On the regularization and matrix Lyapunov functions for fuzzy differential systems with uncertain parameters

1.   Introduction

One of the main assumptions in the classical stability of motion theory ^[1,2] is the assumption of the invariance of the parameters of a system in the course of its movement.

Further development of the stability theory is associated with the investigation of dynamical systems with uncertain parameters. See, for example, ^[3] and some of the references therein, as well as some recent results on models with uncertain parameters ^[4,5,6,7]. The active research on the stability analysis for systems with parameter uncertainties shows the importance of the topic for practice. In fact, many real-world problems exhibit different types of uncertainties. That is why there are numerous studies that have addressed the elaboration of power techniques to investigate of the effect of parameter uncertainties ^{[8,9,10,11,12]}.

One of the main tools in the study of the stability behavior of different classes of systems with uncertain parameters are the scalar and vector Lyapunov-type of functions ^{[13,14,15,16,17]}. The method of matrix-valued Lyapunov functions for the stability analysis of the solutions of continuous systems of differential equations has been developed and outlined in the monographs ^[18,19].

Due to the benefits for theory and applications, the Lyapunov function strategy has attracted much attention from researchers of fuzzy systems of differential equations, and it is intensively applied in their stability analysis. For example, the book ^[20] offers an excellent overview of the state-of-the-art research on the theory of fuzzy differential equations and inclusions and provides a systematic account of the developments in their stability analysis. The stability theory for fuzzy differential equations based on Lyapunov functions has also been developed in numerous articles ^{[21,22,23,24,25]}. However, the method of matrix Lyapunov functions has not been elucidated for such systems, which is one of the main goals of this research.

In addition, for fuzzy systems with uncertain parameters, the development of new methods for the qualitative analysis of stability and boundedness remains an open problem.

In our earlier paper ^[26] we introduced a regularization scheme for systems of fuzzy differential equations with uncertain parameters as a new approach in the study of the properties of such systems. The proposed strategy reduces a collection of fuzzy differential equations to a simple form that allows for analysis of the properties of solutions of both the original fuzzy system of differential equations, as well as the reduced collection of differential equations. The idea is to use a family of mappings and regularize the fuzzy systems with respect to uncertain parameters.

In this paper, using the proposed approach in ^[26], sufficient conditions for the stability and boundedness of the equilibria of the regularized fuzzy system are proposed through the application of the Lyapunov function method on the basis of matrix-valued functions and non-linear integral inequalities. The established results contribute to the development of new methods for the study of fuzzy differential systems with uncertain parameters. This research also adopts the matrix-valued Lyapunov function strategy to their qualitative analysis.

The contributions of our paper are as follows:

(ⅰ) using a new regularization scheme for uncertain fuzzy differential equations with uncertain parameter we establish stability criteria for the regularized system via the matrix-valued Lyapunov function technique;

(ⅱ) stability analysis for autonomous comparison problems is proposed and new stability results for autonomous systems are established;

(ⅲ) boundedness and Lagrange stability criteria for the regularized system are proved;

(ⅳ) the results offered show that the regularization scheme is a very advantageous technique which allows for analysis of the qualitative properties of solutions of both the original fuzzy system of differential equations, as well as the intermediate families of differential equations in a simple way.

The investigation is organized according to the following plan. In Section 2 some important notes on fuzzy sets and functions are provided. Section 3 is devoted to the new regularization approach developed in ^[26]. Matrix-valued Lyapunov functions and some of their properties are considered in Section 4. In Section 5 stability, uniform stability and asymptotic stability results for the regularized system are established. In Section 6, a stability analysis for autonomous comparison problems is conducted. Section 7 includes the boundedness results. Finally, Section 8 offers some comments and future directions for research.

2.   Notes on fuzzy sets and functions

We assume that $X$ is a basic set, and that, for any $x \in X$, $\varphi(x)$ is a membership function that takes its values from the interval $[0, 1]$. Following ^[27,28,29] for a fuzzy set with a membership function $\varphi$ on $X$ its $\kappa$-level sets $[\varphi]^{\kappa}$ are defined as

and its support is given by

We will use the following notation for the Hausdorff distance between two sets ${ \Xi}, { \Pi} \in \mathbb R^n$ and ${ \Xi}, { \Pi} \neq \emptyset$:

where ${ \Pi}_H (0) = \{x \in \mathbb{R}^n : \|x\| < H\}, \ H \geq 0$.

The defined Hausdorff distance $d_H({ \Xi}, { \Pi})$ is a metric for any nonempty closed sets in $\mathbb{R}^n$. In addition, the pair $(C^n, d_H)$ is a metric space, where $C^n$ is the set of all nonempty closed sets in $\mathbb{R}^n$.

We will denote by $E^n$ the space of all functions ${ \varphi}:\mathbb R^n\rightarrow [0, 1]$ such that

$1)$ ${ \varphi}$ is upper semicontinuous in the sense of Baire;

$2)$ there exists an $x_0\in\mathbb{R}^n$ such that ${ \varphi}(x_0) = 1$;

$3)$ $u$ is fuzzy convex, i. e., ${ \varphi}(\lambda x+(1-\lambda)y)\ge \min \{{ \varphi}(x), { \varphi}(y)\}$, $\lambda\in[0, 1]$;

$4)$ the closure of the set $\{x\in\mathbb R^n: { \varphi}(x) > 0\}$ is a compact subset of $\mathbb{R}^n$.

It is well known ^[20,26,29] that, if a fuzzy set with a membership function $\varphi$ is a fuzzy convex set, then $[\varphi]^\kappa$ is convex in $\mathbb{R}^n$ for any $\kappa\in[0, 1]$.

The distance between two sets ${ \varphi}, { \varpi} \in E^n$ will be given as $d({ \varphi}, { \varpi}) = \sup\{| { \varphi}(x)-{ \varpi}(x)|\colon \ x\in \mathbb{R}^n\}$; also, the least upper bound of the metric $d$ on the space $E^n$ is defined by $d[{ \varphi}, { \varpi}] = \sup\{d_H([{ \varphi}]^\kappa, [{ \varpi}]^\kappa)\colon \ \kappa\in[0, 1]\}$ for ${ \varphi}, \, { \varpi}\in E^n$, and it is a metric in $E^n$.

For any two sets ${ \varphi}, { \varpi}\in E^n$ the element ${ \iota} \in E^n$ such that ${ \varphi} = { \varpi}+{ \iota}$, which, if it exists, is the Hukuhara difference of ${ \varphi}$ and ${ \varpi}$ and is denoted by ${ \varphi}-{ \varpi}$.

The family of all nonempty compact convex subsets of $\mathbb{R}^n$ will be denoted by $P_k(\mathbb{R}^n)$.

The integral of a mapping $F$ on a compact interval ${ T = [t_1, t_2], \ t_2 > t_1} > 0$ is denoted by $\int\limits_a^bF(t)\, dt$; for any $0 < \kappa\leq 1$, it is given as

The mapping $F\colon \, T\to E^n$ is said to be differentiable at $t_0\in T$ if the value $F'(t_0)$ exists and $F'(t_0)\in E^n$ is such that both limits $\lim\{[F(t_0+h)-F(t_0)]h^{-1}:h\to 0^+\}$ and $\lim\{[F(t_0)-F(t_0-h)]h^{-1}:h\to 0^+\}$ exist and are equal to $F'(t_0)$. The above limits are considered in the metric space $(E^n, d)$.

If $F_\kappa$ is differentiable, the mapping $F_\kappa$ is differentiable in the sense of Hukuhara for all $\kappa\in[0, 1]$ and $D_HF_\kappa(t) = [\, F'(t)\, ]^\kappa,$ where $D_HF_\kappa$ is the Hukuhara-type derivative of $F_\kappa$.

The family $\{D_H F_{\kappa}(t):\kappa\in[0, 1]\}$ determines an element $F'(t)\in E^n$. Also, if $F:T\rightarrow E^n$ is differentiable at $t\in T$, then the element $F'(t)$ is called the fuzzy derivative of $F(t)$ at the point $t$.

More on the concepts of fuzzy sets and functions is available in ^[30,31,32] and some of the references therein. For important results related to fuzzy differential equations, see ^{[33,34,35,36,37]}.

