Optimal Control Problem for Systems Modelled by Diffusion-Wave Equation

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	DOI: 10.46698/s3949-8806-8270-n Optimal Control Problem for Systems Modelled by Diffusion-Wave Equation Postnov, S. S. Vladikavkaz Mathematical Journal 2022. Vol. 24. Issue 3. Abstract: This paper deals with an optimal control problem for a model system defined by a one-dimensional non-homogeneous diffusion-wave equation with a time derivative of fractional-order. In general case we consider both of boundary and distributed controls which are \(p\)-integrable functions (including \(p=\infty\)). In this case two types of optimal control problem are posed and analyzed: the problem of control norm minimization at given control time and the problem of time-optimal control at given restriction on control norm. The study is based on the use of an exact solution of the diffusion-wave equation, with the help of which the optimal control problem is reduced to an infinite-dimensional \(l\)-moment problem. We also consider a finite-dimensional \(l\)-moment problem obtained in a similar way using an approximate solution of the diffusion-wave equation. Correctness and solvability are analyzed for this problem. Finally, an example of boundary control calculation using a finite-dimensional \(l\)-moment problem is considered. Keywords: optimal control, Caputo derivative, diffusion-wave equation, \(l\)-problem of moments Language: Russian Download the full text For citation: Postnov, S. S. Optimal Control Problem for Systems Modelled by Diffusion-Wave Equation, Vladikavkaz Math. J., 2022, vol. 24, no. 3, pp. 108-119 (in Russian). DOI 10.46698/s3949-8806-8270-n + References 1. Mophou, G. M. Optimal Control of Fractional Diffusion Equation, Computational and Applied Mathematics, 2010, vol. 61, no. 1, pp. 68-78. DOI: 10.1016/j.camwa.2010.10.030. 2. Tang, Q. and Ma, Q. Variational Formulation and Optimal Control of Fractional Diffusion Equations with Caputo Derivatives, Advances in Difference Equations, 2015, article number 283. DOI: 10.1186/s13662-015-0593-5. 3. Zhou, Z. and Gong, W. Finite Element Approximation of Optimal Control Problems Governed by Time Fractional Diffusion Equation, Computational and Applied Mathematics, 2016, vol. 71, no. 1, pp. 301-318. DOI: 10.1016/j.camwa.2015.11.014. 4. Kilbas, A. A., Srivastava, H. M. and Trujillo, J. J. Theory and Applications of Fractional Differential Equations, Amsterdam, Elsevier, 2006, 540 p. 5. Butkovskii, A. G. Distributed Control Systems, New York, American Elsevier, 1969, 446 p. 6. Sandev, T. and Tomovski, Z. The General Time Fractional Wave Equation for a Vibrating String, Journal of Physics A: Mathematical and Theoretical, 2010, vol. 43, no. 5, paper ID 055204. DOI: 10.1088/1751-8113/43/5/055204. 7. Agrawal, O. P. Fractional Variational Calculus in Terms of Riesz Fractional Derivatives, Journal of Physics A: Mathematical and Theoretical, 2007, vol. 40, no. 24, pp. 6287-6303. DOI: 10.1088/1751-8113/40/24/003. 8. Kubyshkin, V. A. and Postnov, S. S. Time-Optimal Boundary Control for Systems Defined by a Fractional Order Diffusion Equation, Automation and Remote Control, 2018, vol. 79, no. 5, pp. 884-896. DOI: 10.1134/S0005117918050090. The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses. ← Contents of issue


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