Variational methods for a class of nonlocal functionals

How to state necessary optimality conditions for control problems with deviating arguments?

“…At time t the value of the control solution of such phenomena depends to its value at time s for any s < t -cf. for instance [1,5,8,9,11,12,16]. For such problems the goal is to optimize, for instance some functional like…”

“…Such problems have been studied by many people: for instance we can see [5,9,11,17] and references of these works. If a solutionū of (4) exists, then we can ask the following natural question: how to establish EulerLagrange equation of (4)?…”

“…We use this terminology by analogy with the classical one. From [5,9,11], it is easy to see that necessary optimality conditions for (4) cannot be obtained in an easy way as the classical case. Let us denote by (x, η, ξ) the respectives arguments of f .…”

“…Let us denote by (x, η, ξ) the respectives arguments of f . In [5] under suitable assumptions, the authors state the following functional differential equation: …”

“…The singular character expressed by the Dirac measure is absent from the previous works. The method used in [5] is based on a formal abstract framework given in [2] -cf. also [3].…”