A general framework for robust control in uid mechanics

The application of optimal control theory to complex problems in uid mechan ics has proven to be quite e ective when complete state information from high resolution numerical simulations is available Moin Bewley Bewley Moin Temam In this approach an iterative optimization algorithm based on the repeated computation of an adjoint eld is used to optimize the controls for nite horizon nonlinear ow problems Abergel Temam In order to extend this in nite dimensional optimization approach to control externally disturbed ows in which the controls must be determined based on limited noisy ow measurements alone it is necessary that the controls computed be insensitive to both state distur bances and measurement noise For this reason robust control theory a generaliza tion of optimal control theory has been examined as a technique by which e ective control algorithms which are insensitive to a broad class of external disturbances may be developed for a wide variety of in nite dimensional linear and nonlinear problems in uid mechanics An aim of the present paper is to put such algorithms into a rigorous mathematical framework The numerical approach proposed to solve the robust control problem is based on computations of an O N adjoint eld where N is the number of grid points used to resolve the continuous PDE for the ow problem Note that N O for problems of engineering interest today and this number may be expected only to increase in the future Computation of the adjoint eld is only as di cult as the computation of the ow itself and thus is a numerically tractable approach to the control problem whenever the computation of the ow itself is numerically tractable In contrast control approaches based on the solution of O N Riccati equations or Hamilton Jacobi Bellman formulations have not been shown to be numerically tractable for discretizations with N O and thus are so far inadequate to treat many of the problems of interest in uid mechanics with a su cient degree of resolution To appear in Physica D April