This paper concerns constrained dynamic optimization problems
governed by delay control systems whose dynamic constraints are described by both
delay-differential inclusions and linear algebraic equations. This is a new class of
optimal control systems that, on one hand, may be treated as a specific type of
variational problems for neutral functional-differential inclusions while, on the other
hand, is related to a special class of differential-algebraic systems with a general
delay-differential inclusion and a linear constraint link between “slow” and “fast”
variables. We pursue a twofold goal: to study variational stability for this class of
control systems with respect to discrete approximations and to derive necessary
optimality conditions for both delayed differential-algebraic systems under consideration
and their finite-difference counterparts using modern tools of variational analysis and
generalized differentiation. The authors are not familiar with any results in these
directions for such systems even in the delay-free case. In the first part of the paper
we establish the value convergence of discrete approximations as well as the strong
convergence of optimal arcs in the classical Sobolev space W-1,2. Then using discrete
approximations as a vehicle, we derive necessary optimality conditions for the initial
continuous-time systems in both Euler-Lagrange and Hamiltonian forms via basic
generalized differential constructions of variational analysis.