Review Topical Sections

Survey on security and privacy issues in cyber physical systems

  • The notion of Cyber-Physical Systems (CPS) is proposed by the National ScientificFoundation to describe a type of systems which combine hardware and software components andbeing the next step in development of embedded systems. CPS includes a wide range of researchtopics ranging from signal processing to data analysis. This paper contains a brief review of the basicinfrastructure for CPS including smart objects and network aspects in relation to TCP/IP stack. AsCPS reflect the processes of the physical environment onto the cyber space, virtualisation as animportant tool for abstraction plays crucial role in CPS. In this context paper presents the challengesassociated with mobility and vritualisation; accordingly three main types of virtualisation, namelynetwork, devices and applications virtualisation are presented in the paper. These aspects are tightlycoupled with security and safety issues. Therefore, different threats, attack types with correspondingsubtypes and possible consequences are discussed as well as analysis of various approaches to copewith existing threats is introduced. In addition threat modelling approaches were also in scope of thiswork. Furthermore, needs and requirements for safety-critical CPS are reviewed. Thus the mainefforts of this paper are directed on introducing various aspects of the CPS with regard to securityand safety issues.

    Citation: Artem A. Nazarenko , Ghazanfar Ali Safdar. Survey on security and privacy issues in cyber physical systems[J]. AIMS Electronics and Electrical Engineering, 2019, 3(2): 111-143. doi: 10.3934/ElectrEng.2019.2.111

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  • The notion of Cyber-Physical Systems (CPS) is proposed by the National ScientificFoundation to describe a type of systems which combine hardware and software components andbeing the next step in development of embedded systems. CPS includes a wide range of researchtopics ranging from signal processing to data analysis. This paper contains a brief review of the basicinfrastructure for CPS including smart objects and network aspects in relation to TCP/IP stack. AsCPS reflect the processes of the physical environment onto the cyber space, virtualisation as animportant tool for abstraction plays crucial role in CPS. In this context paper presents the challengesassociated with mobility and vritualisation; accordingly three main types of virtualisation, namelynetwork, devices and applications virtualisation are presented in the paper. These aspects are tightlycoupled with security and safety issues. Therefore, different threats, attack types with correspondingsubtypes and possible consequences are discussed as well as analysis of various approaches to copewith existing threats is introduced. In addition threat modelling approaches were also in scope of thiswork. Furthermore, needs and requirements for safety-critical CPS are reviewed. Thus the mainefforts of this paper are directed on introducing various aspects of the CPS with regard to securityand safety issues.


    r-d: right dense; l-d: left dense; r-s: right scattered; l-s: left scattered; UPS: uniform practical stability; SUPS: strongly uniform practical stability; LF: Lyapunov function; CFrΔD: Caputo fractional delta derivative; CFrΔDiD: Caputo fractional delta Dini derivative; GLFrΔD: Grunwald–Letnikov fractional delta derivative; GLFrΔDiD: Grunwald–Letnikov fractional delta Dini derivative; CFrDyT: Caputo fractional dynamic equations on time scales; CFrD: Caputo fractional derivative

    In the field of dynamic systems, stability analysis is essential for understanding system responses to various initial conditions and disturbances. Traditional Lyapunov stability theory has made significant contributions here, focusing on convergence to the equilibrium [1,2]. However, this strict convergence requirement can be overly restrictive in real-world applications where slight deviations are often tolerable. UPS offers a more flexible framework, allowing systems to operate within an acceptable range around the equilibrium rather than requiring exact convergence (see [3,4]).

    The recent growth of fractional calculus has introduced valuable tools for stability analysis. By extending differentiation and integration to arbitrary order, fractional calculus has proven to be highly effective in modeling complex systems with memory effects and non-local interactions [5,6,7]. Among these tools, the Caputo fractional derivative (CFrD) has shown practical relevance for various applications (see [8,9]). However, most studies have relied on scalar Lyapunov functions (LFs), which evaluate stability on an individual-variable basis [10,11]. It is widely recognized that as the complexity of a dynamical system increases, identifying a suitable LF becomes more challenging. This difficulty often leads to the use of multiple LFs, forming a vector LF, where each component reveals insights into different parts of the system's behavior. By this approach, certain complex elements can be simplified into interconnected subsystems, allowing for more relaxed conditions. Thus, the method of vector LFs provides a highly adaptable framework for analysis [12,13,14].

