necessary optimality conditions for the calculus of variations (Pinch 1993; Pontryagin et al. 1962). In the past few decades, the interest in the field of fuzzy optimal control has increased and fuzzy optimal control problems have attracted a great deal of attention (Fard et al. 2014; Fard and Zadeh 2012; Fard and Salehi 2014; Farhadinia 2011, 2014; Najariyan and Farahi 2013, 2014; Soolaki et al. 2016; Takagi and Sugeno 1985; Yang and Cai 2010). A number of existing
Communicated by Rosana Sueli da Motta Jafelice.
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J. Soolaki [email protected] O. S. Fard [email protected]; [email protected] Akbar Hashemi Borzabadi [email protected]
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School of Mathematics and Computer Science, Damghan University, Damghan, Iran
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J. Soolaki et al.
schemes of fuzzy optimal control for nonlinear systems are proposed based on the framework of the Takagi–Sugeno (T–S) fuzzy model originated from fuzzy identification (Takagi and Sugeno 1985). Moreover, for most of the T–S-modeled nonlinear systems, fuzzy control design is carried out by the aid of the parallel distributed compensation (PDC) approach (Yang and Cai 2010). However, it is still possible to enumerate all works that establish necessary optimality conditions for the fuzzy calculus of variations or fuzzy optimal control: see Fard et al. (2014), Fard and Zadeh (2012), Fard and Salehi (2014), Farhadinia (2011, 2014), Najariyan and Farahi (2013, 2014), Soolaki et al. (2016). Najariyan and Farahi in their study (Najariyan and Farahi 2013, 2014) obtain necessary optimality conditions of Pontryagin type for a very special case of fuzzy optimal control problems, using r -level sets and representing these in the complex numbers space. Farhadinia in his study (Farhadinia 2014) applies the fuzzy variational approach of Farhadinia (2011) to fuzzy optimal control problems and derives necessary optimality conditions for fuzzy optimal control problems that depend on the Buckley and Feuring derivative (Buckley and Feuring 1999). Here, using the PMP and a novel form of the Hamiltonian approach, we achieve fuzzy solutions (state and control) by solving an appro
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