An optimal control problem in exterior hydrodynamics

Optimal control theory of distributed parameter systems is a rapidly developing subject [3,10,9,11]. So far the impact of this growth to continuum mechanics has been essentially in the branch of solid mechanics. In this paper, we develop some of these ideas for a problem in fluid mechanics which has profound practical applications such as aero/hydro maneuvering of vehicles. We consider the task of finding the optimal way to accelerate an obstacle from rest to a given speed in an infinite medium of viscous fluid. Here the objective is to find the speed trajectory which corresponds to a global minimum for the energy expenditure functional. We note here that the above problem can be transformed into a nonstationary problem in the exterior domain by a simple change of coordinates. We only deal with translatory motions of the obstacle in this paper. The first step towards the theory is a well-posedness theorem which relates the generalised solution of the nonstationary Navier Stokes problem in the exterior domain to the prescribed speed of the obstacle. We have shown that for the two-dimensional problem this correspondence is unique and analytic. The central result of this paper is an existence theorem for the speed trajectory which corresponds to the absolute minimum for the energy expenditure. The Reynolds number in this theorem is arbitrary. The existence theorem for absolute minimum is the fundamental step in establishing a (Pontryagin type) Maximum principle to derive the necessary conditions and in the dynamic programming method (using the Hamilton-JacobiBellman equations) for the control synthesis in the feedback form [14, 13, 2].