Abstract
Abstract We establish Euler-Lagrange equations for a problem of Calculus of Variations where the unknown variable contains a term of delay on a segment
Highlights
We consider the following problem of Calculus of Variations Minimize (P ) whenJ x (∈x)C:=0([−0Tr,FT(]t,Rxnt,)x′(t))dt x|[0,T ] ∈ C1([0, T ], Rn) x0 = ψ, x(T ) = ζ.where r, T ∈ (0, +∞), r < T, F : [0, T ] × C0([−r, 0], Rn) × Rn → R is a functional, ψ ∈ C0([−r, 0], Rn), ζ ∈ Rn, and xt(θ) := x(t + θ) when θ ∈ [−r, 0] and t ∈ [0, T ]
The aim of this paper is to establish a first-order necessary condition of optimality for problem (P ) which is analogous to the Euler-Lagrange equation of the variational problem without delay
In Section, we introduce function spaces and operators which are specific to the delayed functions and we establish several of their properties
Summary
The aim of this paper is to establish a first-order necessary condition of optimality for problem (P ) which is analogous to the Euler-Lagrange equation of the variational problem without delay. Note that in other settings of delay variational problems, the question of the establishment of an Euler-lagrange equation was studied, for instance in [8] (see references therein), [9], [2]. In Section, we introduce function spaces and operators which are specific to the delayed functions and we establish several of their properties.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
