Time-Optimal Boundary Control for Systems Defined by a Fractional Order Diffusion Equation
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We consider the optimal control problem for a system defined by a one-dimensional diffusion equation with a fractional time derivative. We consider the case when the controls occur only in the boundary conditions. The optimal control problem is posed as the problem of transferring an object from the initial state to a given final state in minimal possible time with a restriction on the norm of the controls. We assume that admissible controls belong to the class of functions L∞[0, T ]. The optimal control problem is reduced to an infinite-dimensional problem of moments. We also consider the approximation of the problem constructed on the basis of approximating the exact solution of the diffusion equation and leading to a finitedimensional problem of moments. We study an example of boundary control computation and dependencies of the control time and the form of how temporal dependencies in the control dependent on the fractional derivative index.
Keywordsoptimal control diffusion equation Caputo’s fractional derivative the problem of moments
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- 1.Uchaikin, V.V., Metod drobnykh proizvodnykh (Method of Fractional Derivatives), Ul’yanovsk: Artishok, 2008.Google Scholar
- 12.Tang, Q. and Ma, Q., Variational Formulation and Optimal Control of Fractional Diffusion Equations with Caputo Derivatives, Adv. Diff. Eq., 2015, vol. 283, DOI 10.1186/s13662-015-0593-5.Google Scholar
- 14.Butkovskii, A.G., Teoriya optimal’nogo upravleniya sistemami s raspredelennymi parametrami (Optimal Control Theory for Systems with Distributed Parameters), Moscow: Nauka, 1965.Google Scholar
- 18.Egorov, A.I., Optimal’noe upravlenie teplovymi i diffuzionnymi protsessami (Optimal Control for Heat and Diffusion Processes), Moscow: Nauka, 1978.Google Scholar
- 19.Butkovskii, A.G., Malyi, S.A., and Andreev, Yu.N., Upravlenie nagrevom metalla (Control over the Heating of Metal), Moscow: Metallurgiya, 1981.Google Scholar