3.   Regularization procedure

Throughout the entire paper we consider the following fuzzy system with an uncertain parameter

where $u\in E^{n^2}$; $f\in C(\mathbb{R}_+\times E^{n^2}\times \mathcal S, {E}^{n^2})$; $E^{n^2} = E^n \times E^n$; $\alpha\in\mathcal S$ is an uncertain parameter; $\mathcal S$ is a compact set in $\mathbb{R}^d$.

{As it is stated in ^[26], the parameter vector $\alpha$ represents the uncertainty in system (3.1). It can have a different nature and may represent different characteristics. For example, the uncertainty parameter $\alpha$

$({\rm{a}})$ may describe an uncertain value of a certain physical parameter;

$({\rm{b}})$ may represent an estimate of an external disturbance;

$({\rm{c}})$ may describe an inaccurate measured value of the input effect of one of the subsystems on the other one;

$({\rm{d}})$ may represent some nonlinear elements of the considered mechanical system that are too complicated to be measured accurately;

$({\rm{e}})$ may be an indicator of the existence of some inaccuracies in the system (3.1);

$({\rm{f}})$ may be a union of the characteristics (a)–(e).

To regularize the system (3.1) with respect to the uncertain parameter $\alpha$, in ^[26] we consider a family of mappings $f_\kappa(t, v)$ defined by

where

Now, we propose to consider the following regularized with respect to $\alpha \in \mathcal S$ system of fuzzy differential equations of the type

where $f_\kappa\in C(I\times E^{n^2}, \, E^{n^2})$, $\, I = [t_0, \, t_0+{ \tau}]$, $\, t_0\ge0$, $\, { \tau} > 0$, $\kappa\in[0, 1]$.

For the regularized system (3.5) we assume that $f_m(t, v)$, $f_M(t, v)\in C[\mathbb{R}_+\times E^{n^2}, E^{n^2}]$. The solutions of the initial value problem (IVP) (3.5) are weakly continuous mappings $v: \ I\to E^{n^2}$ which satisfy the integral equation

The proposed regularization scheme is a method that can help to reduce the analysis of the properties of the fuzzy system (3.1) with the uncertain parameter $\alpha \in \mathcal S$ to those of the regularized system (3.5).

4.   Matrix-valued Lyapunov functions

We define a matrix-valued function

with entries $u_{ij} \in \mathbb{R}$ that correspond to the family of regularized equations (3.5) as follows:

$1)$ If $\kappa = 0$, then the entry $u_{11}(t, v_1)\in C(\mathbb{R}_+ \times E^{n^2}, \ \mathbb{R}_+)$ corresponds to the fuzzy equation

$2)$ If $\kappa = 1$, then $u_{22}(t, v_2) \in C(\mathbb{R}_+ \times E^{n^2}, \ \mathbb{R}_+)$ corresponds to the fuzzy equation

$3)$ If $0 < \kappa < 1$, then the entries $u_{12}(t, v) = u_{21}(t, v)\in C(\mathbb{R}_+ \times E^{n^2}, \ \mathbb{R})$ correspond to the fuzzy regularized system

where $f_{\kappa} \in C(\mathbb{R}_+ \times E^{n^2}, \ E^{n^2})$.

This way the fuzzy equations (4.2)–(4.4) become the base in the construction of the matrix-valued Lyapunov-type function (4.1).

Example 4.1. As a very simple example of a matrix-valued Lyapunov function, consider a function $U(t, \cdot)$ with the following entries:

where $v_1, v_2 \in E^{n^2}$ and the state $\theta_0 \in E^{n^2}$ is defined as

Now, on the basis of the matrix function (4.1), and by means of a vector $\eta \in \mathbb{R}_+^2$, we construct a scalar Lyapunov-type function as follows

where $V \in C(\mathbb{R}_+ \times E^{n^2} \times \mathbb{R}_+^2, \ \mathbb{R}_+)$.

Note that, in general, the vector $\eta$ can be defined in one of the following ways:

$\mbox{(a)}$ $\eta = y \in \mathbb{R}^2$, $y \neq 0$;

$\mbox{(b)}$ $\eta = \xi \in C(\mathbb{R}^2, \ \mathbb{R}_+^2)$, $\xi(0) = 0$;

$\mbox{(c)}$ $\eta = \psi \in C(\mathbb{R}_+ \times \mathbb{R}^2, \ \mathbb{R}_+^2)$, $\psi(t, 0) = 0$;

$\mbox{(d)}$ $\eta \in \mathbb{R}_+^2$, $\eta > 0$.

Together with the defined scalar function (4.5), we will also use the following vector function, given by

where $A$ is a constant $(2\times2)$ matrix, $\mathcal L = (\mathcal L_1, \mathcal L_2)^T$ and $\mathcal L \in C(\mathbb{R}_+ \times E^{n^2} \times \mathbb{R}_+^2, \ \mathbb{R}^2_+)$.

We will introduce the following concepts related to the function (4.1) based on the function (4.5).

Definition 4.2. The matrix-valued function $U : \mathbb{R}_+ \times E^{n^2} \rightarrow \mathbb{R}^2$ is said to be

$1)$ positive semi-definite if there exist a neighborhood $D(\rho)$ of the state $\theta_0$, $D(\rho) = \{ v \in E^{n^2}: d[v, \theta_0] < \rho\}$, $0 < \rho < + \infty$ and a vector $\eta \in \mathbb{R}_+^2$ such that

$\mbox{(a)}$ the function $V(t, v, \eta)$ is continuous on $\mathbb{R}_+ \times D(\rho) \times \mathbb{R}_+^2$;

$\mbox{(b)}$ the function $V(t, v, \eta)$ is nonnegative for all $(t, v, \eta)\in \mathbb{R}_+ \times D(\rho) \times \mathbb{R}_+^2$;

$\mbox{(c)}$ $V(t, 0, \eta) = 0$ for all $t \in \mathbb{R}_+$ and $\eta \in \mathbb{R}_+^2$;

$2)$ positive definite if it is positive semi-definite and there exists a positive definite function $w(v): D(\rho) \rightarrow \mathbb{R}_+$ such that $w(v) \leq V(t, v, \eta)$ for all $(t, v, \eta)\in \mathbb{R}_+ \times D(\rho) \times \mathbb{R}_+^2$;

$3)$ decreasing if there exists a positive definite function $z(v): D(\rho) \rightarrow \mathbb{R_+}$, such that $V(t, v, \eta) \leq z(v)$ for all $(t, v, \eta)\in \mathbb{R}_+ \times D(\rho) \times \mathbb{R}_+^2$;

$4)$ radially unbounded if $d[v, \theta_0]\rightarrow \infty$ implies that $V(t, v, \eta) \rightarrow +\infty$ for all $t \in \mathbb{R}_+$ and $\eta \in \mathbb{R}_+^2$.

The notions of negative semi-definite and negative definite matrix-valued functions can be defined analogously.

Together with the consideration of Lyapunov-type functions (4.5) and (4.6), we will also introduce their total derivatives with respect to the regularized system (3.5), as follows:

where $D^+ U(t, v) = \lim \sup \{[U(t+h, \ v+h f_{\kappa}(t, v))-U(t, v)]h^{-1}: h \rightarrow 0^+\}$.

Example 4.3. For the function considered in Example 4.1,

we have

where

The application of the function $U(t, v)$ and its total derivative allows for the investigation of the dynamical properties of the fuzzy system with uncertain parameters, as given by (3.1) on the basis of the simple fuzzy equations (4.2)–(4.4).

4.1.   Scalar comparison principle

In this subsection, we will consider the family of regularized equations (3.5) for $(t, v)\in\mathbb{R}_+\times D(\rho)$. Let $f_\kappa\in C(\mathbb{R}_+\times D(\rho), \, E^{n^2})$ for any $\kappa\in[0, 1]$ and the solution $v(t)$ of the IVP (3.5) be defined on $[t_0, \infty)$. For the regularized system (3.5), we will present a scalar comparison principle by using the function (4.5).

Theorem 4.4. Suppose that for the regularized system (3.5) there exist a matrix-valued function $U(t, \cdot)$ and a vector $\eta\in \mathbb{R}_+^2$ such that

1) The function $V(t, v, \eta)\in C(\mathbb{R}_+\times D(\rho) \times \mathbb{R}_+^2, \, \mathbb{R}_+)$ and there exists a constant $L > 0$ such that

2) There exists a function $g(t, \omega)$, $g\in C(\mathbb{R}_+^2, \, \mathbb{R})$ such that, for any $\kappa\in[0, 1]$

3) The maximal solution $r_M(t; t_0, \omega_0)$ of the scalar comparison equation

exists on $[t_0, \infty)$.

