Receding horizon optimal control for infinite dimensional systems
ESAIM: Control, Optimisation and Calculus of Variations, Volume 8 (2002), p. 741-760

The receding horizon control strategy for dynamical systems posed in infinite dimensional spaces is analysed. Its stabilising property is verified provided control Lyapunov functionals are used as terminal penalty functions. For closed loop dissipative systems the terminal penalty can be chosen as quadratic functional. Applications to the Navier-Stokes equations, semilinear wave equations and reaction diffusion systems are given.

DOI : https://doi.org/10.1051/cocv:2002032
Classification:  49L15,  49N35,  93C20,  93Dx
Keywords: receding horizon control, control Lyapunov function, Lyapunov equations, closed loop dissipative, minimum value function, Navier-Stokes equations
@article{COCV_2002__8__741_0,
     author = {Ito, Kazufumi and Kunisch, Karl},
     title = {Receding horizon optimal control for infinite dimensional systems},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {8},
     year = {2002},
     pages = {741-760},
     doi = {10.1051/cocv:2002032},
     zbl = {1066.49020},
     mrnumber = {1932971},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2002__8__741_0}
}
Ito, Kazufumi; Kunisch, Karl. Receding horizon optimal control for infinite dimensional systems. ESAIM: Control, Optimisation and Calculus of Variations, Volume 8 (2002) pp. 741-760. doi : 10.1051/cocv:2002032. http://www.numdam.org/item/COCV_2002__8__741_0/

[1] F. Allgöwer, T. Badgwell, J. Qin, J. Rawlings and S. Wright, Nonlinear predictive control and moving horizon estimation - an introductory overview, Advances in Control, edited by P. Frank. Springer (1999) 391-449.

[2] T.R. Bewley, Flow control: New challenges for a new Renaissance. Progr. Aerospace Sci. 37 (2001) 21-58.

[3] H. Chen and F. Allgöwer, A quasi-infinite horizon nonlinear model predictive control scheme with guaranteed stability. Automatica 34 (1998) 1205-1217. | Zbl 0947.93013

[4] H. Choi, M. Hinze and K. Kunisch, Instantaneous control of backward facing step flow. Appl. Numer. Math. 31 (1999) 133-158. | MR 1708955 | Zbl 0939.76027

[5] H. Choi, R. Temam, P. Moin and J. Kim, Feedback control for unsteady flow and its application to the stochastic Burgers equation. J. Fluid Mech. 253 (1993) 509-543. | MR 1233904 | Zbl 0810.76012

[6] R.A. Freeman and P.V. Kokotovic, Robust Nonlinear Control Design, State-Space an Lyapunov Techiques. Birkhäuser, Boston (1996). | MR 1396307 | Zbl 0857.93001

[7] W.H. Fleming and M. Soner, Controlled Markov Processes and Viscosity Solutions. Springer-Verlag, New York (1993). | MR 1199811 | Zbl 0773.60070

[8] C.E. Garcia, D.M. Prett and M. Morari, Model predictive control: Theory and practice - a survey. Automatica 25 (1989) 335-348. | Zbl 0685.93029

[9] M. Hinze and S. Volkwein, Analysis of instantaneous control for the Burgers equation. Nonlinear Analysis TMA (to appear). | MR 1904464 | Zbl 1022.49001

[10] K. Ito and K. Kunisch, On asymptotic properties of receding horizon optimal control. SIAM J. Control Optim (to appear). | Zbl 1031.49033

[11] A. Jadababaie, J. Yu and J. Hauser, Unconstrained receding horizon control of nonlinear systems. Preprint. | Zbl 1009.93028

[12] D.L. Kleinman, An easy way to stabilize a linear constant system. IEEE Trans. Automat. Control 15 (1970) 692-712.

[13] D.Q. Mayne and H. Michalska, Receding horizon control of nonlinear systems. IEEE Trans. Automat. Control 35 (1990) 814-824. | MR 1058366 | Zbl 0715.49036

[14] V. Nevistič and J. A. Primbs, Finite receding horizon control: A general framework for stability and performance analysis. Preprint.

[15] J.A. Primbs, V. Nevistič and J.C. Doyle, A receding horizon generalization of pointwise min-norm controllers. Preprint. | Zbl 0976.93024

[16] P. Scokaert, D.Q. Mayne and J.B. Rawlings, Suboptimal predictive control (Feasibility implies stability). IEEE Trans. Automat. Control 44 (1999) 648-654. | MR 1680132 | Zbl 1056.93619

[17] F. Tanabe, Equations of Evolution. Pitman, London (1979). | Zbl 0417.35003

[18] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis. North Holland, Amsterdam (1984). | Zbl 0568.35002