A Penalized Neumann Control Approach for Solving an Optimal Dirichlet Control Problem for the Navier--Stokes Equations

Abstract

We introduce a penalized Neumann boundary control approach for solving an optimal Dirichlet boundary control problem associated with the two-or three-dimensional steady-state Navier-Stokes equations. We prove the convergence of the solutions of the penalized Neumann control problem, the suboptimality of the limit, and the optimality of the limit under further restrictions on the data. We describe the numerical algorithm for solving the penalized Neumann control problem and report some numerical results.

An energy space finite element approach for elliptic Dirichlet boundary control problems

“…Several variational formulations of Dirichlet control problems are discussed in [25]. To include a Dirichlet boundary condition u = z ∈ L 2 (Γ) in a standard variational formulation, alternatively one may consider a penalty approximation of the Dirichlet boundary condition by using a Robin boundary condition, see, e.g., [3,17,19,20]. Again, sufficient smoothness of the boundary Γ has to be assumed.…”

THE h × p FINITE ELEMENT METHOD FOR OPTIMAL CONTROL PROBLEMS CONSTRAINED BY STOCHASTIC ELLIPTIC PDES

“…In tables and figures for computational results, for simplicity, we use DP = ∑ N n=1 p n , where p n is the maximum degree of polynomials in a y n -direction and DOF as the number DP (DOF ) Relative Error for u Relative Error for ξ Relative Error for f 1 (2) 1.198032987982e-01 1.787953125199e-01 1.720684070324e-01 3 (4) 9.562740236029e-03 1.524223854302e-02 1.341233756062e-02 5 (6) 5.570898785792e-04 1.057333186124e-03 8.615989678717e-04 7 (8) 2.795461789913e-05 6.309221819675e-05 4.901139558725e-05 9 (10) 1.286833647831e-06 3.391536066446e-06 2.567358708683e-06 11 (12) 4.560494644988e-08 1.362462543327e-07 1.019093421438e-07 (2) 9.401289567930e-02 1.987307188342e-01 1.752373609462e-01 3 (6) 1.021973093216e-02 2.469886703133e-02 1.951724713049e-02 5 (12) 1.005212637175e-03 3.036986195881e-03 2.380145213483e-03 7 (20) 9.899411004937e-05 3.609536395869e-04 2.891208795498e-04 9 (30) 9.451679011477e-06 3.927211535465e-05 3.149626856263e-05 11 (42) 3.335005157516e-07 2.710535612892e-06 2.420352397715e-06 of degrees of freedom of the discretization with respect to the random parameter space. For instance, if we use p = (p 1 , p 2 ) = (5, 4), then DP = 9 and DOF = 30.…”

“…After that, by using the method of Lagrange multipliers, we derive the optimality system of equations. Then we apply the theory of Brezzi-Rappaz-Raviart (BRR) [16,17,18,19,20] in uncoupling the optimality system, so that we can develop a priori error estimate that gives exponentially fast convergent results for the optimal solution of our optimal control problem.…”

Optimal Control and Shape Optimization of Aorto-Coronaric Bypass Anastomoses

“…After some transformations inequality (4.17) can be written as: 19) where B * is the adjoint of the operator B. A useful identity relating the gradient of cost functional to the solution of the adjoint problem is: …”