Then, $V(t_0, v_0, \eta)\leq\omega_0$ implies that

Proof. Since the solution $v(t)$ of the IVP (3.5) is defined on $[t_0, \infty)$, for $t \in [t_0, \infty)$, we set $m(t) = V(t, v(t), \eta)$ so that $m(t_0)\leq\omega_0$ and evaluate the difference $m(t+h)-m(t)$ for a sufficiently small $h > 0$. We have that

From the above estimate, we obtain

Let $v(t+h) = v(t)+z(t)$, where $z(t)$ is the Hukuhara difference for a sufficiently small $h > 0$. According to the properties of the metric $d[u, v]$, we get

and, hence

Taking the limit in (4.9), we obtain

along any solution $v(t)$ of (3.5).

Taking into account (4.10), the estimate (4.8) takes the following form:

to which we apply Theorem 1.5.1 ^[14] and get

for all $t\geq t_0$. This proves Theorem 4.4.

We will now state some corollaries of Theorem 4.4. The Hahn class $K$ of continuous and strictly increasing functions of the corresponding dimension that are zero at zero will be used ^[14].

Corollary 4.5. Suppose that, in Theorem 4.4, instead of condition 2, we have the following:

$2'$) $D^+V(t, v, \eta)\leq 0, \ (t, v, \eta)\in \mathbb{R}_+\times D(\rho) \times \mathbb{R}_+^2$.

Then,

Corollary 4.6. Suppose that, in Theorem 4.4, instead of condition 2, we have the following:

$2''$) $D^+V(t, v, \eta)\leq -a(\omega(t, v))+g(t, V(t, v, \eta)), \ (t, v, \eta)\in \mathbb{R}_+\times D(\rho) \times \mathbb{R}_+^2$,

where $\omega\in C(\mathbb{R}_+\times D(\rho), \, \mathbb{R}_+)$, $a\in K$ and $g(t, \omega)$ is a function decreasing on $\omega$ for any $t\in \mathbb{R}_+$.

Then, $V(t_0, v_0, \eta)\le\omega_0$ implies that

4.2.   Quasilinear fuzzy systems

We represent the regularized system (3.5) in the following form:

where $g_{\kappa} \in C(\mathbb{R}_+ \times E^{n^2})$ and $A_{\kappa}(t) : [t_0, \infty) \rightarrow E^{n^2}$ for any value of $\kappa \in [0, 1]$ is a semi-linear operator such that

$\mbox{(a)}$ $A_{\kappa}(t)(u+v) = A_{\kappa}(t)u + A_{\kappa}(t)v, \ u, v\in E^{n^2}$;

$\mbox{(b)}$ $A_{\kappa}(t)(\nu u) = \nu A_{\kappa}(t) u$, $\nu \in \mathbb{R}_+$, $u \in E^{n^2}$.

Assume that the solution of the problem described by (4.12) and (4.13) is well defined for $t \geq t_0$ and the operator $A_{\kappa}(t)$ is contracting, i.e., there exists $0 < \gamma < 1$ such that

In addition, for a sufficiently small $h > 0$ the operator

exists for $t \in \mathbb{R}_+$, $\kappa \in [0, 1]$ and $u \in E^{n^2}$, and it satisfies

The comparison principle for the quasilinear regularized problem described by (4.12) and (4.13) is represented as follows.

Theorem 4.7. Suppose that for the regularized system (4.12), there exists a scalar Lyapunov-type function $V(t, v, \eta)$ that satisfies the following conditions:

1) $V(t, v, \eta)\in C(\mathbb{R}_+\times D(\rho) \times \mathbb{R}_+^2, \mathbb{R}_+)$ and there exists a continuous on $\mathbb R_+$ function $L(t) \geq 0$ such that

2) There exists a function $G(t, \omega)$, $G\in C(\mathbb{R}_+^2, \, \mathbb{R})$, such that for any $\kappa\in[0, 1]$

3) The maximal solution $r_M(t; t_0, \omega_0)$ of the comparison equation

exists on $[t_0, \infty)$.

Then, $V(t_0, v_0, \eta) \leq \omega_0$ implies that

The proof of Theorem 4.7 is obtained in a similar way as the proof of Theorem 4.4 taking into account the fact that

From (4.18), we have the following for $m(t) = V(t, v (t), \eta)$:

The above estimate, (4.14), (4.15) and condition 2 of Theorem 4.7 yield

Next, we apply Theorem 3.1.1 ^[14] to obtain (4.17).

Corollary 4.8. Suppose that, in Theorem 4.7, the functions $V(t, v, \eta) = d[v, \theta_0]$ and $G = G(t, d[v, \theta_0])$, $t \in \mathbb R_+$, $v \in E^{n^2}$, $\eta \in \mathbb R_+^2$, where $\theta_0 \in E^{n^2}$ is the state defined in Example 4.1.

Then, $d [v_0, \theta_0]\leq \omega_0$ implies that

4.3.   Vector comparison principle

Recall that a function $G(t, \omega)$, $G: \mathbb R_+ \times \mathbb R_+^2\to \mathbb R^2$ is quasimonotonic with respect to its second variable $\omega$ if $\omega_1\le\omega_2$ and $\omega_{i1} = \omega_{i2}$ for $1\le i\le 2$ imply that $G(t, \omega_1)\le G(t, \omega_2)$ for any two $\omega_1, \omega_2\in \mathbb{R}^2_+$ and $t \in \mathbb R_+$.

In the case that $G(t, \omega) = A\omega$, where $A$ is a $(2\times 2)$ matrix with entries $a_{ij}$, the function $G(t, \omega)$ is quasimonotonic if $a_{ij}\ge0$ for $i\ne j$.

Theorem 4.9. Suppose that, for the regularized system (3.5), there exist a matrix-valued function $U(t, \cdot)$ and a vector $\eta\in \mathbb{R}_+^2$ such that the function (4.6) satisfies the following conditions:

1) $\mathcal L(t, v, \eta)\in C(\mathbb{R}_+\times D(\rho) \times \mathbb{R}_+^2, \, \mathbb{R}_+^2)$ and there exists a constant $L > 0$ such that

where $D[v_1, v_2] = (d[v_1, u_1], \, d[v_2, u_2])^T$ and $||.||$ is the vector norm in $\mathbb{R}^2$;

2) There exists a quasimonotonic with respect to $\omega$ function $G(t, \omega)\in C(\mathbb{R}_+\times \mathbb{R}_+^2, \mathbb{R}^2)$, $G(t, 0) = 0$, such that, for any $\kappa\in[0, 1]$,

3) The maximal solution $r_M(t; t_0, \omega_0)$ of the vector comparison equation

exists on $[t_0, \infty)$.

Then, $\mathcal L(t_0, v_0, \eta)\leq\omega_0$ implies that

The proof of Theorem 4.9 is similar to that of Theorem 4.4, so we omit it here.

Corollary 4.10. Suppose that, in Theorem 4.9, the function $G = G(t, \omega) = A \omega$, $t \in \mathbb R_+$, where $A$ is a $(2\times 2)$ matrix with entries $a_{ij}\ge0$ for $i\ne j$.

Then,

5.   Main stability analysis

Recall that the function $\text{diam}\, [u(t)]^{\kappa}$ is nondecreasing as $t\to\infty$. Hence, the direct application of the metric $\|u(t)\|$ in the stability analysis of the regularized system (3.5) is not suitable for its dynamical properties. For this reason, in our research we will introduce the following assumptions:

$H_1$. For any value of the uncertain parameter $\alpha\in\mathcal S$ the system (3.5) has a steady state $\theta_0$ such that $f_{\kappa}(t, \theta_0) = \theta_0$ for all $t\in [t_0, \infty)$.

$H_2$. For the initial value $v_0\in E^{n^2}$ and any $y_0\in E^{n^2}$ there exists a Hukuhara difference $v_0-y_0 = w_0$.

$H_3$. The solution $v(t) = v(t; t_0, v_0)$ of (3.5) exists for all $t\ge t_0$ and is unique for any $\kappa \in [0, 1]$.

Next, we introduce the following stability notions for the steady state $\theta_0$.

