A general framework for robust control in fluid mechanics

The application of optimal control theory to complex problems in fluid mech anics has proven to be quite effective when complete state information from high-resolution numerical simulations is available [P. Moin, T.R. Bewley, Appl. Mech. Rev., Part 2 47 (6) (1994) S3-S13; T.R. Bewley, P. Moin, R. Tem am, J. Fluid Mech. (1999), submitted for publication]. La this approach, an iterative optimization algorithm based on the repeated computation of an a djoint held is used to optimize the controls for finite-horizon nonlinear h ow problems [F. Abergel, R. Temam, Theoret. Comput. Fluid Dyn. 1 (1990) 303 -325]. In order to extend this infinite-dimensional optimization approach t o control externally disturbed flows in which the controls must be determin ed based on limited noisy flow measurements alone, it is necessary that the controls computed be insensitive to both state disturbances and measuremen t noise. For this reason, robust control theory, a generalization of optima l control theory, has been examined as a technique by which effective contr ol algorithms which are insensitive to a broad class of external disturbanc es may be developed for a wide variety of infinite-dimensional linear and n onlinear problems in fluid mechanics. An aim of the present paper is to put such algorithms into a rigorous mathematical framework, for it cannot: be assumed at the outset that a solution to the infinite-dimensional robust co ntrol problem even exists. Iu this paper, conditions on the initial data, t he parameters in the cost functional, and the regularity of the problem are established such that existence and uniqueness of the solution to the robu st control problem can be proven. Both linear and nonlinear problems are tr eated, and the 2D and 3D nonlinear cases are treated separately in order to get the best possible estimates. Several generalizations are discussed and an appropriate numerical method is proposed. (C) 2000 Elsevier Science B,V , All rights reserved.