    Previous studies, including [15,16,17], have concentrated largely on stability aspects like uniform, asymptotic, and variational stability within delay and impulsive settings, primarily for continuous-time systems. In contrast, research such as [] has examined stability in discrete settings. This paper introduces an innovative approach using vector LFs for Caputo fractional dynamic equations on time scales (CFrDyT), establishing UPS in terms of two measures (m0,m), and presenting a versatile framework well-suited for real-world applications. This approach ensures that systems remain within specific bounds despite small disturbances, reflecting a balance between rigorous stability and practical adaptability.

    With the introduction of time scale calculus in [19], a unified approach for analyzing systems evolving in both discrete and continuous time domains has emerged, extending traditional continuous analysis techniques to discrete cases. Integrating this calculus with fractional dynamics enables a more sophisticated framework for understanding systems that undergo both gradual and abrupt changes, common in hybrid systems. When combined with fractional calculus, time scale calculus provides a versatile platform for analyzing dynamic behaviors in both continuous and discrete settings.

    Building on these advancements, this paper extends the results in [20,21], presenting a novel methodology for examining the (m0,m)-uniform practical stability (UPS) and (m0,m)-strongly uniform practical stability (SUPS) of CFrDyT. To further extend the scope of practical stability analysis [22], this work applies the novel derivatives introduced in [23], the Caputo fractional delta derivative (CFrΔD) and Caputo fractional delta Dini derivative (CFrΔDiD) of order ζ(0,1), which enables unified stability analysis, creating a robust framework for hybrid systems exhibiting both gradual and abrupt changes.

    Let us examine the system

    CΔζy=D(t,y), tT,y(t0)=y0,t00. (1.1)

    Here,

    DCrd[T×RN,RN]

    is a function satisfying

    D(t,0)0,

    and CΔζy denotes the CFrΔD of yRN of order ζ. Let

    y(t)=y(t,t0,y0)Cζrd[T,RN]

    represent a solution of (1.1). Assuming that the function D possesses sufficient smoothness to ensure the existence, uniqueness, and continuous dependence of solutions (see [24,25]), this paper examines the (m0,m)-UPS and (m0,m)-SUPS of the system (1.1).

    To achieve this, we use the comparison system of the form

    CΔζκ=Θ(t,κ),  κ(t0)=κ00, (1.2)

    where

    ΘCrd[T×Rn+,Rn+].

    The system (1.2) is a simpler system of lesser dimension than (1.1), whose qualitative properties including the existence and uniqueness of its solution

    κ(t)=κ(t;t0,κ0)

    and practical stability are already known or easy to find, see [26].

    The structure of this research is outlined as follows: In Section 2, necessary terminologies, key definitions, remarks, and foundational lemmas that support the later developments are given. In Section 3, we develop and give details of core findings and theoretical contributions of our research. In Section 4, we present practical examples to illustrate the relevance and applicability of our approach. Lastly, in Section 5, we present a summary of the main findings and discuss their implications.

    Definition 2.1. [27] If tT, then

    ϖ(t)=inf{sT:s>t}

    is called the forward jump operator and

    ρ(t)=sup{sT:s<t}

    is called the backward jump operator. So that tT is said to be: right dense (r-d) if

    ϖ(t)=t,

    right scattered (r-s) if ϖ(t)>t, left dense (l-d) if ρ(t)=t, and left scattered if ρ(t)<t. Also the function

    τ(t)=ϖ(t)t

    is called the graininess function.

    Definition 2.2. [28] A function

    h:TR

    is said to be r-d continuous represented by Crd, if it remains continuous at every r-d point in T, and has finite left-sided limits at each l-d point of T.

    Definition 2.3. [28] We define the following class of functions:

    K={ψ[[0,r],[0,)]}: ψ(t) is strictly increasing on [0,r] and ψ(0)=0;

    CK={aCrd[T×R+,R+]:a(t,s)K for each t};

    M={mCrd[T×Rn,R+]:inf(t,x)m(t,x)=0}.

    We now define the derivative of LF utilizing the CFrΔDiD as presented in [23].

