**Valery Y. Glizer**

Department of Mathematics, ORT Braude College of Engineering, Karmiel 2161002, Israel; valery48@braude.ac.il or valgl120@gmail.com

Received: 3 February 2019; Accepted: 17 March 2019; Published: 21 March 2019

**Abstract:** A singularly perturbed linear time-dependent controlled system with multiple point-wise delays and distributed delays in the state and control variables is considered. The delays are small, of order of a small positive multiplier for a part of the derivatives in the system. This multiplier is a parameter of the singular perturbation. Two types of the considered singularly perturbed system, standard and nonstandard, are analyzed. For each type, two much simpler parameter-free subsystems (the slow and fast ones) are associated with the original system. It is established in the paper that proper kinds of controllability of the slow and fast subsystems yield the complete Euclidean space controllability of the original system for all sufficiently small values of the parameter of singular perturbation. Illustrative examples are presented.

**Keywords:** singularly perturbed system; multiple state and control delays; controllability

**MSC:** 34K26; 93B05; 93C23

### **1. Introduction**

Differential systems with a small positive multiplier for a part of the highest order derivatives, called singularly perturbed differential systems, are adequate mathematical models for real-life processes with two-time-scale dynamics. In real-life problems, the small multiplier (a parameter of singular perturbation) can be a time constant, a mass, a capacitance, a geotropic reaction, and some other parameters in physics, chemistry, engineering, biology, medicine, etc (see e.g., [1–3] and references therein). An important class of singularly perturbed differential systems represents the systems with small time delays (of order the parameter of singular perturbation). Such systems arise in various real-life applications, for instance, in nuclear engineering [4], in botany [5], in physiology and medicine [6,7], in control engineering [8], and in communication engineering [9,10]. Distributed small delays appear, for instant, in stabilizing controls of singularly perturbed systems with small delays (either point-wise, or distributed, or point-wise and distributed) [11]. In such a case, a closed-loop system contains a distributed small delay. The stabilizing property of a distributed small delay also is used in the present paper (see Sections 3.3, 3.4 and 4.2).

Various topics in theory and applications of singularly perturbed controlled systems, without and with delays in state and control variables, were extensively investigated in the literature (see e.g., [1,12–14] and references therein).

Controllability of a system is one of its basic properties. This property means the ability to transfer the system from any position of a given set of initial positions to any position of a given set of terminal positions in a finite time by a proper choice of the control function. Different types of controllability for systems without or with delays were extensively studied in the literature (see e.g., [15–18] and references therein). To check whether a singularly perturbed system is controllable in a proper sense, the corresponding controllability conditions can be directly applied for any specified value of the small parameter *ε* > 0 of singular perturbation. However, the stiffness, as well as a possible high dimension of the singularly perturbed system, can considerably complicate this application. Moreover, such an application depends on the value of *ε*, and it should be repeated if this parameter changes. Furthermore, in most of real-life problems the current value of *ε* is unknown. These circumstances are crucial in the analysis of the controllability of singularly perturbed systems. They motivate the derivation of conditions, which being independent of *ε*, guarantee the controllability of a singularly perturbed system for all sufficiently small values of this parameter, i.e., robustly with respect to *ε*.

Controllability of singularly perturbed systems was analyzed in a number of works. Thus, in [19–22], the complete controllability of some linear and nonlinear undelayed systems was studied using the separation of time scales concept (see e.g., [1]). In [23] the robust complete Euclidean space controllability, as well as the controllability with respect to the slow state variable and with respect to the fast state variable, were studied for a linear standard singularly perturbed time-invariant system with a single nonsmall pointwise state delay. In [24,25], using the separation of time scales concept, parameter free conditions of complete Euclidean space controllability were obtained for linear standard singularly perturbed systems with pointwise and distributed small state delays. In [26], this result was extended to nonstandard singularly perturbed systems with multiple pointwise and distributed small delays in the state variables. In [27], parameter-free complete Euclidean space controllability conditions, which are not based on the separation of time-scales concept, were derived for a class of linear singularly perturbed systems with small state delays. In [28], a singularly perturbed linear time-dependent controlled system with a single small pointwise delay in the state and control variables was considered. Parameter-free conditions of the complete Euclidean space controllability were established for standard and nonstandard types of this system. In [29], a singularly perturbed linear time-dependent system with small state delays (multiple point-wise and distributed) was studied. Along with the set of time delay differential equations describing the dynamics of this system, a set of delay-free algebraic equations, describing the system's output, also was considered. Based on the separation of time-scales concept, different parameter-free sufficient conditions for the Euclidean space output controllability of this system were established. In [30], the complete Euclidean space controllability for one class of singularly perturbed systems with nonsmall delays (point-wise and distributed) in the state variables was studied. In [31], the defining equations method was used for analysis of the complete Euclidean space controllability of a linear singularly perturbed neutral type system with a single nonsmall pointwise delay. The particular cases of the Euclidean space output controllability, the controllability with respect to the slow state variable and the controllability with respect to the fast state variable, also were studied.

In the present paper, we consider a singularly perturbed linear time-varying system with multiple small point-wise delays and with small distributed delays in the state and control variables. The complete Euclidean space controllability of this system, robust with respect to *ε*, is studied. This study is based on a transformation of the complete Euclidean space controllability of the original system with delays in the state and the control to an equivalent output controllability of a new singularly perturbed system with only state delays. In the new system, the original control variable becomes an additional fast state variable. The Euclidean dimension of the slow mode equation in the new system is the same as in the original system, while the Euclidean dimension of the fast mode equation is larger than such a dimension in the original system. Further analysis is carried out based on the asymptotic decomposition of the original and transformed systems. Each system is decomposed into two much simpler *ε*-free subsystems, slow and fast ones. Equivalence of proper kinds of controllability of the slow subsystems, corresponding to the original and transformed systems, is established. Also, it is established the equivalence of proper kinds of controllability of the fast subsystems. Assuming the controllability of the slow and fast subsystems, associated with the transformed system, the Euclidean space output controllability of the latter is established for all sufficiently small values of *ε* > 0. Then, using the above mentioned equivalence of the controllability

of the original and transformed systems, as well as of their slow and fast subsystems, the complete Euclidean space controllability of the original system, robust with respect to *ε*, is deduced from the assumption on proper kinds of controllability of its slow and fast subsystems. Note that the original system of the present paper is much more general than the original system of [28]. Moreover, in the present paper we propose another, more general, approach to the analysis of the nonstandard case of the original system. Also, we propose here much simpler proof of the Euclidean space output controllability of the transformed system.

The paper is organized as follows. In the next section, the rigorous problem statement, the main definitions and the objective of the paper are formulated. Some auxiliary results, including the transformation of the original system, are presented in Section 3. Section 4 is devoted to main results of the paper. An illustrative example is solved in Section 5. Conclusions are placed in Section 6.

The following main notations are applied in the paper:


$$\left< x(\cdot), y(\cdot) \right>\_{L^2} = \int\_{t\_1}^{t\_2} x^T(t) y(t) dt;$$

the norm of any *<sup>x</sup>*(·) <sup>∈</sup> *<sup>L</sup>*2[*t*1, *<sup>t</sup>*2; *<sup>E</sup>n*] is defined as:

$$\|\|\mathbf{x}(\cdot)\|\|\_{L^2} = \left(\int\_{t\_1}^{t\_2} \mathbf{x}^T(t)\mathbf{x}(t)dt\right)^{1/2}.$$