Definition 5.1. The state $\theta_0$ of (3.5) is:

$\mathcal{S}_1$ equi-stable if, for any $t_0\in\mathbb{R}_+$ and $\varepsilon > 0$ there exists $\delta = \delta(\varepsilon, t_0)$ such that for any initial data $v_0 \in E^{n^2}$ with $d[v_0, \theta_0] < \delta$ we have that $d[v(t; t_0, v_0), \theta_0] < \varepsilon$ for all $t\ge t_0$ and all $\kappa \in [0, 1]$;

$\mathcal{S}_2$ uniformly stable if $\delta$ in $\mathcal{S}_1$ does not depend on $t_0$;

$\mathcal{S}_3$ quasi-equi-asymptotically stable if for any $t_0\in\mathbb{R}_+$ and any $\xi > 0$ there exist $\delta_0(t_0, \xi) > 0$ and $\tau(t_0, \xi)\in\mathbb{R}_+$ such that $d[v_0, \theta_0] < \delta_0(t_0, \xi)$ implies that $d[v(t; t_0, v_0), \theta_0] < \xi$ for all $t\ge t_0+\tau(t_0, \xi)$ and $\kappa \in [0, 1]$;

$\mathcal{S}_4$ uniformly quasi-asymptotically stable if $\delta_0$ and $\tau$ in $\mathcal{S}_3$ are independent on $t_0$;

$\mathcal{S}_5$ equi-asymptotically stable if $\mathcal{S}_1$ and $\mathcal{S}_3$ hold simultaneously;

$\mathcal{S}_6$ uniformly asymptotically stable if $\mathcal{S}_2$ and $\mathcal{S}_4$ hold simultaneously;

$\mathcal{S}_7$ uniformly exponentially stable if for an arbitrary solution $v(t; t_0, v_0)$, we have

where $\beta(d):[0, { R}]\rightarrow \mathbb{R}_+$ is nondecreasing on $d$ for some ${ R} > 0$ and $\lambda > 0$ is a constant.

Example 5.2. Consider on $E^2$ the following fuzzy equation

where $(\mu\neq 0)\in[-1, 1]$, $\mu = \mu(\kappa)$, which we represent as follows:

for $\kappa\in[0, 1]$. At the same time, for an initial value $v_0\in E^2$ on the level $[v_0]^\kappa = [v_{10}, v_{20}]^\kappa$ for $\kappa\in[0, 1]$ the general solution of (5.2) has the form

for all $0\le \kappa \le1$ and $t\ge 0$.

It follows from (5.3) that at all $\kappa$-levels, the zero solution of (5.1) is unstable, even for sufficiently small values of $v_{10}$, $v_{20}$ for any $\kappa \in [0, 1]$. At the same time, the zero solution is stable under some additional conditions for the initial data $[v_0]^\kappa = [v_{10}, v_{20}]^\kappa$. Particularly, if $v_{10}+v_{20} = 0$ for $0 < \mu < 1$ or if $v_{10}-v_{20} = 0$ for $-1 < \mu < 0$, and any $\kappa\in[0, 1]$, the relations given by (5.3) imply that $\left[v_1(t)\right]^{\kappa}$ and $\left[v_2(t)\right]^{\kappa}$ are decreasing for $t\rightarrow \infty$. These conditions are equivalent to the existence of the Hukuhara difference for $[v_{10}, v_{20}]^\kappa$, $\kappa\in[0, 1]$.

We will now establish some stability criteria for the stationary state $\theta_0$ of (3.5) based on a scalar auxiliary function of the type (4.5). Functions of the class $CK = \{ a\in{C[\mathbb R_+^2, \mathbb R_+]}: a(t, u) \in K$ for each $t \in \mathbb R_+$ and $a(t, { \wp}) \to \infty$ as ${ \wp} \to \infty \}$ will also be used.

Theorem 5.3. Suppose that, for the regularized system (3.5), conditions $H_1$–$H_3$ hold, and that there exist a matrix-valued function $U(t, \cdot)$ and a vector $\eta\in \mathbb{R}_+^2$ such that the function (4.5) satisfies the following conditions:

1) $V(t, v, \eta)$ satisfies condition 1 of Theorem 4.4;

2) There exist functions $a \in K$ and $b \in CK$ and constant positive definite $(2 \times 2)$ matrices $A_1$ and $A_2$ such that

for $(t, v, \eta)\in \mathbb{R}_+\times D(\rho)\times \mathbb{R}_+^2;$

3)

Then the state $\theta_0$ of (3.5) is equi-stable.

Proof. Under the condition 2) of Theorem 5.3, the estimate (5.4) can be represented as

for $(t, v, \eta)\in \mathbb{R}_+\times D(\rho) \times \mathbb{R}_+^2$, where $\lambda_m(A_1) > 0$ and $\lambda_M(A_2) > 0$ are the minimal and maximal eigenvalues of the matrices $A_1$ and $A_2$, respectively, and the comparison functions $\overline{a}\in K$, $\overline{b}\in CK$ exist such that

and

Let $0 < \varepsilon < \rho$ and $t_0\in \mathbb{R}_+$ be given. We choose $\delta = \delta(t_0, \varepsilon)$ so that

We will show that, for such a choice of $\delta$, the stationary state $\theta_0$ of (3.5) is equi-stable. If this is not true, then there exist a solution $v^*(t; t_0, v_0)$ and a $t_1 > t_0$ such that, for $\kappa \in [0, 1]$,

According to condition 3 of Theorem 5.3 and Corollary 4.5, we have

Hence, taking into account the estimates (5.5) and (5.6), we get

The obtained contradiction shows that $d[v_0, \theta_0] < \delta$ implies that $d[\, v(t; t_0, v_0), \theta_0\, ] < \varepsilon$ for all $t\geq t_0$ and all $\kappa \in [0, 1]$, which proves the theorem.

Theorem 5.4. Suppose that, for the regularized system (3.5), conditions $H_1$–$H_3$ hold, there exists a function of the type (4.5) for which conditions 1 and 2 of Theorem 5.3 are satisfied and, instead of 3 we have the following:

$3'$) There exists a constant $\beta > 0$ such that

Then, the state $\theta_0$ of (3.5) is equi-asymptotically stable.

Proof. Since the conditions of Theorem 5.4 follow from the conditions of Theorem 5.3, the steady state $\theta_0$ is equi-stable.

Let $\varepsilon = \rho$ and $\delta_0 = \delta_0(t_0, \rho)$. From Theorem 5.3, we have that $d[v_0, \theta_0] < \delta_0$ implies that $d[v(t), \theta_0] < \rho$ for all $t\geq t_0$ and $\kappa \in [0, 1]$.

From condition $3'$ of Theorem 5.4, we obtain

For the given $\varepsilon > 0$ we choose

Hence, for any $\kappa \in [0, 1]$

From the above inequalities, for any initial data $v_0 \in E^{n^2}$ with $d[v_0, \theta_0] < \delta_0$, we have that $d[\, v(t; t_0, v_0), \theta_0\, ] < \varepsilon$ for $t\geq t_0+\tau(t_0, \varepsilon)$ and any $\kappa \in [0, 1]$, which proves Theorem 5.4.

Theorem 5.5. Suppose that, for the regularized system (3.5), conditions $H_1$–$H_3$ hold, and that there exist a matrix-valued function $U(t, \cdot)$ and a vector $\eta\in \mathbb{R}_+^2$ such that the function (4.5) satisfies the following conditions:

1) $V(t, v, \eta)$ satisfies condition 1 of Theorem 4.4 on $\mathbb{R}_+\times (D(\rho) \cap D^c(\sigma))\times \mathbb{R}_+^2$, where $D^c(\sigma)$ is the complement of $D(\sigma)$ for $0 < \sigma < \rho$;

2) $V(t, v, \eta)$ satisfies condition 2 of Theorem 5.3 on $\mathbb{R}_+\times (D(\rho) \cap D^c(\sigma))\times \mathbb{R}_+^2$ for the functions $a, b \in K$;

3) $V(t, v, \eta)$ satisfies condition 3 of Theorem 5.3 on $\mathbb{R}_+\times (D(\rho) \cap D^c(\sigma))\times \mathbb{R}_+^2$.

Then the state $\theta_0$ of (3.5) is uniformly stable.