    Definition 2.4. Let

    [t0,T)T=[t0,T)T,

    where T>t0. Then a Lyapunov-like function

    (t,y)Crd[T×RN,RN+]

    is defined as the generalized CFrΔDiD relative to (1.1) as follows: given ϵ>0, there exists a neighborhood P(ϵ) of tT such that

    1τζ{(ϖ(t),y(ϖ(t))(t0,y0)[tt0τ]ι=1(1)ι+1(ζCι)[(ϖ(t)ιτ,y(ϖ(t)τζD(t,y(t))(t0,y0)]}<CΔζ+(t,y)+ϵ,

    for each sP(ϵ) and s>t, and

    y(t)=y(t,t0,y0)

    is any solution of (1.1), where 0<ζ<1, ϖ is the forward jump operator as defined in Definition 2.1,

    τ=ϖt,   ζCι=ζ(ζ1)...(ζι+1)ι!

    and [(tt0)τ] denotes the integer part of the fraction (tt0)τ.

    Definition 2.5. [23] Using Definition 2.6, we state the CFrΔDiD of the LF (t,y) as

    CΔζ+(t,y)=lim supτ0+1τζ[[tt0τ]ι=0(1)ι(ζCι)[(ϖ(t)ιτ,y(ϖ(t))τζD(t,y(t))L(t0,y0)]], (2.1)

    which is equivalent to

    CΔζ+(t,y)=lim supτ0+1τζ{(ϖ(t),y(ϖ(t))(t0,y0)[tt0τ]ι=1(1)ι+1(ζCι)[(ϖ(t)ιτ,y(ϖ(t))τζD(t,y(t))(t0,y0)]}

    and

    CΔζ+(t,y)=lim supτ0+1τζ{(ϖ(t),y(ϖ(t))+[tt0τ]ι=1(1)ι(ζCι)[(ϖ(t)ιτ,y(ϖ(t))τζD(t,y(t))]}(t0,y0)(tt0)ζΓ(1ζ), (2.2)

    where tT, and y,y0Rn, and

    y(ϖ(t))τζD(t,y)RN.

    For discrete times, Eq (2.1) becomes

    CΔζ+(t,y)=1τζ[[tt0τ]ι=0(1)ι(ζCι)((ϖ(t),y(ϖ(t)))(t0,y0))],

    and for continuous times (T=R), Eq (2.1) becomes

    CΔζ+(t,y)=CDζ+(t,y)=lim supκ0+1κζ{(t,y(t))(t0,y0)[tt0κ]ι=1(1)ι+1(ζCι)[(tικ,y(t))κζD(t,y(t))(t0,y0)]}. (2.3)

    Notice that (2.3) is the same in [5], where κ>0.

    Definition 2.6. Let

    y(t)=y(t;t0,y0)

    be any solution of (1.1), then system (1.1) is said to be:

    (UP1) (m0,m)-uniformly practically stable if, given (λ,A)R+ with 0<λ<A, we have m0(t0,y0)<λ implies m(y(t))<A, t0T;

    (UP2) (m0,m)-uniformly practically quasi-stable if given (λ,B,T)>0 and t0T, we have m0(t0,y0)<λ m(y(t))<B, for tt0+T;

    (UP3) (m0,m)-SUPS if (UP1) and (UP2) hold together.

    Definition 2.7. Corresponding to Definition 2.6, the solution

    κ(t)=κ(t;t0,κ0)

    of (1.2), is called uniformly practically stable if given 0<λ<A, we have

    ni=1κ0<λni=1κi(t;t0,κ0)<A,  tt0, (2.4)

    for some t0TR+.

    Lemma 2.1. [23] Let

    R,BCrd[T,Rn].

    Assume t1>t0, with

    t1T:R(t1)=B(t1)

    and

    R(t)<B(t)

    for t0t<t1. If the CFrΔDiD of R and B are defined at t1, then the inequality

    CΔζ+R(t1)>CΔζ+B(t1)

    holds.

    Lemma 2.2. (Comparison theorem) Let

    (i) ΘCrd[T×Rn+,Rn+] and Θ(t,κ)τ is non-decreasing in κ.

    (ii) (t,y)Crd[T×RN,RN+] be locally Lipschitzian in y and

    CΔζ+(t,y)Θ(t,(t,y)),(t,y)T×RN.

    (iii) ϱ(t)=ϱ(t;t0,κ0) be the maximal solution of (1.2) existing on T.

    Then

    (t,y(t))ϱ(t),tt0 (2.5)

    provided that

    (t0,y0)κ0,

    where

    y(t)=y(t;t0,y0)

    is any solution of (1.1), tT, tt0.

