#### Abstract

This research investigates differential-algebraic equations with higher index (index four). Specifically, a functional analytic approach is proposed to find the solution of (index four) Hessenberg differential-algebraic equations (DAEs). The approximate solution of the proposed functional approach for a suitable separable Hilbert space is obtained with the help of the direct optimisation technique and Ritz basis functions method. Illustrations have been ranked from an 8 × 8 test system of index-four Hessenberg linear DAEs to a 4 × 4 DAE of rotating masses as well as an 8 × 8 differential-algebraic generator model, where it reformulated index-four linear Hessenberg DAEs. Their approximate solutions were obtained using the present approach with comparisons. The numerical results demonstrate the simplicity of the proposed approach and express suitable accuracy and efficiency.

#### 1. Introduction

Many life applications can be modelled as differential-algebraic systems, including optimal control problems, constrained robotic systems, and constrained electrical networks [1–3]. In the literature, the index property, of which several definitions exist, is usually utilised to characterise the differential-algebraic system [4, 5]. To determine an equivalent ordinary differential equation, the minimum number of times that the algebraic constraint of DAEs must be differentiated to time is represented with the frequently used differentiation index [5]. The higher index DAEs present a greater challenge in being solved than those in the lower index system because it makes applications more cumbersome [6]. In general DAEs, with a special structure, Hessenberg DAEs are an exceedingly important class implemented in various engineering and scientific applications [7, 8].

The authors in [2, 3, 9] presented numerical methods to solve index two and index three. These methods utilize the index reduction method to reduce the higher DAE index to be lower one [10–12].

To solve higher index (index four) Hessenberg linear DAE systems, this article develops an analytical approximation method to be efficiently and easily implemented. It is based on the functional analytical theory, which will help to determine a functional with critical points that serve as solutions for the fourth index Hessenberg DAEs and vice versa. To present the solution as a linear combination of the basic elements of the given setting space (i.e., separable Banach space), this work introduces a parameterisation approach inserted into the obtained variational formulation. Subsequently, to determine the solution of the variational formulation for unknown parameters, the method uses a direct method of calculus of variation. Finally, solving the linear algebraic system obtains the unknown parameters.

#### 2. Fourth-Order Hessenberg DAEs Problem Formulation

Index-four linear Hessenberg DAEs with control input are considered as follows:where(1) is a square invertible matrix, and denotes independent state vectors(2) are constant matrices of appropriate dimension, and is not required to be invertible or a square matrix(3) are suitable constant vectors, and is the given single control input with as the class of admissible control. ; and denotes differentiable functions,

Using the implicit function theorem, which can be observed in [13, 14], (1d) can be differentiated with respect to *t*:

From equation (1c), the following is obtained:

Now, in equation (3), a new constraint is conjugated with equations (1a)–(1d) (i.e., differential-algebraic system). The differentiability of equation (3) continues until the state is estimated to have a class of differential equations with states defined on a manifold of the resulting algebraic constraints.

Furthermore, (state variable) does not exist in equation (3); accordingly, equation (1c) can be differentiated (with respect to *t*) again.

Hence, another algebraic constraint (inserted into the differential-algebraic system (1a)–(1d)) is represented by equation (5). To determine the state variable and until the index-four condition () is satisfied, the derivation is continued:

Then, since is invertible,where

Since system (1) is the index-four Hessenberg DAEs system, the initial condition must be selected in such a way as to satisfy all algebraic constraints in equations (1d), (3), and (5). From the additional two hidden constraint equations (1c)–(1d) and from equation (7), over the class of , the following form is the reduced type index-four Hessenberg DAEs (1) with control

Set ={where (12)–(14) are satisfied. Ultimately, the following represents the class of consistent initial condition:

