Value functional and optimal feedback control in linear-quadratic optimal control problem for fractional-order system

Mikhail I. Gomoyunov

doi:10.3934/mcrf.2023002

Article Contents

2024, Volume 14, Issue 1: 215-254. Doi: 10.3934/mcrf.2023002

This issue Previous Article Optimal robust reinsurance contracts with investment strategy under variance premium principle Next Article Zero-sum stochastic games in continuous-time with risk-sensitive average cost criterion on a countable state space

Value functional and optimal feedback control in linear-quadratic optimal control problem for fractional-order system

Mikhail I. Gomoyunov^{1, 2,}

1.
Krasovskii Institute of Mathematics and Mechanics, Ural Branch of Russian Academy of Sciences, Russia
2.
Ural Federal University, Russia

Received: August 2022

Revised: January 2023

Early access: February 2023

Published: March 2024

This work is supported by RSF grant 19-11-00105, https://rscf.ru/en/project/19-11-00105/.

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Abstract

In this paper, a finite-horizon optimal control problem involving a dynamical system described by a linear Caputo fractional differential equation and a quadratic cost functional is considered. An explicit formula for the value functional is given, which includes a solution of a certain Fredholm integral equation. A step-by-step feedback control procedure for constructing $ \varepsilon $-optimal controls with any accuracy $ \varepsilon > 0 $ is proposed. The basis for obtaining these results is the study of a solution of the associated Hamilton–Jacobi–Bellman equation with so-called fractional coinvariant derivatives.

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Mathematics Subject Classification: Primary: 49N10, 34A08; Secondary: 49L12, 49N35.