    Proof. We shall make this proof by the principle of induction. For an assertion

    S(t):(t,y(t))ϱ(t),tT,

    the following holds:

    (i) S(t0) is true since (t0,y0)κ0.

    (ii) Assuming t is r-s and S(t) is true. Then we prove that S(ϖ(t)) is true; i.e.,

    (ϖ(t),y(ϖ(t)))ϱ(ϖ(t)), (2.6)

    set

    ϰ(t)=(t,y(t)),

    and then

    ϰ(ϖ(t))=(ϖ(t),y(ϖ(t))),

    then from Definition 2.6, we obtain

    CΔζ+ϰ(t)=lim supτ0+1τζ{[(tt0)τ]ι=0(1)ιζCι[ϰ(ϖ(t)ιτ)ϰ(t0)]},

    also,

    CΔζ+ϱ(t)=lim supτ0+1τζ{[(tt0)τ]ι=0(1)ιζCι[ϱ(ϖ(t)ιτ)ϱ(t0)]},tt0,

    then,

    CΔζ+ϱ(t)CΔζ+ϰ(t)=lim supτ0+1τζ{[(tt0)τ]ι=0(1)ιζCι[ϱ(ϖ(t)ιτ)ϱ(t0)]}lim supτ0+1τζ{[(tt0)τ]ι=0(1)ιζCι[ϰ(ϖ(t)ιτ)ϰ(t0)]},(CΔζ+ϱ(t)CΔζ+ϰ(t))τζ=lim supτ0+{[(tt0)τ]ι=0(1)ιζCι[[ϱ(ϖ(t)ιτ)ϱ(t0)][ϰ(ϖ(t)ιτ)ϰ(t0)]]}[ϱ(ϖ(t))ϱ(t0)][ϰ(ϖ(t))ϰ(t0)][ϱ(ϖ(t))ϰ(ϖ(t))][ϱ(t0)ϰ(t0)],

    then,

    [ϱ(ϖ(t))ϰ(ϖ(t))](CΔζ+ϱ(t)CΔζ+ϰ(t))τζ+[ϱ(t0)ϰ(t0)],

    so that,

    [ϰ(ϖ(t))ϱ(ϖ(t))](CΔζ+ϱ(t)CΔζ+ϰ(t))τζ+[ϰ(t0)ϱ(t0)](Θ(t,ϰ(t))Θ(t,ϱ(t)))τζ+[ϰ(t0)ϱ(t0)].

    By the non-decreasing property of Θ(t,κ)τ and since S(t) is true, then

    ϰ(ϖ(t))ϱ(ϖ(t))0,

    so Eq (2.6) holds.

    (iii) For r-d points tU, where U is a right neighborhood tT. We can clearly see that S(t) is true immediately from [5, Lemma 3] since at every r-d point of T, (t,y(t)) is continuous and the domain is R.

    (iv) Let t be l-d, and let S(s) be true for all s>t. We want to show that S(t) is true. This immediately follows r-d continuity of (t,y(t)), y(t), and the maximal solution ϱ(t). Thus, by the principle of induction, the assertion S(t) holds for all tT, thereby concluding the proof.

    The proof is completed.

    Now, we present the (m0,m)-UPS and (m0,m)-SUPS results for the fractional dynamic system (1.1).

    Theorem 3.1. ((m,m0)-UPS) Assume that

    (m1) 0<λ<A;

    (m2) m0,mM, and m0 is uniformly finer than m, implying m(t,y)ϕ(m0(t,y)), ϕK, whenever m0(t,y)<λ;

    (m3) There exists

    Crd[T×RN,RN+]

    and

    QCrd[Rn+,Rn+],

    such that if

    Q((t,y))V(t,y),

    V(t,y) is locally Lipschitz in y, and for

    (t,y)S(m,A)={(t,y)T×RN:m(t,y)<A},

    then,

    b(m(t,y))ni=0Vi(t,y),ifm(t,y)<A0Vi(t,y)a(m0(t,y)),ifm0(t,y)<λ; (3.1)

    (m4) for (t,y)S(m,A)

    CΔζ+V(t,y)Θ(t,V(t,y)), (3.2)

    where

    ΘCrd[T×Rn+,Rn+],

    Θ(t,κ) is quasimonotone nondecreasing in κ with

    Θ(t,κ)0,

    aCK, and for each i, 1in, Θi(t,κ)τ(t)+κi is nondecreasing in κ for all tT;

    (m5) ϕ(λ)<A and a(λ)<b(A) hold;

    Then the UPS properties of (1.2) imply the corresponding (m0,m)-UPS properties of (1.1).