#### 3. Fourth Index Hessenberg DAE Solution via Variational Formulation Technique

The present approach is step by step represented as follows: Step 1: define the operator , with , where is an appropriate Hilbert space (e.g., , ) by . The operator L refers to a linear differential and algebraic operator can be written as follows: where , ,, ,, ,and , for all By using the variational formulation of obtained functional corresponding to (13) on D(*L*)={ satisfy (13)}. In almost everywhere in , the DAEs are satisfied by the linearly invariant (reduced type) index Hessenberg DAEs system. Furthermore, almost everywhere over the setting class , the solution should be a continuous function that satisfies equations (12)–(14)) Step 2: in equation (16), the operator is not symmetric to the usual bilinear form , because of the existence of the operator . To satisfy the nondegenerate condition on the range and the domain of operator , the bilinear form needs to be chosen (i.e., for every , then , and for every (then . Therefore, a usual variational formulation exists only if the usual bilinear (inner product) or the linear operator is redefined form using , where with , and is the complete Hilbert space (where *L* refers to the asymmetric linear operator and separable Hilbert space is denoted by ). The reader is recommended to [10, 11] for more details. is defined if *L* (on ) is positive definite; otherwise, it is needed to determine (the critical points). Over the class of the consistent initial conditions and , the augmented functional of can be defined as follows: where , and , , ,, , and Thus, the critical points of is guaranteed by the functional as the solution of equations (12)–(14) and vice versa when is chosen in the form of nondegenerate bilinear. Step 3: the critical points of equations (12)–(14) are the solutions of equation (12). Furthermore, the critical points are a solution of equations (12)–(14) by the nondegenerate property on . A countable linearly independent (bases) system , , ... etc.) (complete set of functions; base) can be used to approximate the solution, where positive number and is determined so that is extremized. Step 4: with respect to , can be derived. To determine these critical points as a function of parameters, *n* is given a suitable positive number. To obtain the system of a solvable linear algebraic, one can set . Furthermore, is a nonsingular matrix, where the coefficients can be determined from , if . Accordingly, , and the unique solution is guaranteed. Else, if is chosen as arbitrary, there may be infinitely solutions or no solution.

#### 4. Illustration Examples

##### 4.1. Test Example

The following test example is designed with a suitable exact solution in mind to confirm the validity of the aforementioned approach, where(1)The system is DAEs with control(2)The system is represented as fourth order Hessenberg linear DAE with controlwith

Note that is an invertible matrix.

Significantly, in the DAE system, variable does not appear. Therefore, the algebraic constraint equations (18)–(23) can be differentiated with respect to *t*. Then, to obtain the following, substitute the differential constraints to equations (24) and (25).

Thus, equation (27) is considered a new constraint inserted to equations (18)–(25). In the last constraint, variable is not exist. Therefore, to obtain the following, equation (27) can be differentiated one more time with respect to *t*.where .

Next, one can differentiate equation (28) again for the *t* set: . Step 1: is estimated as , where the derivative of the class of the consistent initial conditions can be Step 2: as can be seen in the following, the definition of the variational function with the class of the consistent initial conditions can be written as follows: where , , , , . Step 3: one can set , with to obtain Step 4: as mentioned above, the functional is a function of . As is extremized, one can set , , , , and It leads to the linear algebraic equation , which is solvable directly for . By deriving with respect to ,, and , (the matrix coefficient) can be determined. Hence, the approximate solution is obtained. The numerical values of the unknown coefficients are listed in Table 1.

Figure 1 depicts the numerical solutions with equality states compared with the exact solution , for a given

##### 4.2. Application of the Generator Model

Consider the model of a generator described with the following parameters. On the right axis, the input is the angle connected to mass with inertia which is rotated about the angle with angular velocity The torque acting on the right and left sides are and , respectively. The mass is then connected to the second axis with a generator. The electrical quantities and the variables described in the second axis are then assumed as in [15]. The rest of the electrical circuit includes one inductor and resistor. The model is illustrated in Figure 2.

The system is now modelled using the equations representing the different parts and their connections.with ,and is an invertible matrix.

In the DAE system, variable does not appear. Therefore, the algebraic constraint equation (31) can be differentiated with respect to *t*. Then, three times until is yielded in the following:where .

Next, one can differentiate equation (27) again for the *t* and set .: Step 1: it is mentioned above that is obtained as , The derivative of the class of the consistent initial conditions is as follows: Step 2: the following is the definition of the variational function with the class of the consistent initial condition. where , , , , . Step 3: one can set , with to obtain Step 4: is a function of , and as is extremized, one can set , , , , and . It is led to the linear algebraic equation , which is solvable directly for . Hence, the approximate solution is determined. Lastly, the derivative of with respect to ,, and is used to obtain the matrix coefficient . Step 5: to evaluate the solution’s accuracy without knowing the exact solution, the model utilises -norm by substituting the values of , , , and in system (31). The error presented in the test example at each point along time and for different kinds of are displayed (Figures 3–6).

##### 4.3. Application of the Rotating Masses System

Consider a system of two rotating masses that are described by the torques and the angular velocities and . The masses have the moments of inertia and . Figure 7 illustrates this scenario.

The system to matrically describes this model is provided as