Optimization or flow control problems have the usual three ingredients. First, one has an objective, a reason why one wants to control the flow. Mathematically such an objective functional is expressed as a cost, or performance functional. Next, one has controls or design parameters are expessed in terms of unknown data in the mathematical specification of the problem. Finally, one has contraints that determine what type of flow one is interested in and that place direct or indirect limited on candidate optimizers. Mathematically, the type of flow is in terms of a specific set of partial differential equations.
   Actually, In this talk, we concern the Stokes equations as the type of folw. In our case, the optimization problem we consider is to find an optimal state $(\mathbf{u}, p)$ and an optimal control $\mathbf{f}$ which minimize the $L^2$-norm distance between $\mathbf{u}$ and target velocity $\hat{\mathbf{u}}$ subject to $(\mathbf{u},p,\mathbf{f})$ satisfying the Stokes system. Such constraint optimization problem can be converted to the unconstrained optimization problems by the Lagrange multiplier rule. It is a standard approach for solving finite-dimensional constrained optimization problem([3,4]). As a result, we have a coupled optimality system related to two Stokes type equations associate with state variables and adjoint variables. In this system, we use only state variables and adjoint variables but controls. This optimality system can be dealt with mixed finite element approaches. But, in our case, we use first-order system least-squares(FOSLS) technique by using the state flux $'\mathbf{U} = \nabla{\mathbf{u}}^t$ of the state velocity $\mathbf{u}$ and the adjoint flux $\mathbf{V} = \nabla{\mathbf{v}}^t$ of the adjoint velocity $\mathbf{v}$. We reformulate the coupled second-order optimality system as the coupled first-order optimality system. Then, we define the least-squares functionals based on the uncontrained optimal system in terms of appropriate weights and norms for the system. In particular, we impose a weight $\frac{1}{\sigma^2}$ on adjoint system to get a good control result. We show that the least-squares functional is equivalent to $H^1$ product norms by proving the $H^2$-regularity of that optimality system[2]. This principles result in symmetric and positive definite algebraic system. We show that a least-squares principle based on $L^2$ norms applied to this system yields optimal discretization error estimates $H^1$ norm in each variable.
    For the numerical test, we use multigrid V-cycle method using Gauss-Seidel smoothing iteration on the unit square domain. And we also use the single approximating space of continuous piecewise linear polynomials for the approximations of all unknowns. Note that we use a simple target velocity that is a solution of the Stokes system. We examine the effects of changes in the parameters \sigma and $\nu$. Generally, effecting good control requires small values of penalty parameter $\sigma$. So, we observe the $L^2$ norm of between controlled flow and target flow. That is, we show the controlled flow behavior when $\sigma =1,0.1,0.01$, and $0.001$ for fixed kinematic viscosity $\nu=1$. Then, we see that the controlled flow matches the target flow very well for small values of penalty parameter $\sigma$. Now, we consider the controlled flow for different kinematic viscosity $\nu =1,0.1,0.01$ and $0.001$ when $\sigma=1$ is fixed. In this case, we see the norms of |u-hat{u}| and |f| are very similar to the ones of above varying $\sigma$ case. These computational results mean that if $\nu$ is small, then the flow is easily controlled by small control flow $\mathbf{f}$. Finally, we consider the dependence of mesh size $h=1/4,1/8,1/16,1/32$ for fixed $\nu=1$ and $\sigma=0.001$. We wee here that the norm of $\|\mathbf{u}-\hat{\mathbf{u}}\|$ and the value of objective functional are decreased.
   As a conclusion, we applied on optimal control problem with Stokes equations as contraints by using the earler FOSLS work[1] on Stokes system. Moreover, we imposed weights $\frac{1}{\sigma^2}$ and $\nu^2$ for good control effecting good control of small penalty parameter and viscosity. In this sense, if we find the best weights then, a least-squares principles seems to be proper for optimal control problems.

References
[1] Z. Cai, R. Lazarov, T. A. Manteufel, and S. F. Mccormick, First-order system least squares for the Stokes equation, with application to linear elasticity, SIAM J. Numer. Anal. Vol. 34 (1997), pp. 1727-1741.

[2] S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II, Comm. Pure Appl. Math. 17 (1964), 35-92.

[3] P. Bochev and M. Gunzburger. Least-squares finite element methods for optimality systems arising in optimization and cotrol problems, SIAM J. Numer. Anal. Vol. 43 (2006), pp. 2517-2543.

[4] M. Gunzburger, Perspectives in Flow Control and Optimization, SIAM, Philadelphia, 2003.



(This work was supported by KRF-2005-070-C00017 and BK21 MCDI)