    Proof. Since (1.2) is UPS, then we deduce from (2.4) that t0T and 0<λ<A,

    ni=0κ0i<a(λ)ni=1κi(t;t0,κ0)<b(A),  tt0. (3.3)

    We can easily assert that (1.1) is (m,m0)-UPS with respect to (λ,A).

    However, if the assertion were false, then for any solution

    y(t)=y(t;t0,y0)

    of (1.1) with

    m0(t0,y0)<λ,

    there would be a point t1>t0 such that

    m(t1,y(t1))=Aandm(t,y(t))Afort[t0,t1). (3.4)

    Setting

    κ0i=Vi(t,y0),

    and from (2.5), we have

    V(t,y(t))ϱ(t,t0,κ0), (3.5)

    where ϱ(t) is the maximal solution of (1.2).

    Also, from assumptions (m2) and (m5), it is clear to see that

    m(t0,y0)ϕ(m0,(t0,y0))<ϕ(λ)<A.

    Then, we deduce that,

    ni=1κ0i<0Vi(t0,y0)a(m0(t0,y0))<a(λ). (3.6)

    Combining (3.3)–(3.6), we obtain

    b(A)=b(m(t1,y(t1)))ni=1Vi(t1,y(t1))ni=1ϱi(t1;t0,κ0)<b(A). (3.7)

    Equation (3.7) is a contradiction, so the statement that (1.1) is (m0,m)-UPS is true.

    Theorem 3.2. ((m,m0)-SUPS) Assume that conditions (m1)(m4) of Theorem 3.1 is satisfied, and Eq (1.2) is SUPS; then, Eq (1.1) is (m,m0)-SUPS.

    Proof. To make this proof, we need (1.1) to be (m0,m)-UPS and (m0,m)-uniformly practically quasi-stable together. Now, (1.1) is (m0,m)-UPS by Theorem 3.1, so we need to prove only (m0,m)-uniform practical quasi-stability. By the assumption of the theorem, Eq (1.2) is (m0,m)-SUPS for

    (a(λ),b(λ),b(B),T)>0,

    so that for all t0T,

    ni=1κ0i<a(λ)ni=1κi(t,t0,κ0)<b(B),tt0+T, (3.8)

    where κi(t,t0,κ0) is any solution of (1.2). By the UPS of (1.2), we can comfortably make the assumption that

    m0(t0,y0)<λ,

    so that

    m(t,y(t))<A

    for tt0. Setting

    κ0i=Vi(t0,y0),

    and by Lemma 2.2, we have the estimate

    V(t,y(t))ϱ(t,t0,κ0), (3.9)

    where ϱ(t) is the maximal solution of (1.2). Following this argument closely, and from (3.8) and (3.9), we obtain

    b(m(t,y(t)))ni=1Vi(t,y(t))ni=1ϱi(t,t0,κ0)<b(B),tt0+T.

    So that

    m(t,y(t))<B

    whenever

    m0(t0,y0)<λ,

    for tt0+T. Therefore, we conclude that (1.1) is (m0,m)-SUPS.

    Given the following system

    CΔζ1(t)=2121exp(1)22exp(1),CΔζ2(t)=21exp(2)2222exp(2), (4.1)

    with initial conditions

    1(t0)=10

    and

    2(t0)=20,

    for

    (1,2)R2.

    Choose the LF for (4.1) to be

    =(V1,V2)T,

    where

    V1(t,1,2)=|1|

    and

    V2(t,1,2)=|2|.

    Using (2.5), the CFrΔDiD for

    V1(t,1,2)=|1|

    is computed as

    CΔζ+V1(t,1)=lim supτ0+1τζ{V1(ϖ(t),1(ϖ(t))+[tt0τ]ι=1(1)ι(ζCι)[V1(ϖ(t)ιτ,1(ϖ(t))τζD1(t,1(t))]}V1(t0,10)(tt0)ζΓ(1ζ)=lim supτ0+1τζ{|1(ϖ(t))|+[tt0τ]ι=1(1)ι(ζCι)[|1(ϖ(t))τζD1(t,1)|]}|10|(tt0)ζΓ(1ζ)lim supτ0+1τζ{|1(ϖ(t))|+[tt0τ]ι=1(1)ι(ζCι)[|1(ϖ(t))|+|τζD1(t,1)|]}|10|(tt0)ζΓ(1ζ)lim supτ0+1τζ{|1(ϖ(t))|+[tt0τ]ι=1(1)ι(ζCι)|1(ϖ(t))|+[tt0τ]ι=1(1)ι(ζCι)|τζD1(t,1)|}|10|(tt0)ζΓ(1ζ)|1(ϖ(t))|lim supτ0+1τζ[tt0τ]ι=0(1)ι(ζCι)+|D1(t,1)|lim supτ0+[tt0τ]ι=1(1)ι(ζCι)|10|(tt0)ζΓ(1ζ).

