A variational technique for optimal boundary control in a hyperbolic problem Murat Subas ßı a,⇑ , Yes ßim Saraç a , Ahmet Kaçar b a Ataturk University, Science Faculty, Department of Mathematics, Erzurum, Turkey b Kastamonu University, Faculty of Education, Department of Elementary Mathematics Education, Kastamonu, Turkey article info Keywords: Variational methods Optimal boundary control Hyperbolic problem abstract We investigate the problem of controlling the boundary functions in a one dimensional hyperbolic problem by minimizing the functional including the final state. After proving the existence and uniqueness of the solution to the given optimal control problem, we get the Frechet differential of the functional and give the necessary condition to the opti- mal solution in the form of the variational inequality via the solution of the adjoint prob- lem. We constitute a minimizing sequence by the method of projection of the gradient and prove its convergence to the optimal solution. Ó 2011 Elsevier Inc. All rights reserved. 1. Introduction We study the optimal control problem by minimizing the regularized functional J a ðwÞ¼ Z l 0 ½uðx; T ; wÞ yðxÞ 2 dx þ akw  mk 2 W ð1Þ for a > 0 associated with the following hyperbolic problem: u tt ¼ a 2 u xx þ F ðx; tÞ; ðx; tÞ2 X ¼ð0; lÞð0; T  ð2Þ uðx; 0Þ¼ uðxÞ; u t ðx; 0Þ¼ wðxÞ; x 2ð0; lÞ ð3Þ u x ð0; tÞ¼ gðtÞ; u x ðl; tÞ¼ hðtÞ; t 2ð0; T Þ: ð4Þ Here w(t): = {g(t),h(t)} is the control pair and we look for this element in the following admissible controls set: W :¼fw ¼fgðtÞ; hðtÞg : kwk W 6 bg L 2 ð0; T Þ L 2 ð0; T Þ: ð5Þ This set determines a Hilbert space with the following inner product: hw 1 ; w 2 i W ¼ Z T 0 ðg 1 g 2 þ h 1 h 2 Þdt; 8w 1 :¼fg 1 ; h 1 g; w 2 :¼fg 2 ; h 2 g2 W and the norm on it defined by the following equality: kwk W ¼ Z T 0 ðg 2 þ h 2 Þ dt  1=2 ; 8w :¼fg; hg2 W: 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.12.053 ⇑ Corresponding author. E-mail address: msubasi@atauni.edu.tr (M. Subas ßı). Applied Mathematics and Computation 218 (2012) 6629–6636 Contents lists available at SciVerse ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc