This study focuses on developing an efficient and easily implemented novel technique to solve the index- k Hessenberg differential-algebraic equation (DAE) with input control. The implicit function theorem is first applied to solve the algebraic constraints of having unknown state differential variables to form a reduced state-space representation of an ordinary differential (control) system defined on smooth manifold with consistent initial conditions. The variational formulation is then developed for the reduced problem. A solution of the reduced problem is proven to be the critical point of the variational formulation, and the critical points of the variational formulation are the solutions of the reduced problem on the manifold. The approximate analytical solution of the equivalent variational formulation is represented as a finite number of basis functions with unknown parameters on a suitable separable Hilbert setting solution space. The unknown coefficients of the solution are obtained by solving a linear algebraic system. The different index problems of linear Hessenberg differential-algebraic control systems are approximately solved using this approach with comparisons. The numerical results reveal the good efficiency and accuracy of the proposed method. This technique is applicable for a large number of applications like linear quadratic optima, control problems, and constrained mechanical systems.

This paper deals with finding an approximate solution to the index-2 time-varying linear differential algebraic control system based on the theory of variational formulation. The solution of index-2 time-varying differential algebraic equations (DAEs) is the critical point of the equivalent variational formulation. In addition, the variational problem is transformed from the indirect into direct method by using a generalized Ritz bases approach. The approximate solution is found by solving an explicit linear algebraic equation, which makes the proposed technique reliable and efficient for many physical problems. From the numerical results, it can be implied that very good efficiency, accuracy, and simplicity of the present approach are obtained.

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The concentration of this paper on studying the functional properties and the solvability of Boundary-Value Linear Composed time-varying Descriptor operator system defined on suitable reflexive Hilbert space, using (Variational) approach, which is based on constructing a appropriate functional form whose the critical-points are the solution of the proposed operator system and vice versa. The validity of this (Variational) approach is depends on the positivity and symmetry of the Boundary-Value Linear Composed time-varying Descriptor operator system on Hilbert space using Composed Cartesian product bilinear. The needful mathematical requirements are extracted and proved. Arithmetic Algorithms are proposed. Some illustration and computations between the exact solution and the suggested formula are also involved.