Variation formulas of solutions and optimal control problems for differential equations with retarded argument

Noether's symmetry Theorem for variational and optimal control problems with time delay

“…For a gentle introduction to control problems with time delay we refer the reader to the classical references [15,18]. Optimal control problems with delays in the state and control variables, subject to mixed-state and control-state constraints, are studied in [4,10,16].…”

Variation Formulas for the Solution of Delay Differential Equations with Mixed Initial Conditions and Delay Perturbations

“…Theorems 1.1 and 1.2 are proved by the method given in [1]. The following assertion is a corollary of Theorems 1.1 and 1.…”

“…If t 00 C 0 D t 10 ; then Theorem 1.1 is valid on the interval OEt 10 ; t 10 C ı 2 and Theorem 1.2 is valid on the interval OEt 10 ı 2 ; t 10 : Finally, we note that variation formulas for the solution of various classes of functional-differential equations without perturbations of delay can be found in [1][2][3][4][5][6][7][8].…”

Formulas of variation for solutions for some classes of functional differential equations and their applications

“…t 00−σ Y (ξ + σ ; t)B(ξ + σ )δg(ξ )dξ + β(t; δµ); the matrix functions Y (ξ ; t), Ψ (ξ ; t) satisfy the system Ψ ξ (ξ ; t) = −Y (ξ ; t)F x [ξ ] − (Y (ξ + τ ; t)F y 1 [ξ + τ ] Y (ξ + σ ; t)F z 1 [ξ + σ ]), Y (ξ ; t) = Ψ (ξ ; t) + (Y (ξ + τ ; t)A(ξ + τ ) Y (ξ + σ ; t)B(ξ + σ )), ξ ∈ [t 00 , t], Y (t; t) = Ψ (t; t) = I n×n ; Y (ξ ; t) = Ψ (ξ ; t) = Θ n×n , ξ > t. Theorems 1.1 and 1.2 are proved by the method described in [1,2].…”