Optimal LQ-feedback control for a class of first-order hyperbolic distributed parameter systems
ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 4, pp. 897-908.

The Linear-Quadratic (LQ) optimal control problem is studied for a class of first-order hyperbolic partial differential equation models by using a nonlinear infinite-dimensional (distributed parameter) Hilbert state-space description. First the dynamical properties of the linearized model around some equilibrium profile are studied. Next the LQ-feedback operator is computed by using the corresponding operator Riccati algebraic equation whose solution is obtained via a related matrix Riccati differential equation in the space variable. Then the latter is applied to the nonlinear model, and the resulting closed-loop system dynamical performances are analyzed.

DOI : https://doi.org/10.1051/cocv:2008015
Classification : 49J20,  93B52,  34K30,  47H06,  34K20
Mots clés : first-order hyperbolic PDE's, infinite-dimensional systems, LQ-optimal control, stability, optimality
@article{COCV_2008__14_4_897_0,
author = {Aksikas, Ilyasse and Winkin, Joseph J. and Dochain, Denis},
title = {Optimal {LQ-feedback} control for a class of first-order hyperbolic distributed parameter systems},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {897--908},
publisher = {EDP-Sciences},
volume = {14},
number = {4},
year = {2008},
doi = {10.1051/cocv:2008015},
zbl = {1148.49033},
mrnumber = {2455389},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv:2008015/}
}
TY  - JOUR
AU  - Aksikas, Ilyasse
AU  - Winkin, Joseph J.
AU  - Dochain, Denis
TI  - Optimal LQ-feedback control for a class of first-order hyperbolic distributed parameter systems
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2008
DA  - 2008///
SP  - 897
EP  - 908
VL  - 14
IS  - 4
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv:2008015/
UR  - https://zbmath.org/?q=an%3A1148.49033
UR  - https://www.ams.org/mathscinet-getitem?mr=2455389
UR  - https://doi.org/10.1051/cocv:2008015
DO  - 10.1051/cocv:2008015
LA  - en
ID  - COCV_2008__14_4_897_0
ER  - 
Aksikas, Ilyasse; Winkin, Joseph J.; Dochain, Denis. Optimal LQ-feedback control for a class of first-order hyperbolic distributed parameter systems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 4, pp. 897-908. doi : 10.1051/cocv:2008015. http://www.numdam.org/articles/10.1051/cocv:2008015/

[1] H. Abou-Kandil, G. Freiling, V. Ionescu and G. Jank, Matrix Riccati Equations in Control and Systems Theory, Series: Systems & Control: Foundations & Applications. Birkhauser (2003). | MR 1997753 | Zbl 1027.93001

[2] I. Aksikas, Analysis and LQ-Optimal Control of Infinite-Dimensional Semilinear Systems: Application to a Plug Flow Reactor. Ph.D. thesis, Université Catholique de Louvain, Belgium (2005).

[3] I. Aksikas, J. Winkin and D. Dochain, Stability analysis of an infinite-dimensional linearized plug flow reactor model, in Proceedings of the 43rd IEEE Conference on Decision and Control, CDC (2004) 2417-2422.

[4] I. Aksikas, J. Winkin and D. Dochain, LQ-optimal feedback regulation of a nonisothermal plug flow reactor infinite-dimensional model. Int. J. Tomography & Statistics 5 (2007) 73-78. | MR 2393755

[5] I. Aksikas, J. Winkin and D. Dochain, Optimal LQ-feedback regulation of a nonisothermal plug flow reactor model by spectral factorization. IEEE Trans. Automat. Control 52 (2007) 1179-1193. | MR 2332744

[6] I. Aksikas, J. Winkin and D. Dochain, Asymptotic stability of infinite-dimensional semilinear systems: application to a nonisothermal reactor. Systems Control Lett. 56 (2007) 122-132. | MR 2288527 | Zbl 1112.93054

[7] V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems. Boston: Academic Press (1993). | MR 1195128 | Zbl 0776.49005

[8] A. Bressan, Hyperbolic Systems of Conservation Laws: The One-Dimensional Cauchy Problem. Oxford University Press (2000). | MR 1816648 | Zbl 0997.35002

[9] H. Brezis, Opéateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, Mathematics Studies. North-Holland (1973). | MR 348562 | Zbl 0252.47055

[10] F.M. Callier and C.A. Desoer, Linear System Theory. Springer-Verlag, New York (1991). | MR 1123479 | Zbl 0744.93002

[11] F.M. Callier and J. Winkin, LQ-optimal control of infinite-dimensional systems by spectral factorization. Automatica 28 (1992) 757-770. | MR 1168933 | Zbl 0776.49023

[12] P.D. Christofides, Nonlinear and Robust Control of Partial Differential Equation Systems: Methods and Application to Transport-Reaction Processes. Birkhauser, Boston (2001). | MR 1933756 | Zbl 1018.93001

[13] R.F. Curtain and H.J. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory. Springer-Verlag, New York (1995). | MR 1351248 | Zbl 0839.93001

[14] C.M. Dafermos and M. Slemrod, Asymptotic behavior of nonlinear contraction semigroups. J. Funct. Anal. 13 (1973) 97-106. | MR 346611 | Zbl 0267.34062

[15] D. Dochain, Contribution to the Analysis and Control of Distributed Parameter Systems with Application to (Bio)chemical Processes and Robotics. Thèse d'Agrégation de l'Enseignement Supérieur, Université Catholique de Louvain, Louvain-la-Neuve, Belgium (1994).

[16] G.F. Froment and K.B. Bischoff, Chemical Reactor Analysis and Design. 2nd edition, John Wiley, New York (1990).

[17] M. Ikeda and D.D. Siljak, Optimality and robustness of linear quadratic control for nonlinear systems. Automatica 26 (1990) 499-511. | MR 1056135 | Zbl 0717.93021

[18] M. Laabissi, M.E. Achhab, J. Winkin and D. Dochain, Trajectory analysis of nonisothermal tubular reactor nonlinear models. Systems Control Lett. 42 (2001) 169-184. | MR 2007047 | Zbl 0985.93030

[19] V. Lakshmikantham and S. Leela, Nonlinear Differential Equations in Abstract Spaces. Pergamon, Oxford (1981). | MR 616449 | Zbl 0456.34002

[20] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories, Volume II: Abstract Hyperbolic-like Systems over a Finite Time Horizon. Cambridge University Press (2000). | MR 1745476 | Zbl 0961.93003

[21] Z. Luo, B. Guo and O. Morgül, Stability and Stabilization of Infinite Dimensional Systems with Applications. Springer-Verlag, London (1999). | MR 1745384 | Zbl 0922.93001

[22] R.H. Martin, Nonlinear Operators and Differential Equations in Banach spaces. John Wiley & Sons, New York (1976). | MR 492671 | Zbl 0333.47023

[23] A. Pazy, Semigroups of Linear Operators and Application to Partial Differential Equations, Appl. Math. Sci. 44. Springer-Verlag, New York (1983). | MR 710486 | Zbl 0516.47023

[24] W.H. Ray, Advanced Process Control, Series in Chemical Engineering. Butterworth, Boston (1981).

[25] L.M. Silverman and H.E. Meadows, Controllability and observability in time-variable linear systems. J. SIAM Control 5 (1967) 64-73. | MR 209043 | Zbl 0163.11001

Cité par Sources :