Proof. Condition 2) of Theorem 5.5 leads to

Let $0 < \varepsilon < \rho$ and $t_0\in \mathbb{R}_+$ be given. We can choose $\delta = \delta(\varepsilon) > 0$ so that

We will show that for the above choice of $\delta > 0$ the stationary state $\theta_0$ of the regularized system (3.5) is uniformly stable. If this is not true, then there exist a solution $v(t)$ of (3.5) and $t_1, t_2$, where $t_2 > t_1 > t_0$, such that $d[v(t_1), \theta_0] = \delta$, $\, d[v(t_2), \theta_0] = \varepsilon$ and $\delta \leq d[v(t), \theta_0]\leq\varepsilon < \rho$ for $t\in [t_1, t_2]$ and $\kappa \in [0, 1]$.

Set $\sigma = \delta$, and, according to the condition 3 of Theorem 5.5, we have

From the above inequality, we get

The obtained contradiction proves Theorem 5.5.

Theorem 5.6. Suppose that, for the regularized system (3.5), conditions $H_1$–$H_3$ hold, there exists a function of the type (4.5) for which conditions 1 and 2 of Theorem 5.5 are satisfied and, instead of 3 we have the following:

$3''$) There exists a function $c \in K$ such that

Then, the state $\theta_0$ of (3.5) is uniformly asymptotically stable.

Proof. Since all conditions of Theorem 5.5 are satisfied, the state $\theta_0$ of (3.5) is uniformly stable. Then, for $\varepsilon = \rho$ and $\delta_0 = \delta_0(\rho)$, $d[v_0, \theta_0] < \delta_0$ implies that $d[v(t), \theta_0] < \rho$ for all $t\geq t_0$ and $\kappa \in [0, 1]$.

To prove Theorem 5.6, we have to show that the state $\theta_0$ is attractive, i.e., that there exists a $t^*\geq t_0$ such that $d[v(t^*), \theta_0] < \delta$ for $t_0\leq t^*\leq t_0+\tau$, where $\tau = 1+\frac{\lambda_M(A_2)\overline{b}(\delta_0)}{\lambda_m(A_1)\overline{a} (\delta)}$.

Suppose that the above is not true and $\delta\leq d[v(t), \theta_0]$ for $t_0\le t\le t_0+\tau$. Then, it follows from $3''$ that

From the last inequality for the given choice of $\tau$, we have

The obtained contradiction proves that $d[v_0, \theta_0] < \delta$ implies that $d[v(t; t_0, v_0), \theta_0] < \varepsilon$ for $t\geq t_0+\tau$, i.e., the state $\theta_0$ is attractive. Theorem 5.6 is proved.

In the next result, we will need the following concept.

Definition 5.7. Two functions $a, b \in K$ are said to be of the same order of magnitude if there exist positive constants $k_1$ and $k_2$ such that $k_1a(r) \leq b(r) \leq k_2a(r)$ for all $r\in \mathbb R_+$.

Theorem 5.8. Suppose that, for the regularized system (3.5), conditions $H_1$–$H_3$ hold, and that there exist a matrix-valued function $U(t, \cdot)$ and a vector $\eta\in \mathbb{R}_+^2$ such that the function (4.5) satisfies the following conditions:

1) $V(t, v, \eta)$ satisfies condition 1 of Theorem 4.4;

2) There exist a comparison function $\sigma_1 \in K$, a positive constant $\Delta_1$ and a $(2 \times 2)$ matrix function $F_1(\theta)$, $\theta \in \mathbb R$ such that

for $(t, v, \eta)\in \mathbb{R}_+\times D(\rho)\times \mathbb{R}_+^2$ and $\rho > 1$;

3) There exists a comparison function $\sigma_2 \in K$ and a $(2 \times 2)$ matrix function $F_2(\theta)$, $\theta \in \mathbb R$ such that

Then, if the matrix $F_1(\theta)(\theta\neq 0)\in \mathbb{R}_+^2$ is positive definite, the matrix $F_2(\theta)(\theta\neq 0)\in \mathbb{R}_+^2$ is negative definite and the comparison functions $\sigma_1, \sigma_2$ are of the same order of magnitude, the state $\theta_0$ of (3.5) is uniformly exponentially stable.

Proof. We can represent the upper estimate of $V(t, v, \eta)$ in condition 2) of Theorem 5.8 as follows

where $\lambda_M(F_1) > 0$ is the maximal eigenvalue of the matrix $F_1$.

Since the function $\sigma_1 \in K$, there exists a function $\gamma \in K$ such that

From (5.10) and (5.11), it follows that condition 2 of Theorem 5.8 takes the following form:

We represent condition 3 of Theorem 5.8 in the following form:

where the function $\pi \in K$ exists such that

Since the functions $\gamma$ and $\pi$ are of the same order of magnitude, there exist constants $K_1 > 0$ and $K_2 > 0$ such that

From (5.12) and (5.13) for any $\kappa \in [0, 1]$, we obtain

where $\chi = \lambda_M(F_2) \lambda_M^{-1}(F_1), \ \chi < 0$.

Next, the estimate (5.15) and condition 2 of Theorem 5.8 imply that

Hence, for

we have an exponential type convergence of an arbitrary solution $v_{\kappa}(t)$ of (3.5) to the state $\theta_0$, which proves the theorem.

We will next establish stability criteria by means of the vector comparison function of type (4.6).

Theorem 5.9. Suppose that, for the regularized system (3.5), conditions $H_1$–$H_3$ hold, and that there exist a matrix-valued function $U(t, \cdot)$ and a vector $\eta\in \mathbb{R}_+^2$ such that the function (4.6) satisfies the following conditions:

1) $\mathcal L(t, v, \eta)$ satisfies all conditions of Theorem 4.9;

2) There exist vector functions $a, b \in K$ and constant positive definite $(2 \times 2)$ matrices $A_1$ and $A_2$ such that for the function

the inequalities

hold for $(t, v, \eta)\in \mathbb{R}_+\times D(\rho)\times \mathbb{R}_+^2.$

Then, the stability properties of the zero solution of the vector comparison equation (4.21) imply the corresponding stability properties of the state $\theta_0$ of (3.5).

Proof. We will study the equi-asymptotic stability of the steady state $\theta_0$ of the regularized system (3.5).

Let $0 < \varepsilon < \rho$ and $t_0\in \mathbb{R}_+$ be given. Suppose that the zero solution of (4.21) is equi-asymptotically stable. Hence, it is stable, and for given $\lambda_m(A_1)\overline{a}(\varepsilon) > 0$ and $t_0\in \mathbb{R}_+$ there exists $\delta_1 = \delta_1(t_0, \varepsilon) > 0$ such that

implies that

where $r_M(t; t_0, \omega_0) = (r_{M1}(t; t_0, \omega_0), r_{M2}(t; t_0, \omega_0))^T$ and $\overline{a} \in K$ exists such that $a^T (d[v, \theta_0])a(d[v, \theta_0])\geq \overline{a}(d[v, \theta_0])$ on $D(\rho)$.

We set $\omega_0 = V_0(t_0, v_0, \eta)$ and choose $\delta = \delta(t_0, \varepsilon) > 0$ so that

We will prove that if $d[\omega_0, \theta_0] < \delta$, then $d[v(t; t_0, v_0), \theta_0] < \varepsilon$ for all $t\geq t_0$ and all $\kappa \in [0, 1]$, where $v(t; t_0, v_0)$ is an arbitrary solution of (3.5).

If the above assertion is not true, then there exists $t_1 > t_0$ such that

Since $\mathcal L(t, v, \eta)$ satisfies all conditions of Theorem 4.9, we have

From (5.18), we have

and, hence

where $\, r_0(t_1;t_0, \omega_0) = \sum\limits_{i = 1}^2r_{Mi}(t; t_0, \omega_0)$.

The contradiction (5.22) proves that the state $\theta_0$ of the regularized system (3.5) is equi-stable.