    Applying (2.12) and (2.14) in [23], we obtain

    CΔζ+V1(t,1)=|1(ϖ(t))|(tt0)ζΓ(1ζ)|D1(t;1)||10|(tt0)ζΓ(1ζ),CΔζ+V1|1(ϖ(t))|(tt0)ζΓ(1ζ)|D1(t;1)|.

    When

    t,   |1(ϖ(t))|(tt0)ζΓ(1ζ)0,

    then

    CΔζ+V1|D1(t;1)|=[|2121exp(1)22exp(1)|]=[|21(21+22)exp(1)|][|21|][2|1|],CΔζ+V12V1+0V2. (4.2)

    Similarly, compute the CFrΔDiD for

    V2(t,1,2)=|2|=lim supτ0+1τζ{|2(ϖ(t))|+[tt0τ]ι=1(1)ι(ζCι)[|2(ϖ(t))τζD2(t,2)|]}|20|(tt0)ζΓ(1ζ)lim supτ0+1τζ{|2(ϖ(t))|+[tt0τ]ι=1(1)ι(ζCι)[|2(ϖ(t))|+|τζD2(t,2)|]}|20|(tt0)ζΓ(1ζ)lim supτ0+1τζ{|2(ϖ(t))|+[tt0τ]ι=1(1)ι(ζCι)|2(ϖ(t))|+[tt0τ]ι=1(1)ι(ζCι)|τζD2(t,2)|}|20|(tt0)ζΓ(1ζ)|2(ϖ(t))|lim supτ0+1τζ[tt0τ]ι=0(1)ι(ζCι)+|D2(t,2)|lim supτ0+[tt0τ]ι=1(1)ι(ζCι)|20|(tt0)ζΓ(1ζ).

    Applying (2.12) and (2.14) in [23], we obtain

    CΔζ+V2|2(ϖ(t))|(tt0)ζΓ(1ζ)|D2(t;2)|.

    When

    t,   |2(ϖ(t))|(tt0)ζΓ(1ζ)0,

    then,

    CΔζ+V2|D2(t;2)|=[|21exp(2)2222exp(2)|]=[|22(2122)exp(2)|][|22|]2|2|.

    Therefore,

    CΔζ+V20V12V2. (4.3)

    From (4.2) and (4.3), it follows that

    CΔζ+(2002)(V1V2)=Θ(t,V). (4.4)

    Next, we choose a comparison system

    CΔζ+κ=Θ(t,κ)=Mκ (4.5)

    with

    M=(2002).

    Clearly, Figure 1 below shows the stability of system (4.5). The vector inequality (4.4), along with all the other requirements of Theorem 3.1, holds when the matrix M has eigenvalues with negative real parts. Given that the eigenvalues of M are both 2, it can be concluded that (4.1) is not only (m,m0)-UPS but also (m,m0)-SUPS.

    Figure 1.  Stability of system (4.5).

    Given the following system

    CΔζ1(t)=4122+3231,CΔζ2(t)=321+2+42232,CΔζ3(t)=213+33, (4.6)

    with initial conditions

    1(t0)=10,2(t0)=20,and3(t0)=30,

    for

    =(1,2,3)R3.

    Choose a vector LF

    =(V1,V2,V3)T:V1=21,   V2=22,   V3=23

    and

    0(1,2,3)=2i=1Vi(1,2,3)=21+22+23.

