Relaxation Problems Involving Second-Order Differential Inclusions. (10th April 2013)

• Record Type: Journal Article
• Authors: Gomaa, Adel Mahmoud
• Other Names: Zizovic Malisa R. Academic Editor.
• Is Part Of: Abstract and applied analysis. Volume 2013(2013)
• Publication Date: 2013-04-10
• Subjects: Mathematical analysis -- Periodicals; Mathematical analysis; Applied Mathematics; Mathematical Analysis; Periodicals; 515.05
• DOI: 10.1155/2013/792431
• Languages: English
• ISSNs: 1085-3375

Abstract

Abstract : We present relaxation problems in control theory for the second-order differential inclusions, with four boundary conditions, u ¨ ( t ) ∈ F ( t, u ( t ), u ˙ ( t ) ) a.e. on [ 0, 1 ] ; u ( 0 ) = 0, u ( η ) = u ( θ ) = u ( 1 ) and, with m ≥ 3 boundary conditions, u ¨ ( t ) ∈ F ( t, u ( t ), u ˙ ( t ) ) a.e. on [ 0, 1 ] ; u ˙ ( 0 ) = 0, u ( 1 ) = ∑ i = 1 m - 2 ‍ a i u ( ξ i ), where 0 < η < θ < 1, 0 < ξ 1 < ξ 2 < ⋯ < ξ m - 2 < 1 and F is a multifunction from [ 0, 1 ] × ℝ n × ℝ n to the nonempty compact convex subsets of ℝ n . We have results that improve earlier theorems.