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This work studied some classes of linear time-varying and constant coefficients composed of descriptor operator equations with consistent initial conditions as well as consistent boundary conditions using suitable Hilbert spaces. The composed descriptor operator is standing for coupled pair of differential-algebraic operator equations together with the consistent initial operator or consistent boundary operator defined in this work on reflexive Cartesian Hilbert spaces.We have proved that this composed descriptor operator is extendable to closed bounded densely defined operator, and has adjoint as well as second adjoint with natural extension property. Due to the importance of the symmetry of this operator for solvability, the symmetry has been also discussed and developed with respect to Cartesian bilinear form using the functional (Variational) approach. This approach is based on finding a suitable functional form whose critical point is the solution of the proposed problem (composed descriptor operator) and the solution of the proposed problem is a critical point of the obtained Variational functional.
