Abstract
An original numerical matrix algorithm aimed at solving the optimal control problems for linear systems of ordinary differential equations with constant coefficients is proposed. The work of the algorithm is demonstrated with the problem, which consists in generating a given small disturbance of the Poiseuille flow in an infinite duct by blowing and suction through the walls. The costs of creating the leading modes and optimal disturbances are compared, which is of independent interest.
Funding statement: The work was supported by the Russian Science Foundation (Project 22–71–10028).
Acknowledgment
The authors are grateful to the reviewer for useful comments that allowed significantly improve this paper.
References
[1] V. M. Alekseev, V. M. Tikhomirov, and S. V. Fomin, Optimal Control. Springer, New York, 1987.10.1007/978-1-4615-7551-1Search in Google Scholar
[2] E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, and D. Sorensen, LAPACK Users Guide. SIAM, Philadelphia, 1992.Search in Google Scholar
[3] A. V. Boiko, A. V. Dovgal, G. R. Grek, and V. V. Kozlov, Physics of Transitional Shear Flows. Springer, Berlin, 2012.10.1007/978-94-007-2498-3Search in Google Scholar
[4] A. V. Boiko and Yu. M. Nechepurenko, Numerical spectral analysis of temporal stability of laminar duct flows with constant cross sections. J. Comput. Math. Math. Phys. 48 (2008), No. 10, 1–17.10.1134/S0965542508100011Search in Google Scholar
[5] A. V. Boiko and Yu. M. Nechepurenko, Numerical study of stability and transient phenomena of Poiseuille flows in ducts of square cross-sections. Russ. J. Numer. Anal. Math. Modelling 24 (2009), No. 3, 193–205.10.1515/RJNAMM.2009.013Search in Google Scholar
[6] C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods. Fundamentals in Single Domains. Springer, Berlin, 2006.10.1007/978-3-540-30726-6Search in Google Scholar
[7] K. V. Demyanko and Yu. M. Nechepurenko, Linear stability analysis of Poiseuille flow in a rectangular duct. Russ. J. Numer. Anal. Math. Modelling 28 (2013), No. 2, 125–148.10.1515/rnam-2013-0008Search in Google Scholar
[8] G. H. Golub and C. F. Van Loan, Matrix Computations. Johns Hopkins University Press, Baltimore, MD, 3rd ed., 1996.Search in Google Scholar
[9] N. J. Higham, The scaling and squaring method for the matrix exponential revisited. SIAM J. Matrix Anal. Appl. 26 (2005), 1179–1193.10.1137/04061101XSearch in Google Scholar
[10] E. B. Lee and L. Marcus, Foundations of Optimal Control Theory. Wiley, New York, 1967.Search in Google Scholar
[11] N. N. Moiseev, Numerical Methods in the Theory of Optimal Systems. Nauka, Moscow, 1971 (in Russian).Search in Google Scholar
[12] Yu. M. Nechepurenko, On the dimension reduction of linear differential-algebraic control systems. Doklady Math. 86 (2012), 457–459.10.1134/S1064562412040059Search in Google Scholar
[13] Yu. M. Nechepurenko and M. Sadkan, A low-rank approximation for computing the matrix exponential norm. SIAM J. Matr. Anal. Appl. 32 (2011), No. 2, 349–363.10.1137/100789774Search in Google Scholar
[14] T. Tatsumi and T. Yoshimura, Stability of the laminar flow in a rectangular duct. J. Fluid Mech. 212 (1990), 437–449.10.1017/S002211209000204XSearch in Google Scholar
© 2024 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Numerical solution of optimal control problems for linear systems of ordinary differential equations
- Nitrogen cycle module for INM RAS climate model
- Evaluation of 2010 heatwave prediction skill by SLNE coupled model
- Two-phase flow simulation algorithm for numerical estimation of relative phase permeability curves of porous materials
Articles in the same Issue
- Frontmatter
- Numerical solution of optimal control problems for linear systems of ordinary differential equations
- Nitrogen cycle module for INM RAS climate model
- Evaluation of 2010 heatwave prediction skill by SLNE coupled model
- Two-phase flow simulation algorithm for numerical estimation of relative phase permeability curves of porous materials