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References

[1]	O. P. Agrawal, A quadratic numerical scheme for fractional optimal control problems, J. Dyn. Syst. Meas. Contr., 130 (2008), Art. 011010, 6 pp. doi: 10.1115/1.2814055.
[2]	O. Baghani, Solving state feedback control of fractional linear quadratic regulator systems using triangular functions, Commun. Nonlinear Sci. Numer. Simulat., 73 (2019), 319-337. doi: 10.1016/j.cnsns.2019.01.023.
[3]	A. H. Bhrawy, E. H. Doha, J. A. T. Machado and S. S. Ezz-Eldien, An efficient numerical scheme for solving multi-dimensional fractional optimal control problems with a quadratic performance index, Asian J. Control, 17 (2015), 2389-2402. doi: 10.1002/asjc.1109.
[4]	L. Bourdin, Cauchy–Lipschitz theory for fractional multi-order dynamics: State-transition matrices, Duhamel formulas and duality theorems, Differential Integral Equations, 31 (2018), 559-594. doi: 10.57262/die/1526004031.
[5]	A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, AIMS Series on Applied Mathematics, 2. American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2007.
[6]	A. E. Bryson and Y. C. Ho, Applied Optimal Control: Optimization, Estimation, and Control, Hemisphere, Washington, DC, 1975.
[7]	A. Dabiri, L. K. Chahrogh and J. A. T. Machado, Closed-form solution for the finite-horizon linear-quadratic control problem of linear fractional-order systems, Proceedings of the 2021 American Control Conference, New Orleans, LA, USA, (2021), 3864-3869. doi: 10.23919/ACC50511.2021.9483119.
[8]	L. Debnath and P. Mikusiński, Introduction to Hilbert Spaces with Applications, 3$^{rd}$ edition, Elsevier, Amsterdam, 2005.
[9]	K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition using Differential Operators of Caputo Type, Lecture Notes in Mathematics, 2004. Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-14574-2.
[10]	M. I. Gomoyunov, Approximation of fractional order conflict-controlled systems, Progr. Fract. Differ. Appl., 5 (2019), 143-155. doi: 10.18576/pfda/050205.
[11]	M. I. Gomoyunov, Dynamic programming principle and Hamilton–Jacobi–Bellman equations for fractional-order systems, SIAM J. Control Optim., 58 (2020), 3185-3211. doi: 10.1137/19M1279368.
[12]	M. I. Gomoyunov, On representation formulas for solutions of linear differential equations with Caputo fractional derivatives, Fract. Calc. Appl. Anal., 23 (2020), 1141-1160. doi: 10.1515/fca-2020-0058.
[13]	M. I. Gomoyunov, Optimal control problems with a fixed terminal time in linear fractional-order systems, Arch. Control Sci., 30 (2020), 721-744. doi: 10.24425/acs.2020.135849.
[14]	M. I. Gomoyunov, Solution to a zero-sum differential game with fractional dynamics via approximations, Dyn. Games Appl., 10 (2020), 417-443. doi: 10.1007/s13235-019-00320-4.
[15]	M. I. Gomoyunov, To the theory of differential inclusions with Caputo fractional derivatives, Diff. Equat., 56 (2020), 1387-1401. doi: 10.1134/S00122661200110014.
[16]	M. I. Gomoyunov and N. Yu. Lukoyanov, Differential games in fractional-order systems: Inequalities for directional derivatives of the value functional, Proc. Steklov Inst. Math., 315 (2021), 65-84. doi: 10.4213/tm4227.
[17]	M. I. Gomoyunov, N. Yu. Lukoyanov and A. R. Plaksin, Path-dependent Hamilton–Jacobi equations: The minimax solutions revised, Appl. Math. Optim., 84 (2021), S1087-S1117. doi: 10.1007/s00245-021-09794-4.
[18]	S. Han, P. Lin and J. Yong, Causal state feedback representation for linear quadratic optimal control problems of singular Volterra integral equations, Math. Control Relat. Fields, (2022). doi: 10.3934/mcrf.2022038.
[19]	D. Idczak and S. Walczak, On a linear-quadratic problem with Caputo derivative, Opuscula Math., 36 (2016), 49-68. doi: 10.7494/OpMath.2016.36.1.49.
[20]	A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.
[21]	A. V. Kim, Functional Differential Equations: Application of $i$-Smooth Calculus, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999. doi: 10.1007/978-94-017-1630-7.
[22]	A. N. Krasovskii and N. N. Krasovskii, Control under Lack of Information, Systems & Control: Foundations & Applications. Birkhäuser Boston, Inc., Boston, MA, 1995. doi: 10.1007/978-1-4612-2568-3.
[23]	N. N. Krasovskii and A. I. Subbotin, Game-Theoretical Control Problems, Springer Series in Soviet Mathematics. Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-3716-7.
[24]	E. B. Lee and L. Markus, Foundations of Optimal Control Theory, John Wiley & Sons, Inc., New York-London-Sydney, 1967.
[25]	Y. Li and Y. Chen, Fractional order linear quadratic regulator, Proceedings of the 2008 IEEE/ASME International Conference on Mechtronic and Embedded Systems and Applications, Beijing, China, (2008), 363-368. doi: 10.1109/MESA.2008.4735696.
[26]	S. Liang, S.-G. Wang and Y. Wang, Representation and LQR of exact fractional order systems, Proceedings of the 53$^{rd}$ IEEE Conference on Decision and Control, Los Angeles, CA, USA, (2014), 6908-6913. doi: 10.1109/CDC.2014.7040474.
[27]	S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, Switzerland, 1993.
[28]	J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, NY, 1972.
[29]	J. Yong, Differential Games: A Concise Introduction, World Scientific, Hackensack, NJ, 2015. doi: 10.1142/9121.
[30]	A. C. Zaanen, Linear Analysis. Measure and Integral, Banach and Hilbert space, Linear Integral Equations, Interscience Publishers Inc., New York, North-Holland Publishing Co., Amsterdam, P. Noordhoff N. V., Groningen, 1953.
[31]	B. Zhou and J. L. Speyer, Fractional linear quadratic regulators using Wiener–Hopf spectral factorization, SIAM J. Control Optim., 57 (2019), 4011-4032. doi: 10.1137/19M1239520.