Necessary conditions of optimality in a problem of optimal control of moving sources for singular heat equation.

*(English)*Zbl 1438.49014Summary: A problem of optimal control of processes described by a singular heat equation and systems of ordinary differential equations with moving sources is investigated in this paper. In spite of applied importance of problems with moving sources controls, they have not been studied enough so far [J. L. Lions, Optimal control of systems governed by partial differential equations. Springer, Berlin (1971; Zbl 0203.09001); A. G. Butkovskii and L. M. Pustylnikov, Theory of moving control of distributed parameter systems. Moskau: Nauka (1980); V. A. Kubyshkin and V. I. Finyagina, Mobile controls in distributed parameter systems. Moskau: Sinteg (2005)], [the author, Proc. Inst. Math. Mech., Natl. Acad. Sci. Azerb. 31, 219–224 (2009; Zbl 1228.49007); “Necessary conditions of optimality in optimal control problem with moving sources”, Rep. Nat. Acad. Sci. Azerb. 68, No. 4, 10–15 (2012)]. Sufficient conditions of Fréchet differentiability of quality test and an expression for its gradient are obtained, necessary conditions of optimality in the form of point wise and integral maximum principles are established for an optimal control problem considered below.

##### MSC:

49J20 | Existence theories for optimal control problems involving partial differential equations |

49K20 | Optimality conditions for problems involving partial differential equations |

35K20 | Initial-boundary value problems for second-order parabolic equations |

##### Keywords:

moving sources; maximum principles; integral identity; reduced problem; necessary conditions of optimality
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\textit{R. A. Teymurov}, Casp. J. Appl. Math. Ecol. Econ. 1, No. 1, 104--113 (2013; Zbl 1438.49014)

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##### References:

[1] | J.L. Lions Optimal control of systems described by partial equations. M.: Mir, 1972, 416 p. (Russian). |

[2] | A.G. Butkovskii, L.M. Pustylnikov Theory of moving control of distributed parameter systems. M.: Nauka, 1980, 384 p. (Russian). |

[3] | Kubyshkin V.A., Finyagina V.I. Mobile controls in distributed parameter systems. M: Sinteg, 2005, 420 p. · Zbl 0845.93041 |

[4] | Ladyzhenskaya O.A., Solonnikov V.A. Uraltseva N.N. Linear and quasilinear equations of parabolic type. M.: Nauka , 1967, 736 p. (Russian). |

[5] | Ladyzhenskaya O.A. Boundary value problems of mathematical physics. M.: Nauka, 1973, 408 p. (Russian). |

[6] | Vasil‘ev F.P. Solution methods of extremal problems. M.: Nauka, 1981, 400 p. (Russian). Necessary conditions of optimality in a problem of optimal control of moving sources113 |

[7] | Teymurov R.A. On existence and uniqueness of the solution of mobile source optimal control problem.Proceedings of Institute of Mathematics and Mechanics, 2009, vol. XXXI, pp. 219-224. · Zbl 1228.49007 |

[8] | Teymurov R.A. Necessary conditions of optimality in optimal control problem with moving sources.Reports of National Academy of Sciences of Azerbaijan, 2012, v. LXVIII, No. 4, pp. 10-15. Rafig A. |

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