Sivaguru S. Sritharan

Optimal control of viscous flow

Preface Contributors 1. An introduction to deterministic and stochastic control of viscous flow, S. S. Sritharan 2. Optimal control problems for a class of nonlinear equations with an application to control of fluids Max D. Gunzburger, L. Steven Hou and Thomas Svobodny 3. Feedback control of time dependent stokes flows V. Barbu 4. An optimal control problem governed by the evolution Navier-Stokes equations E. Casas 5. Optimal control of turbulent flows Frederic Abergal and Roger Temam 6. Optimal control problems for Navier-Stokes system with distributed control function A. V. Fursikov 7. Design of feedback compensators for viscous flow Y. R. Ou 8. Numerical approximation of optimal flow control problems by SQP method S. S. Ravindran.

Existence of optimal controls for viscous flow problems

A class of optimal control problems in viscous flow is studied. Main result is the existence theorem for optimal control. Three typical flow control problems are formulated within this general class.

Dynamic programming of the Navier-Stokes equations

A class of optimal control problems in viscous flow is studied. Main results are the Pontryagin maximum principle and the verification theorem for the Hamilton-Jacobi-Bellman equation.

An optimal control problem in exterior hydrodynamics

In this paper we consider the problem of accelerating an obstacle in an incompressible viscous fluid from rest to a given speed in a given time with minimum energy expenditure. An existence theorem for the speed trajectory which corresponds to the absolute minimum is provided. The results are valid for arbitrary Reynolds numbers.

LARGE DEVIATIONS FOR THE STOCHASTIC SHELL MODEL OF TURBULENCE

In this work, we first prove the existence and uniqueness of a strong solution to stochastic GOY model of turbulence with a small multiplicative noise. Then using the weak convergence approach, Laplace principle for solutions of the stochastic GOY model is established in certain Polish space. Thus a Wentzell–Freidlin type large deviation principle is established utilizing certain results by Varadhan and Bryc.

THE STOCHASTIC MAGNETO-HYDRODYNAMIC SYSTEM

The magneto-hydrodynamic system perturbed by white noise is cast as a stochastic partial differential equation taking values in an appropriate Lusin space. The martingale problem associated with this SPDE is shown to be well-posed under suitable conditions.

Optimal chattering controls for viscous flow

Optimal control theory of viscous flow has many important applications in engineering science. During the past few years several fundamental advances have been reported for flow control problems with convex cost. The main questions addressed were the existence theorem for optimal control [l-4], necessary conditions for the free terminal state problem [4-71 as well as the full Pontryagin maximum principle for problems with terminal constraint [8] and feedback synthesis using the Hamilton-Jacobi-Bellman equation [5,8,91. Finite element methods for the maximum principle with free end state are analysed in [lo]. See also the book [ll] for reports of progress by various authors of this field. In this paper we study a class of flow control problems where the fluid is controlled by a distributed forcing at a portion of the boundary and the cost functional is nonconvex with respect to the control variable. Our study is motivated in part by the nonconvex flow control problems that arise in practice where we have to minimize nonconvex functionals such as average lift to drag ratio by distributed boundary control [ll, 121. In such examples, even if the nonconvexity is in the total velocity dependence in the cost functional, if the control is distributed on the boundary, then the boundary extension procedures we use lead to nonconvexity in state as well as control variable (see further discussion below and also the subsequence section where we formulate an example). The fundamental problem of existence of optimal control is resolved by a suitable generalization of the classical concept of Young measures. Young measures were introduced in [13,14] (see also [15]) to deal with nonconvex problems in the calculus of variations. During the sixties, several authors applied similar ideas to finite dimensional nonconvex control problems [16-201. For other applications and theoretical developments see [21,22]. Most of these works are concerned with Young measures defined on finite dimensional sets. However, for the control problems of this paper we need to define probability measures on control sets which are infinite dimensional. The probability measures used in this paper are of finitely additive type in contrast to the sigma (countably) additive Young measures. When the control set is finite dimensional and

Optimal Feedback Control of Hydrodynamics: a Progress Report

In this article we review some of the recent results in the mathematical theory of optimal feedback control of viscous flow. Main results are existence of ordinary and chattering controls, Pontryagin maximum principle and feedback synthesis using infinite dimensional Hamilton-Jacobi equation of dynamic programming. Some preliminary results on stochastic control also presented.

Impulse control of stochastic Navier-Stokes equations

Optimal control theory of 2uid dynamics has numerous applications such as aero= hydrodynamic control, combustion control, Tokomak magnetic fusion as well as ocean and atmospheric prediction. During the past decade several fundamental advances have been made by a number of researchers as documented in Sritharan [21,22]. In this paper we develop a new direction to this subject, namely we mathematically formulate and resolve impulse and stopping time problems. Impulse control of Navier–Stokes equations has signi7cance beyond control theory. In fact, in optimal weather prediction the task of updating the initial data optimally at strategic times can be reformulated precisely as an impulse control problem for the primitive cloud equations (which consist of the Navier–Stokes equation coupled with temperature and species evolution equations, cf. [11]), see [1,9,15]. For the study of optimal stopping problem alone, it is possible to impose regularity assumptions on the stopping cost. However, in our case, optimal stopping problems are used as intermediate steps to treat the impulse control problem through an iteration process. This dictates that we must work with stopping costs which have only continuity property. Optimal stopping and impulse control problems are very well known, particularly for di;usion processes (e.g., see the books of Bensoussan and Lions [3,4]), for degenerate

On the acceleration of viscous fluid through an unbounded channel

Viscous incompressible time-dependent flow through a two-dimensional channel of finite cross section is considered. The well-posed theorem for this problem has remained open in the past because flux carrying velocity fields possess infinite energy. We resolve this issue by constructing a suitable infinite energy vector field and seeking the solution of the Navier-Stokes problem as a finite-energy perturbation of this vector field.