We will next prove that it is attractive. Let $\varepsilon = \rho$ and $\hat{\delta}_0 = \delta(t_0, \rho) > 0$. We choose $0 < \sigma < \rho$ and for given $\lambda_m(A_1)\overline{a}(\sigma)$ and $t_0\in \mathbb{R}_+$ choose $\delta_1^* = \delta_1^*(t_0, \sigma) > 0$ and $\tau = \tau(t_0, \sigma) > 0$ so that

implies that

Let $\omega_0 = V_0(t_0, v_0, \eta)$. Determine $\delta_0^* = \delta_0^*(t_0, \sigma) > 0$ so that $\lambda_M(A_2) \overline{b}(\delta_0^*) < \delta_1^*$. Choose $\delta_0 = \min(\delta_1^*, \delta_0^*)$ and assume that $d[\omega_0, \theta_0] < \delta_0$. Hence, $d[v(t; t_0, v_0), \theta_0] < \rho$ for all $t\geq t_0$ and $\kappa \in [0, 1]$, and therefore, the estimate (5.21) is satisfied for all $t\geq t_0$, $\kappa \in [0, 1]$. Suppose that there exists a sequence $\{t_k\}$, $\, t_k\ge t_0+\tau$, $\, t_k\to +\infty$ as $k\to+\infty$ and $\sigma\leq d[v(t_k), \theta_0]$, where $v(t)$ is an arbitrary solution of (3.5) with initial data $v_0 \in E^n$ such that $d[v_0, \theta_0] < \delta_0$, and such that the Hukuhara difference $u_0-w_0 = v_0$ exists.

Finally, from (5.18) and (5.23), we have

The obtained above contradiction (5.24) proves that the stationary state $\theta_0$ of (3.5) is attractive and, therefore, quasi-asymptotically stable.

The other stability properties can be proved similarly by linking the dynamic properties of the regularized fuzzy system (3.5) and these of the comparison system (4.21).

Remark 5.10. The established stability criteria show that the idea of using a family of mappings and regularized fuzzy systems of type (3.1) with respect to uncertain parameters greatly benefits their stability analysis. In fact, due to some complications in the study of fuzzy differential systems with uncertain parameters, the proposed results in this direction are very few ^[20]. Hence, the proposed regularization procedure complements such published accomplishments and, due to the offered advantages is more appropriate for applications. In addition, the modifications of the Lyapunov theory-named matrix-valued Lyapunov functions further extend the advantages of the proposed strategy over the classical Lyapunov function strategies.

Example 5.11. We consider a fuzzy Cohen–Grossberg neural network of Lotka-Volterra type with two interacting species

where $t \geq 0$; $u_1(t)$ and $u_2(t)$ are the populations of the two species at time $t$, respectively, $r_1$ and $r_2$ are intrinsic growth rates; $K_1$ and $K_2$ are the carrying capacities of the environment; $e_{12}$ and $e_{21}$ are inter-specific coefficients. All parameters $r_1$, $r_2$, $K_1$, $K_2$ and $e_{12}$ and $e_{21}$ are positive numbers. The uncertain parameters are $\alpha_{12}$ and $\alpha_{21}$, which can take values from the interval $[0, 1]$ and represent the interaction strength between the species.

Introduce the notations

Then, the model (5.25) has the following form:

where $u\in E^2$ and $f\in C(\mathbb{R}_+\times E^2\times \mathcal{S}, E^2)$, $\mathcal{S} = [0, 1] \times [0, 1]$.

It is easy to show that for (5.25) there exists an equilibrium $u^{\varepsilon}$ at

which is positive for all permissible values of $\alpha_{12}$ and $\alpha_{21}$ whenever the carrying capacity ratio $K_1/K_2$ satisfies the condition

Now, we consider a regularized system that corresponds to (5.26) and is given by

where $f_\kappa\in C(\mathbb R_+\times E^2, \, E^2)$ and $f_m(t, v) = \emptyset$ and $f_\kappa(t, v) = f_M(t, v)\kappa$, $\kappa\in[0, 1]$.

Denote the steady state of (5.28) by $\theta_0$. Suppose that there exists a function $\tilde c(t)$ such that

$({\rm{i}})$ $d\left[f_\kappa(t, v), \theta_0\right]\le \tilde c(t)d[v, \theta_0]$ for all $\kappa\in [0, 1]$ and $(t, v) \in \mathbb R_+ \times E^2$;

$({\rm{ii}})$ $\int\limits_0^\infty \tilde c(s)ds\le+\infty.$

If for the corresponding regularized equation (5.8) all conditions $H_1$–$H_3$ are satisfied, then (ⅰ) and (ⅱ) guarantee the uniform stability of its stationary solution $\theta_0\in E^2$.

In fact, the Lyapunov-type function $V(t, v, \eta) = d[v, \theta_0]$ satisfies all conditions of Theorem 5.5. More precisely, $V(t, v, \eta) = \eta^T U(t, v)\eta$, where the matrix $U(t, v)$ has entries $u_{ij}(t, v)$, $i, j = 1, 2$ defined as $u_{11}(t, v) = u_{22}(t, v) = \frac 1 2 d[v, \theta_0]$, $u_{12}(t, v) = u_{21}(t, v) = 0$ and $\eta = (1, 1)^T$.

In addition, for any sufficiently small $h > 0$ we have

From the definition of the derivative $D^+V(t, v, \eta)$ and (ⅰ), we get

Since, condition (ⅱ) is sufficient ^[18,19] to guarantee the uniform stability of the zero solution of

then according to the comparison principle, the state $\theta_0$ of the regularized system (5.28) is uniformly stable, too.

6.   Stability analysis for autonomous comparison problems

In this section, first, under the conditions of Theorem 5.9, we will analyze the stability of the zero solution of the comparison system (4.21) for the case that the vector function $G(t, \omega)$ is autonomous. In this case, the comparison system is in the following form:

where $\bar G\in C(\mathbb{R}_+^2, \mathbb{R}^2)$, $\bar G = (\bar G_1, \bar G_2)^T$, $\omega = (\omega_1, \omega_2)^T$ and the stability problems of the zero solution have effective resolutions under the following assumptions:

$({\rm{i}})$ The function $\bar{G}$ is quasimonotonic and nondecreasing with respect to $\omega$ on

$({\rm{ii}})$ There exists a local solution $\omega(t)$ of the IVP (6.1) which is uniquely determined by the given initial data;

$({\rm{iii}})$ There exists a neighborhood $D^*$ of the state $\omega = 0$ such that, for any $\omega\in \overline{D}^*$, we have that $\, \bar{G}(\omega)\neq 0$ for $\omega \neq 0$ and $\bar{G}(0) = 0$.

In what follows, uniform asymptotic stability criteria will be established.

Theorem 6.1. Suppose that, for the regularized system (3.5), conditions $H_1$–$H_3$ hold, and that there exist a matrix-valued function $U(t, \cdot)$ and a vector $\eta\in \mathbb{R}_+^2$ such that the function (4.6) satisfies the following conditions:

1) $\mathcal L(t, v, \eta)$ satisfies condition 1 of Theorem 4.9 and condition 2 of Theorem 5.9;

2) $\mathcal L(t, v, \eta)$ is such that, for any $\kappa\in[0, 1]$, we have

3) For any $\delta > 0$, the system of inequalities

has a unique solution $\overline{\omega}_1, \; \overline{\omega}_2$ such that $0 < \overline{\omega}_i < \delta$ for $i = 1, 2$.

Then the state $\theta_0$ of (3.5) is uniformly asymptotically stable.

Proof. Under condition 3 of Theorem 6.1, the isolated zero solution of the comparison system (6.1) is uniformly asymptotically stable ^[18,19]. Further, applying the reasoning from the proof of Theorem 5.9, we complete the proof of Theorem 6.1.

Theorem 6.1 and Corollary 4.10 imply the validity of the following result.

Corollary 6.2. Suppose that, in Theorem 6.1, the function $\bar G = \bar G(\omega) = A \omega$ and $t \in \mathbb R_+$, where $A$ is a $(2\times 2)$ matrix with entries $a_{ij}\geq 0$ for $i\ne j$.

Then, if the system of inequalities

has a unique solution $\overline{\omega}_1$, $\overline{\omega}_2$ such that $0 < \overline{\omega}_j$ for all $j = 1, 2$, then the state $\theta_0$ of the regularized fuzzy system (3.5) is uniformly asymptotically stable

Next, we suppose that

for $n_i\neq n_j$, $i, j\in[1, m]$. We will represent an autonomous regularized system of the type (3.5) on $E^{n_1}\times\ldots\times E^{n_m}$ in the following form:

where $f_\kappa^i\in C(E^{n_i}, E^{n_i})$ and $g_\kappa^i\in C(E^{n_1}\times\ldots\times E^{n_m}, E^{n_i})$ for $\kappa\in[0, 1]$, $i = 1, 2, \ldots, m$.

