Optimal Mobile Control in Inverse Problem for Barenblatt–Zheltov–Kochina Type Fractional Order Equation

• Published in: Lobachevskii journal of mathematics Vol. 46; no. 1; pp. 488 - 501
• Main Authors: Ramazanova, A. T., Abdullozhonova, A. N., Yuldashev, T. K.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01.01.2025; Springer Nature B.V
• Subjects: Algebra; Analysis; Fourier series; Functionals; Geometry; Integral equations; Inverse problems; Mathematical Logic and Foundations; Mathematics; Mathematics and Statistics; Nonlinear control; Operators (mathematics); Optimal control; Optimization; Probability Theory and Stochastic Processes; inverse problem; nonlinear control; minimization of the functional; necessary conditions for optimal mobile control; Hilfer fractional operator; nonlinear Barenblatt–Zheltov–Kochina differential equation; generalized solvability; 2010 Mathematics Subject Classification: 35B50, 35D30, 35K55, 35R30, 49J20, 49N45
• ISSN: 1995-0802; 1818-9962
• DOI: 10.1134/S1995080224608087

Nonlinear optimal mobile control of thermal processes in a mixed inverse problem for a Barenblatt–Zheltov–Kochina differential eauation with a Hilfer fractional operator is studied. The inverse problem is considered with spectral and final conditions. The necessary optimality conditions for nonlinear mobile control are formulated. The determination of the optimal control function is reduced to solve system of Fredholm functional-integral equations, which consists product of two integrals of nonlinear functions. The unique solvability of this nonlinear functional-integral equation is proved by the method of contracting mapping. Approximate calculations for the state function of the controlled process, for the redefinition function, and for the optimal control function are obtained. The absolute and uniform convergence of the obtained Fourier series are proved.