# Optimal LQ-feedback control for a class of first-order hyperbolic distributed parameter systems

Ilyasse Aksikas; Joseph J. Winkin; Denis Dochain

ESAIM: Control, Optimisation and Calculus of Variations (2008)

- Volume: 14, Issue: 4, page 897-908
- ISSN: 1292-8119

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topAksikas, Ilyasse, Winkin, Joseph J., and Dochain, Denis. "Optimal LQ-feedback control for a class of first-order hyperbolic distributed parameter systems." ESAIM: Control, Optimisation and Calculus of Variations 14.4 (2008): 897-908. <http://eudml.org/doc/250316>.

@article{Aksikas2008,

abstract = { 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.
},

author = {Aksikas, Ilyasse, Winkin, Joseph J., Dochain, Denis},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {First-order hyperbolic PDE's; infinite-dimensional systems; LQ-optimal control; stability; optimality; first-order hyperbolic PDE's},

language = {eng},

month = {2},

number = {4},

pages = {897-908},

publisher = {EDP Sciences},

title = {Optimal LQ-feedback control for a class of first-order hyperbolic distributed parameter systems},

url = {http://eudml.org/doc/250316},

volume = {14},

year = {2008},

}

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

DA - 2008/2//

PB - EDP Sciences

VL - 14

IS - 4

SP - 897

EP - 908

AB - 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.

LA - eng

KW - First-order hyperbolic PDE's; infinite-dimensional systems; LQ-optimal control; stability; optimality; first-order hyperbolic PDE's

UR - http://eudml.org/doc/250316

ER -

## References

top- 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).
- 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).
- 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.
- I. Aksikas, J. Winkin and D. Dochain, LQ-optimal feedback regulation of a nonisothermal plug flow reactor infinite-dimensional model. Int. J. Tomography & Statistics5 (2007) 73–78.
- I. Aksikas, J. Winkin and D. Dochain, Optimal LQ-feedback regulation of a nonisothermal plug flow reactor model by spectral factorization. IEEE Trans. Automat. Control52 (2007) 1179–1193.
- 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.
- V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems. Boston: Academic Press (1993).
- A. Bressan, Hyperbolic Systems of Conservation Laws: The One-Dimensional Cauchy Problem. Oxford University Press (2000).
- H. Brezis, Opéateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, Mathematics Studies. North-Holland (1973).
- F.M. Callier and C.A. Desoer, Linear System Theory. Springer-Verlag, New York (1991).
- F.M. Callier and J. Winkin, LQ-optimal control of infinite-dimensional systems by spectral factorization. Automatica28 (1992) 757–770.
- P.D. Christofides, Nonlinear and Robust Control of Partial Differential Equation Systems: Methods and Application to Transport-Reaction Processes. Birkhauser, Boston (2001).
- R.F. Curtain and H.J. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory. Springer-Verlag, New York (1995).
- C.M. Dafermos and M. Slemrod, Asymptotic behavior of nonlinear contraction semigroups. J. Funct. Anal.13 (1973) 97–106.
- 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).
- G.F. Froment and K.B. Bischoff, Chemical Reactor Analysis and Design. 2nd edition, John Wiley, New York (1990).
- M. Ikeda and D.D. Siljak, Optimality and robustness of linear quadratic control for nonlinear systems. Automatica26 (1990) 499–511.
- 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.
- V. Lakshmikantham and S. Leela, Nonlinear Differential Equations in Abstract Spaces. Pergamon, Oxford (1981).
- 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).
- Z. Luo, B. Guo and O. Morgül, Stability and Stabilization of Infinite Dimensional Systems with Applications. Springer-Verlag, London (1999).
- R.H. Martin, Nonlinear Operators and Differential Equations in Banach spaces. John Wiley & Sons, New York (1976).
- A. Pazy, Semigroups of Linear Operators and Application to Partial Differential Equations, Appl. Math. Sci.44. Springer-Verlag, New York (1983).
- W.H. Ray, Advanced Process Control, Series in Chemical Engineering. Butterworth, Boston (1981).
- L.M. Silverman and H.E. Meadows, Controllability and observability in time-variable linear systems. J. SIAM Control5 (1967) 64–73.

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