We will apply the following two metrics

and

where $D\in\mathbb{R}_+^m$ and $u, v\in E^{n^m}$. Note that the use of the measure (6.4) is related to the following condition

where $g \in C(\mathbb{R}_+, \mathbb{R}_+)$.

The comparison principle for the autonomous regularized system (6.3) is represented as follows.

Theorem 6.3. Suppose that, for the regularized system (6.3), there exists a Lyapunov-type function $\mathcal L(v, \eta)$ that satisfies the following conditions:

1) $\mathcal L(v, \eta)\in C(E^{n^m} \times \mathbb R_+^2, \mathbb{R}_+^m)$ and there exists a constant $(m \times m)-$matrix $P$ with real entries such that

for $v_1, v_2\in E^{n^m}$, $\eta \in \mathbb R_+^2$, where $||.||$ is the norm in $R^m$ and $|.|$ is the corresponding matrix norm;

2) There exists a family $G(\omega)$, $G\in C(\mathbb{R}_+^m, \, \mathbb{R}^m)$, such that, for any $\kappa\in[0, 1]$

3) The maximal solution $r_M(t, t_0, \omega_0)$ of the comparison equation

exists on $[t_0, \infty)$.

Then, $\mathcal L(v_0, \eta) \leq \omega_0$ implies that

The steps of the proof of Theorem 6.3 are identical to those of the proof of Theorem 2.17 ^[19].

For the family of equations (6.3), the following result, whose proof is similar to the proof of Theorem 5.9, is valid.

Theorem 6.4. Suppose that for the system (6.3) there exists a vector Lyapunov-type function $\mathcal L(v, \eta)$ such that the following holds:

1) $\mathcal L(v, \eta)$ satisfies all conditions of Theorem 6.4 for $v\in { \tilde D}(\rho)$, where ${ \tilde D}(\rho) = \{v \in E^{n^m}:\, d_0[v, \theta_0] < \rho\}$ and $\eta \in \mathbb R_+^2$;

2) There exist vector functions $a, b \in K$ and constant positive definite $(m \times m)$ matrices $A_1$ and $A_2$ such that, for the function

for $v\in { \tilde D}\rho)$, $\eta \in \mathbb R_+^2$, we have

Then the stability properties of the zero solution of the vector comparison equation (6.7) with $G_\kappa(0) = 0$ imply the corresponding stability properties of the state $\theta_0$ of (6.3).

Example 6.5. In this example, we will use functions $a \in K$ such that $a \to \infty$ as their variable approaches $\infty$. Such a class of functions will be denoted as $KR$. Consider Eq (6.3) under the following assumptions:

$1)$ There exist functions $\mathcal L_i(v, \eta)\in C(E^{n^m} \times \mathbb R_+^2, \mathbb{R}_+)$ and $\psi_{i1}$, $\psi_{i2}\in KR$, $i = 1, 2, \ldots, m$, such that

$2)$ For all $i, j = 1, 2, \ldots, m$, there exist constants $\sigma_i(\kappa)\in\mathbb{R}$ and $a_{ij}(\kappa)$ such that

$({\rm{a}})$ $D^+\mathcal L_i(v, \eta)\leq \sigma_i(\kappa)\psi_{i3}(d_0[v_i, \theta_0])$ with respect to the solutions of the family of equations

$({\rm{b}})$ $D^+\mathcal L_i(v, \eta)\leq \psi_{i3}^{\frac 1 2}(d_0[v_i, \theta_0])\sum\limits_{j = 1}^ma_{ij}(\kappa)\psi_j^{\frac 1 2}(d_0[v_i, \theta_0])$ on the connection function $g_\kappa^i(v_1, \ldots, v_m)$ between subsystems of (6.11) for any $\kappa\in[0, 1]$, $i, j = 1, 2, \ldots, m$.

From 1) and 2), for the function $V(v, \eta) = b^T \mathcal L(v, \eta)$, $b\in\mathbb{R}_+^m$, we have

where $\psi_3 = \left(\psi_{13}^{\frac 1 2}(d_0[v_1, \theta_0]), \ldots, \psi_{m3}^{\frac 1 2}(d_0[v_m, \theta_0])\right)^T$, $\widetilde{S} = [s_{ij}(\kappa)]$, $i, j = 1, 2, \ldots, m$, $\kappa\in[0, 1]$,

If the matrix $\overline{M} = \frac 1 2 (\widetilde{S}^T(\kappa)+\widetilde{S}(\kappa))$ is positive definite for all $\kappa\in[0, 1]$, then

where $\lambda_M(\overline{M})$ is the maximal eigenvalue of $\overline{M}$ for any $\kappa\in[0, 1]$ and the function $\psi\in K$ exists such that $\psi_{3}^T(d_0[v, \theta_0])\psi_{3}(d_0[v, \theta_0])\leq \psi(d_0[v, \theta_0])$. Condition 1) and (6.12) imply that the stationary state $\theta_0$ of (6.3) is uniformly asymptotically stable.

7.   Boundedness results

Consider the fuzzy system (3.1) and by means of the regularization process transform it to a regularized system of the following type:

where $f_{\kappa} (t, v)$ is defined by (3.2) for all $\kappa \in [0, 1]$, $g_{\kappa}, \ h_{\kappa} \in C(\mathbb{R}_+ \times E^{n^2}, \ E^{n^2})$, $h_{\kappa}(t, v)\neq 0$ for $v = 0$ and $t \in \mathbb{R}_+$.

For the above fuzzy system (7.1), assume that there exist a unique steady state $\theta_0$ and functions $\tilde f(t)$ and $\tilde m(t)$ that are positive and integrable on any finite interval in $\mathbb R_+$ such that the following holds:

$H_4$. For any $\kappa \in [0, 1]$ and $(t, v)\in \mathbb{R}_+ \times E^{n^2}$,

$H_5$. For any $\kappa \in [0, 1]$ and $(t, v)\in \mathbb{R}_+ \times E^{n^2}$ and $k > 1$

$H_6$. For all $t \geq t_0$, $t_0 \in \mathbb{R}_+$

Now, the goal of our investigations is to obtain boundedness criteria for the family of solutions $v(t)$ of the set of fuzzy differential equations (7.1) for any $\kappa \in [0, 1]$.

Lemma 7.1. Suppose that, for the system (7.1), conditions $H_4$–$H_6$ hold for all $t \in [t_0, a)$. Then, for the function $V(t, v, \eta) = d[v, \theta_0]$, the estimate

is satisfied for all $t \in [t_0, a)$.

Proof. Without loss of generality, we can consider the function $V(t, v, \eta) = d[v, \theta_0]$ for $d[v, \theta_0]\geq K^*$, where $K^*$ is sufficiently large. We have that

for any $0 < K_1 < K_2$ and all $v \in E^{n^2}$.

For the total derivative of the function $V(t, v, \eta)$ with respect to system (7.1) from $H_4$ and $H_5$, we obtain

for any $(t, v)\in \mathbb{R}_+ \times E^{n^2}$, $\eta = (1, 1)^T$ and $k > 1$.

From the above estimate, we get

Set $z(t) = V(t, v(t), \eta) = d[v(t), \theta_0]$. Then, $z(t_0) = d[v_0, \theta_0]$ and the estimate (7.6) has the following form:

We apply the Gronwall-Bellman inequality to the above estimate and get

The estimate (7.8) leads to

We multiply both sides of (7.9) by the negative expression

and we obtain

Hence,

From (7.10), we obtain

The assumption H$_6$ and (7.11) lead to

Considering (7.9) and (7.10), we find that

Finally, (7.8) and (7.12) imply (7.3).

We introduce the following boundedness definitions ^[14,20].

