# zbMATH — the first resource for mathematics

Analysis of instantaneous control for the Burgers equation. (English) Zbl 1022.49001
The authors study the distributed control of the one-dimensional Burgers equation by means of instantaneous controls for a tracking type functional. The process consists in choosing at time $$t_k$$ a control which minimizes a functional taking into account a tracking type problem involving the state and the target at time $$t_{k+1}$$. The authors prove that this procedure is equivalent to writing the controlled Burgers equation in a closed loop form. In the minimization procedure at time $$t_k$$ the choice of the step size for the gradient method is crucial in the analysis of the stability of the closed loop system. The authors establish different stability results for the corresponding closed loop systems and they illustrate their theoretical results by different numerical simulations.

##### MSC:
 49J20 Existence theories for optimal control problems involving partial differential equations 35Q53 KdV equations (Korteweg-de Vries equations)
Full Text:
##### References:
 [1] Abergel, F.; Temam, R., On some control problems in fluid mechanics, Theoret. comput. fluid dyn., 1, 303-325, (1990) · Zbl 0708.76106 [2] Berggren, M., Numerical solution of a flow control problem: vorticity reduction by dynamic boundary action, SIAM J. sci. comput., 19, 3, 829-860, (1998) · Zbl 0946.76016 [3] T.R. Bewley, P. Moin, R. Temam, DNS-based predictive control of turbulence: an optimal benchmark for feedback algorithms. Preprint, 2000, J. Fluid Mech, to appear. · Zbl 1036.76027 [4] H. Choi, Suboptimal control of turbulent flow using control theory, Proceedings of the International Symposium on Mathematical Modelling of Turbulent Flows, 1995. Tokyo, Japan. [5] H. Choi, M. Hinze, K. Kunisch, Instantaneous control of backward-facing-step flows, Applied Numer. Math., 31 (1999) 133-158. see also Preprint No. 571/1997, Fachbereich Mathematik, Technische Universität Berlin, Deutschland. · Zbl 0939.76027 [6] Choi, H.; Temam, R.; Moin, P.; Kim, J., Feedback control for unsteady flow and its application to the stochastic Burgers equation, J. fluid mech., 253, 509, (1993) · Zbl 0810.76012 [7] Dautray, R.; Lions, J.L., Functional and variational methods, Mathematical analysis and numerical methods for science and technology, Vol. 2, (1990), Springer Berlin [8] García, C.E.; Prett, D.M.; Morari, M., Model predictive control: theory and practice—a survey, Automatica, 25, 3, 335-348, (1989) · Zbl 0685.93029 [9] M.D. Gunzburger, S. Manservisi, Analysis and approximation for linear feedback control for tracking the velocity in Navier-Stokes flows, Preprint, 1999, to appear. · Zbl 0969.76025 [10] Gunzburger, M.D.; Manservisi, S., Analysis and approximation of the velocity tracking problem for navier – stokes flows with distributed control, SIAM J. numer. anal., 37, 1481-1512, (2000) · Zbl 0963.35150 [11] D.C. Hill, Drag reduction strategies, CTR Annual Research Briefs, 1993. Center for Turbulence Research, Stanford University/NASA Ames Research Center, 3-14. [12] M. Hinze, Optimal and instantaneous control of the instationary Navier-Stokes equations, Habilitationsschrift, 1999, Fachbereich Mathematik, Technische Universität Berlin. [13] M. Hinze, A. Kauffmann, A new class of feedback control laws for dynamical systems, Preprint No. 602/1998, 1998, Fachbereich Mathematik, Technische Universität Berlin. [14] Hinze, M.; Kunisch, K., Control strategies for fluid flows—optimal versus suboptimal control, (), 351-358 · Zbl 1068.76512 [15] Hou, L.S.; Yan, Y., Dynamics for controlled navier – stokes systems with distributed controls, SIAM J. control optim., 35, 654-677, (1997) · Zbl 0871.49008 [16] Kauffmann, A.; Kunisch, K., Optimal control of a solid fuel ignition model, ESAIM: Proceedings, 8, 65-76, (2000) · Zbl 0966.49022 [17] Lee, C.; Kim, J.; Choi, H., Suboptimal control of turbulent channel flow for drag reduction, J. fluid mech., 358, 245-258, (1998) · Zbl 0907.76039 [18] Min, C.; Choi, H., Suboptimal feedback control of vortex shedding at low Reynolds numbers, J. fluid mech., 401, 123-156, (1999) · Zbl 0968.76017 [19] V. Nevistić, J.A. Primbs, Finite receding horizon control: a general framework for stability and performance analysis, Preprint, 1997. Automatic Control Laboratory, ETH Zürich, Switzerland. [20] Rawlings, J.B.; Muske, K.R., The stability of constrained receding horizon control, IEEE trans. automat. control, 38, 10, 1512-1516, (1993) · Zbl 0790.93019 [21] Temam, R., Navier – stokes equations, (1979), North-Holland Amsterdam · Zbl 0454.35073 [22] Tröltzsch, F.; Unger, A., Fast solution of optimal control problems in selective cooling of steel, Zamm, 81, 447-456, (2001) · Zbl 0993.49024 [23] Volkwein, S., Augmented Lagrangian-SQP techniques and optimal control problems for the stationary Burgers equation, Computational optimization and applications, 18, 133-158, (2001) [24] Volkwein, S., Distributed control problems for the Burgers equation. 1999, Computational optimization and applications, 16, 57-81, (2001) · Zbl 0974.49020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.