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Robust control of the Kuramoto-Sivashinsky equation. (English) Zbl 0987.49003
Summary: Robust control theory, a generalization of optimal control theory, has been proposed as an effective technique when control algorithms are sensitive to a broad class of external disturbances. In [{\it T. R. Bewley}, {\it R. Teman} and {\it M. Ziane}, Physica D 138, No. 3-4, 360-392 (2000; Zbl 0981.76026)], a general framework for the robust control of the Navier-Stokes equations in finite time horizon was developed. In this article the robust boundary control for the Kuramoto-Sivashinsky equation is considered in the same spirit: a robust boundary control problem is formulated, and the existence and uniqueness for the robust control problem are proved. A data assimilation problem corresponding to the Kuramoto-Sivashinsky equation is considered, existence and uniqueness of solution are derived. This approach is also applicable as well to other equations with a structure similar to that of the Kuramoto-Sivashinsky equation.

49J20Optimal control problems with PDE (existence)
35Q35PDEs in connection with fluid mechanics
34K35Functional-differential equations connected with control problems
76R50Diffusion (fluid mechanics)