Properties and computation of continuous-time solutions to linear systems
Introduction
According to the traditional notation, (resp. ) indicate complex (resp. real) matrices. Further, and denote the rank, the conjugate transpose, the range (column space) and the null space of . The index of is the minimal determined by and termed as .
About the notation and main properties of generalized inverses, we suggest monographs [2], [30], [42]. The Drazin inverse of is the unique which fulfills the matrix equationsThe group inverse coincides with in the case . The Moore-Penrose (M.P.) inverse of is the unique satisfying
A matrix which fulfilsis an outer inverse of with predefined range and null space and termed as .
The M.P. inverse the weighted M.P. inverse the Drazin inverse and the group inverse are particular outer inverses :where are positive definite and . The next statements are fulfilled with (see [2], [30]):
Applications of generalized inverses have been investigated in numerous studies. The Drazin inverse has been used in finite Markov chains, in the study of differential and singular linear difference equations [3], cryptography [18] etc. Also, generalized inverses show useful properties in solving system of linear equations (SoLE). It is a common fact that the minimum-norm least-squares solution of inconsistent linear equations is given by the M.P. solution (see [24]). This fact caused a renaissance in studying generalized inverses. The correlation between the generalized inverses and least squares solution is established via the well known fact that is smallest if and only if where . Later, this extremely important property was generalized to another types of generalized inverses. The minimum-norm least-squares solution to inconsistent system is given by the weighted M.P. inverse solution [7]. The unique solution to the restricted linear equations in the case is the Drazin inverse solution (see [2]). Moreover, the unique solution to the restricted linear equations is where is the weighted Drazin inverse, and (see [39]). In the most general case, is the solution to the restricted linear equations [5, Lemma 3.1], [6].
In all aspects of optimization and numerical analysis, the M.P. inverse solution to a linear system is indeed an important tool, which is commonly used in many practical fields, such as: image processing, economy, medicine etc. Later, the authors in [23] generalized the idea on Hilbert spaces and showed how it can be applied in order to compute -inverse or the M.P. inverse.
In the following, we restate results from [44] that establish Drazin-inverse solution’s minimal properties. Theorem 1.1 [44] Suppose that with matrix index equal to . The unique solution in ofis . Theorem 1.2 [43,44] Suppose that with matrix index equal to . All solutions of (1.3) are given by
Since the linear system (1.3) is analogous to the normal equationwe will call it the normal generalized equations ofThe vector is known as the Drazin-inverse solution to the system (1.6). Since the Drazin-inverse provides the system with a solution, the system (1.6) can be considered as Drazin-consistent system.
To that goal, applications of generalized inverses in solving linear systems have often been studied. The method of conjugate gradients was applied in computing the Drazin inverse solution of inconsistent linear system in the case when is Hermitian positive semidefinite [15]. Various semi-iterative methods for solving inconsistent linear systems were proposed in [9], [13] Moreover, a number of Krylov subspace methods for solving linear systems were considered in [1], [10], [11], [15], [17], [26], [27], [53]. A unified framework for Krylov subspace methods for arbitrary linear system was given in [25]. In addition, the Drazin inverse solution can be obtained using index splitting methods [40] as well as the extended Cramer rule [29]. The determinantal representation of was investigated in [46].
In the present paper, it is our purpose to investigate the properties of the inverse, in particular the -inverse solution to a consistent or inconsistent linear system.
The general structure of sections is as follows. Next Section 2 gives some preliminaries about the continuous-time approach in solving time-invariant linear systems (TILS) and time-varying linear systems (TVLS). The third section is aimed at the definition of the new method for finding the -inverse solution to a given linear system under some constraints. Various dynamical systems for computing -inverse solution are considered in Section 4. Main properties of the -inverse solution as well as the convergence result for our method are claimed. Properties of Zhang neural network (ZNN) models for finding -inverse solution of time-varying linear systems are considered in Section 5. Simulation examples are reported in the last Section 6.
Section snippets
Dynamical systems approach in solving linear systems
Our goal is to investigate application of generalized inverses in solving the TILS
The parallel-processing computational gradient neural network (GNN) model for solving (2.1) where is regular, was proposed in [31]. The model is also known as Wang neural network (WNN) model. It is necessary to consider the error function in vector form and the scalar error-monitoring goal functionwhere denotes the Frobeinus norm. Now, the GNN dynamic
Minimal properties of -inverse solution
We consider and which is generated by the request Lemma 3.1 Let and be generated according to conditions (3.1). Then the following statements can be stated: (a) The TILSis always consistent. (b) The solution set of (3.2) is defined by Proof (a) Since the TILS (3.2) is always consistent [21]. (b) The solution set of homogeneous SoLE is equal to . In addition, since
Dynamical systems for computing -inverse solution
This section is aimed to investigating various dynamical systems for computing -inverse solution. Section 4 applicability of -inverse solution is justified by theoretical results given in Section 3. Moreover, such solutions are the only possible approach in solving singular rectangular SoLE.
ZNN Models for solving TVLS
In this section we intend to apply the ZNN design to solve time-varying linear systems, both regular and singular, both consistent and inconsistent.
Numerical results
In order to demonstrate the efficiency of the proposed models, we will present several simulation examples in this section. Simulation are performed on the developed GNN and ZNN models, both in the time-invariant and in the time-varying case.
Conclusion
Our general scope is to solve the general singular or rectangular TVLS or TILS using both GNN or ZNN models.
- •
TILS case, i.e., and . a) is an singular matrix. Then we have two directions:
i) Usage of ZNN models is not possible, since they can only be applied to nonsingular systems. But, we can use the Tikhonov regularization of the system first, and then we can use the ZNN models in the regularized linear system.
ii) We can use GNN models.
Keep in mind that in order
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Cited by (0)
- 1
Predrag Stanimirović is supported from the Ministry of Education, Science and Technological Development, Republic of Serbia, Grant No. 174013.
- 2
Dijana Mosić is supported by the Ministry of Education, Science and Technological Development, Republic of Serbia, Grant No. 174007.