Global Structure of Determining Matrices for a Class of Differential Control Systems

This paper developed and established unprecedented global results on the structure of determining matrices of generic double time-delay linear autonomous functional differential control systems, with a view to obtaining the controllability matrix associated with the rank condition for the Euclidean controllability of the system. The computational process and implementation of the controllability matrix were demonstrated on the MATLAB platform to determine the controllability disposition of a small-problem instance. Finally, the work examined the computing complexity of the determining matrices.


Introduction 1
Literature is awash with research activities on the determining matrices of autonomous linear hereditary control systems, due to the fact that they constitute the most efficient mechanism and vehicle for the investigation of the Euclidean controllability of the above class of control systems, with considerable savings in computational time, using the rank condition on the corresponding controllability matrices, being the least computationally intensive when compared to indices of control systems matrices and controllability Grammians. Amongst the works of notable authors. Ukwu (2016) Gabasov and Kirillova (1976) and eliminated any ambiguity that could arise in its application. The proof in his work relied on the results in  and (Ukwu, 2016a), which also incorporated the characterization of Euclidean controllability in terms of the indices of control systems and appropriated Taylor's theorem as an indispensable tool. The article provided an illustrative example of the computation of the controllability matrices and stated the implication of Euclidean controllability.  obtained the functional form of the determining matrices for the class of single-delay linear neutral autonomous control systems of the form: The control , is in the space, 0, , . 1  3  0  1 2  0  1 2  2  2  3  0  1  2  3  0  0  1  1 3   2  3  2 3 3      hh  This article adds to the body of knowledge by obtaining the structure of the afore-mentioned determining matrices for all pertinent global parameters and examining their computing complexity, thus filling the yawning gaps in Ikeh and Ukwu (2021).

Main Result of This Work
The main result of this research is given in the theorem below: where the permutations in expressions (2.1) and (2.2) are all feasible.

Proof:
To prove (2.1), (2.2) and (2.4) for the first three sets under the union operation in (2.4) of the theorem, it is enough to prove and then weed out all summation infeasibilities by setting infeasible summations to zero, in accordance with mathematical convention. Assertion 1 of (Ikeh & Ukwu, 2021) may then be invoked to conclude the proof of (2.

4). Expression (2.3) is simply a restatement of the initial condition (1.4).
For notational optimality, expression (2.5) can also be written in the form:   Page | 92 For the rest of the proof, the only remaining cases are constituted by the set These two results correspond to (i) and (iv) of the established preliminary results (3.1) in (Ikeh & Ukwu, 2021). Also, The first component of the right hand side of (2.9) can be rewritten as: The third component of (4.1) can be rewritten in the form:   oof.  Combining the above conditions, we deduce that the determining matrices for the free part of the system This result is consistent with the results for the same single-delay system in (Gabasov & Kirillova, 1976), (Manitus & Olbrot, 1976) and (Ukwu, 1992 This theorem is a modified version of that stated and proved in [7] to suit the system under consideration.

Illustrations of Non-Electronic Computations of
( )       ,2 The manual computations of this result are quite cumbersome. In general, there is an inherent computational intractability associated with these determining matrices. A resort to electronic implementation could be contemplated, but devising appropriate code for such an undertaking would task the ingenuity of even a professional computer programmer. However for small parameter instances, the implementation may be carried out on the MATLAB platform. The following section illustrates the implementation of determining matrices, controllability matrices and rank condition for Euclidean controllability for small parameter instances on the MATLAB platform.

Electronic Implementation of Controllability Matrix and Rank Condition for System (1)
This section is concerned with the prosecution of the above task for small problem instances on the MATLAB platform.