A distributed optimal control problem for an elliptic problem is considered. A penaltyrleast-squares method for optimal control problems is used wherein the constraint equations are enforced via penalization. First, the existence of the optimal control problem is proved. The convergence, as the penalty parameter tends to zero, of the solution to the penalized optimal control problem to that unpenalized one is demonstrated as is the convergence of a gradient method for determining solutions of the penalized optimal control problem. Finally, finite element approximations of the penalized optimal control problem are studied and optimal error estimates are obtained. ᮊ

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Scholarly studies of U.S. legislators’ voting behavior have concluded that constituent interests exercise only limited influence, but these conclusions may result from inadequate measurement. I develop new measures of economic interests that emphasize import/export (sectoral) cleavages in addition to business/labor (factoral) cleavages and, in the process, transcend geographic boundaries. Results of logistic regression analysis suggest that the interests of economic and nongeographic constituencies, as reflected in campaign contributions, were highly significant predictors of voting in the U.S. Congress on the U.S.–Korea Free Trade Agreement and that the import/export cleavage was more salient than the business/labor cleavage. In addition, legislators’ ideological positions with respect to national security were more significant than their partisan affiliations and more significant than their positions on other dimensions of ideology.

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Abstract. First-order least-squares method of a distributed optimal control problem for the incompressible Stokes equations is considered. An optimality system for the optimal solution are reformulated to the equivalent first-order system by introducing the vorticity and then the leastsquares functional corresponding to the system is defined in terms of the sum of the squared H −1 and L 2 norms of the residual equations of the system. Finite element approximations are studied and optimal error estimates are obtained. Resulting linear system of the optimality system is symmetric and positive definite. The V-cycle multigrid method is applied to the system to test computational efficiency.

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The purpose of this paper is to construct an unconstrained optimal control problem by using a least-squares approach for the constrained distributed optimal control problem associated with incompressible Stokes equations. The constrained equations are reformulated to the equivalent first-order system by introducing vorticity, and then the least-squares functional corresponding to the system is enforced via a penalty term to the objective functional. The existence of a solution of the unconstrained optimal control problem is proved, and the convergence of this solution to that of unpenalized one is demonstrated as the penalty parameter tends to zero. Finite element approximations with error estimates are studied, and the relevant computational experiments are presented.