Relaxation Problems Involving Second-Order Differential Inclusions. (10th April 2013) - Gomaa, Adel Mahmoud
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.