Definition 7.2 The family of solutions $v(t) = v(t; t_0, v_0)$ of the initial value problem (7.1) and (7.2) is

$\mathcal{B}_1$ bounded if there exists a constant $r > 0$ such that, for any $t_0\in\mathbb{R}_+$ and $v_0 \in E^{n^2}$, we have that $d[v(t; t_0, v_0), \theta_0] < r$ for all $t\ge t_0$ and all $\kappa \in [0, 1]$, where $r$ may depend on any solution of the set of equations given by (7.1);

$\mathcal{B}_2$ equi-bounded if, for any $t_0\in\mathbb{R}_+$ and $\delta > 0$, there exists $r = r(t_0, \delta)$ such that, for any initial data $v_0 \in E^{n^2}$, $d[v_0, \theta_0] < \delta$ implies that $d[v(t; t_0, v_0), \theta_0] < r$ for all $t\ge t_0$ and all $\kappa \in [0, 1]$;

$\mathcal{B}_3$ uniformly bounded if $r$ in $\mathcal{B}_2$ does not depend on $t_0$;

$\mathcal{B}_4$ quasi-equi-ultimately bounded with a bound $\bar r$ if there exists $\bar r > 0$ and, for any $\delta_0 > 0$, there exists $\tau = \tau(t_0, \delta) > 0$ such that $d[v_0, \theta_0] < \delta_0$ implies that $d[v(t; t_0, v_0), \theta_0] < \bar r$ for all $t\ge t_0+\tau$ and $\kappa \in [0, 1]$;

$\mathcal{B}_5$ quasi-uniformly ultimately bounded with a bound $\bar r$ if $\tau$ in $\mathcal{B}_4$ is independent on $t_0$;

$\mathcal{B}_6$ equi-ultimately bounded with a bound $\bar r$ if $\mathcal{B}_2$ and $\mathcal{B}_4$ hold simultaneously;

$\mathcal{B}_7$ uniformly ultimately bounded with a bound $\bar r$ if $\mathcal{B}_3$ and $\mathcal{B}_5$ hold simultaneously;

$\mathcal{B}_8$ equi-stable in the Lagrange sense if $\mathcal{B}_2$ and $\mathcal{S}_3$ hold simultaneously;

$\mathcal{B}_9$ uniformly stable in the Lagrange sense if $\mathcal{B}_3$ and $\mathcal{S}_4$ hold simultaneously.

Theorem 7.3. Assume that, for the set of fuzzy equations given by (7.1), the conditions of Lemma 7.1 are satisfied and

Then, the family of solutions $v(t)$ is bounded.

Proof. If we suppose that the family of solutions $v(t)$ of (7.1) is unbounded, then, for any $r > 0$ there exists a $t_1 \in [t_0, \infty)$ such that

For the function $V(t, v, \eta) = d[v, \theta_0]$, the estimate (7.3) and condition (7.13) imply that $d[v(t_1), \theta_0] < r$ for all $\kappa \in [0, 1]$. The obtained contradiction proves the assertion of Theorem 7.3.

Next, we will need the following assumption:

$(H_6^*).$ For any $\delta > 0$, $k > 1$ and $d[v_0, \theta_0] < \delta$, we have

Theorem 7.4. Assume that, for the set of fuzzy equations given by (7.1), the conditions $H_4$–$H_6^*$ hold for all $t \in [t_0, a)$, and that, for any $\delta > 0$ and $t_0 \in \mathbb{R}_+$ there exists $r = r(t_0, \delta) > 0$ such that

for all $t \in [t_0, a)$.

Then, the family of solutions $v(t)$ is equi-bounded.

Proof. Analogously to Lemma 7.1, using conditions $H_4$–$H_6^*$, we obtain

From the estimate (7.15) and condition (7.14), we obtain that $d[v(t), \theta_0] < r (t_0, \delta)$ for any $t \in [t_0, a)$ and $\kappa \in [0, 1]$ whenever $d[v_0, \theta_0] < \delta$. This proves Theorem 7.4.

The proofs of the following boundedness results are similar to the proof of Theorem 7.4.

Theorem 7.5. Assume that, for the set of fuzzy equations given by (7.1), the conditions of Theorem 7.4 hold, and that, instead of (7.14), for any $\delta > 0$ there exists $r^* = r^*(\delta) > 0$ such that

for all $t \in [t_0, a)$ uniformly on $t_0 \in \mathbb R_+$.

Then, the family of solutions $v(t)$ is uniformly bounded.

Theorem 7.6. Assume that, for the set of fuzzy equations given by (7.1), the conditions of Theorem 7.4 hold, and that, instead of (7.14), for given $\bar {\beta} > 0$ and $\tau = \tau(t_0, \delta) \in \mathbb{R}_+$ we have

Then, the family of solutions $v(t)$ is quasi-equi-ultimately bounded.

Theorem 7.7. Assume that, for the set of fuzzy equations given by (7.1), the conditions of Theorem 7.4 hold, and that, instead of (7.14), for given $\bar r > 0$ and $\tau^* = \tau^*(\delta) \in \mathbb{R}_+$ we have

for all $t \geq t_0 + \tau^*$ uniformly on $t_0 \in \mathbb R_+$.

Then, the family of solutions $v(t)$ is quasi-uniformly ultimately bounded with a bound $\bar r$.

The next result follows directly from Theorems 7.4 and 7.6.

Theorem 7.8. Assume that, for the set of fuzzy equations given by (7.1), the conditions of Theorems 7.4 and 7.6 hold simultaneously.

Then, the family of solutions $v(t)$ is equi-ultimately bounded with a bound $\bar{\beta}$.

The proof of the next result follows from the proofs of Theorems 7.5 and 7.7.

Theorem 7.9. Assume that, for the set of fuzzy equations given by (7.1), the conditions of Theorems 7.5 and 7.7 hold simultaneously.

Then, the family of solutions $v(t)$ is uniformly ultimately bounded with a bound $\bar{r}$.

Finally, we will establish Lagrange stability results for the steady state $\theta_0$ of (7.1).

Theorem 7.10. Assume that, for the set of fuzzy equations given by (7.1), the conditions of Theorem 7.4 hold, and that, for any $t_0 \in \mathbb{R}_+$ and $\xi > 0$, there exist $\delta_0 (t_0, \xi) > 0$ and $\tau = \tau(t_0, \xi)\in \mathbb{R}_+$ such that

Then, the family of solutions $v(t)$ is equi-stable in the Lagrange sense.

Proof. Since all conditions of Theorem 7.4 are satisfied, the family of solutions $v(t)$ is equi-bounded.

In addition, (7.3) and (7.16) imply that $d[v(t), \theta_0] < \xi$ for any $t \geq t_0 + \tau (t_0, \xi)$ and $\kappa \in [0, 1]$, whenever $d[v_0, \theta_0] < \delta_0(t_0, \xi)$. This proves that the steady state $\theta_0$ of (7.1) is quasi-equi-asymptotically stable, which proves Theorem 7.10.

The proof of the last result is similar to the proof of Theorem 7.10.

Theorem 7.11. Assume that, for the set of fuzzy equations given by (7.1), the conditions of Theorem 7.5 hold, and that, for any $\xi > 0$, there exist $\delta^* (\xi) > 0$ and $\tau^* = \tau^*(\xi)\in \mathbb{R}_+$ such that, uniformly on $t_0 \in \mathbb R_+$,

Then, the family of solutions $v(t)$ is uniformly stable in the Lagrange sense.

Remark 7.12. The proposed boundedness results extend and generalize some existing boundedness and Lagrange stability results for fuzzy systems ^[38] to the uncertain case. Also, they complement some recently published results ^[39] in the fuzzy case because of the use of the proposed regularization method.

8.   Conclusions

In this paper, two approaches are proposed for the study of the stability and boundedness of solutions of fuzzy systems of differential equations with uncertain parameters. These approaches are based on the method of matrix-valued Lyapunov-type functions and an integral method based on nonlinear integral inequalities. By applying a new scheme of regularization of fuzzy equations with respect to the inaccuracy parameter ^[26], numerous new criteria for the stability and boundedness of solutions for regularized equations have been established via the proposed methods. The results obtained can be applied to various fuzzy systems and real-world models in which the effects of some uncertain parameters cannot be neglected. In addition, the proposed methods for the qualitative analysis of fuzzy equations are of decisive importance for many applications, in particular in the theory of motion control. The proposed approaches have significant potential for further generalizations and applications to important classes of fuzzy systems, including systems with delays and impulsive perturbations. An interesting topic for future research is the application of the proposed methods to fuzzy systems with fractional-order dynamics and systems whose dynamics are modeled by fuzzy equations with conformable derivatives of the state vector.

Use of AI tools declaration

The authors declare that they have not used artificial intelligence tools in the creation of this article.

Conflict of interest

The authors declare no conflict of interest.