Geodesic
Optimal control problem for systems modelled by diffusion-wave equation
Vladikavkazskij matematičeskij žurnal, Tome 24 (2022) no. 3, pp. 108-119 
S. S. Postnov. Optimal control problem for systems modelled by diffusion-wave equation. Vladikavkazskij matematičeskij žurnal, Tome 24 (2022) no. 3, pp. 108-119. http://geodesic.mathdoc.fr/item/VMJ_2022_24_3_a8/
@article{VMJ_2022_24_3_a8,
     author = {S. S. Postnov},
     title = {Optimal control problem for systems modelled by diffusion-wave equation},
     journal = {Vladikavkazskij matemati\v{c}eskij \v{z}urnal},
     pages = {108--119},
     year = {2022},
     volume = {24},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMJ_2022_24_3_a8/}
}
TY  - JOUR
AU  - S. S. Postnov
TI  - Optimal control problem for systems modelled by diffusion-wave equation
JO  - Vladikavkazskij matematičeskij žurnal
PY  - 2022
SP  - 108
EP  - 119
VL  - 24
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/VMJ_2022_24_3_a8/
LA  - ru
ID  - VMJ_2022_24_3_a8
ER  - 
%0 Journal Article
%A S. S. Postnov
%T Optimal control problem for systems modelled by diffusion-wave equation
%J Vladikavkazskij matematičeskij žurnal
%D 2022
%P 108-119
%V 24
%N 3
%U http://geodesic.mathdoc.fr/item/VMJ_2022_24_3_a8/
%G ru
%F VMJ_2022_24_3_a8

Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Mophou, G. M., “Optimal Control of Fractional Diffusion Equation”, Computational and Applied Mathematics, 61:1 (2010), 68–78 | DOI

[2] Tang, Q. and Ma, Q., “Variational Formulation and Optimal Control of Fractional Diffusion Equations with Caputo Derivatives”, Advances in Difference Equations, 2015, 283 | DOI

[3] Zhou, Z. and Gong, W., “Finite Element Approximation of Optimal Control Problems Governed by Time Fractional Diffusion Equation”, Computational and Applied Mathematics, 71:1 (2016), 301–318 | DOI

[4] Kilbas, A. A., Srivastava, H. M. and Trujillo, J. J., Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006, 540 pp.

[5] Butkovskii, A. G., Distributed Control Systems, American Elsevier, New York, 1969, 446 pp.

[6] Sandev, T. and Tomovski, Z., “The General Time Fractional Wave Equation for a Vibrating String”, Journal of Physics A: Mathematical and Theoretical, 43:5 (2010), 055204 | DOI

[7] Agrawal, O. P., “Fractional Variational Calculus in Terms of Riesz Fractional Derivatives”, Journal of Physics A: Mathematical and Theoretical, 40:24 (2007), 6287–6303 | DOI

[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, 79:5 (2018), 884–896 | DOI