    Computing the CFrΔDiD for

    V1=21,

    we obtain:

    CΔζ+V1=lim supτ0+1τζ{[(1(ϖ(t)))2][(10)2]+[tt0τ]ι=1(1)ι(ζCι)[(1(ϖ(t))τζD1(t,))2][((10)2]}=lim supτ0+1τζ{[(1(ϖ(t)))2][(10)2]+[tt0τ]ι=1(1)ι(ζCι)[(1(ϖ(t)))221(ϖ(t))τζD1(t,1,2,3)+τ2ζ(D1(t,1,2,3))2][(10)2]}=lim supτ0+1τζ{[tt0τ]ι=0(1)ι(ζCι)[(10)2]}+lim supτ0+1τζ{[tt0τ]ι=0(1)ι(ζCι)[(1(ϖ(t)))2]}lim supτ0+{[tt0τ]ι=1(1)ι(ζCι)[21(ϖ(t))τζD1(t,1,2,3)]}.

    From (2.12) and (2.14) in [23], we obtain

    CΔζ+V1(tt0)ζΓ(1ζ)[(1(ϖ(t)))2][21(ϖ(t))D1(t,1,2,3)].

    When

    t,   (tt0)ζΓ(1ζ)[(1(ϖ(t)))2]0,

    so that we obtain

    CΔζ+V12[1(ϖ(t))D1(t,1,2,3)].

    Using

    (ϖ(t))τCΔζ(t)+(t),

    we obtain

    CΔζ+V1=2[τ(t)D21(t,1,2,3)+1(t)D1(t,1,2,3)]=2[τ(t)(4122+3231)2+1(4122+3231)]=2τ(t)[(4122+3231)2]21[4122+3231]. (4.7)

    If T=R then τ=0, reducing (4.7) to the form

    CΔζ+V1(1,2,3)=21[4122+3231]=821+222+623=(8  2  6)(V1  V2  V3)T. (4.8)

    When T=N0 then, τ=1, reducing (4.7) to

    CΔζ+V1(1,2,3)=2[(41υ22+3231)2]21[4122+3231]21[4122+3231],

    resulting in the same outcome as (4.8). Evidently, this applies to any other discrete time as well.

    Also, computing the CFrΔDiD for

    V2()=22,

    we obtain:

    CΔζ+V2()=lim supτ0+1τζ{[(2(ϖ(t)))2][(20)2]+[tt0τ]ι=1(1)ι(ζCι)[(2(ϖ(t))τζD2(t,))2][(20)2]}=lim supτ0+1τζ{[(2(ϖ(t)))2][(20)2]+[tt0τ]ι=1(1)ι(ζCι)[(2(ϖ(t)))222(ϖ(t))τζD2(t,1,2,3)+τ2ζ(D2(t,1,2,3))2][(20)2]}=lim supτ0+1τζ{[tt0τ]ι=0(1)ι(ζCι)[(20)2]}+lim supτ0+1τζ{[tt0τ]ι=0(1)ι(ζCι)[(2(ϖ(t)))2]}lim supτ0+{[tt0τ]ι=1(1)ι(ζCι)[22(ϖ(t))τζD2(t,1,2,3)]}.

    From (2.12) and (2.14) in [23], we obtain

    CΔζ+V2()(tt0)ζΓ(1ζ)[(2(ϖ(t)))2][22(ϖ(t))D2(t,1,2,3)].

    When

    t,   (tt0)ζΓ(1ζ)[(2(ϖ(t)))2]0,

    so that

    CΔζ+V2()2[2(ϖ(t))D2(t,1,2,3)],

    using

    (ϖ(t))τCΔζ(t)+(t),

    we obtain

    CΔζ+V2()=2[τ(t)D22(t,1,2,3)+2(t)D2(t,1,2,3)]=2[τ(t)(321+2+42232)2+2(321+2+42232)]=2τ(t)[(321+2+42232)2]21[321+2+42232]. (4.9)

    If T=R, then τ=0, reducing (4.9) to

    CΔζ+V2(1,2)=22[321+2+42232]=621822+23=(6  8  1)(V1  V2  V3)T. (4.10)

    When T=N0, then τ=1, and (4.9) reduces to:

    CΔζ+V2()=2[(321+2+42232)2]21[321+2+42232]21[321+2+42232],

    resulting in (4.10).

    Finally, for the CFrΔDiD of

    V3(1,2,3)=23,

    we obtain

    CΔζ+V3=lim supτ0+1τζ{[(3(ϖ(t)))2][(30)2]+[tt0τ]ι=1(1)ι(ζCι)[(3(ϖ(t))τζD3(t,))2][(30)2]}=lim supτ0+1τζ{[(3(ϖ(t)))2][(30)2]+[tt0τ]ι=1(1)ι(ζCι)[(3(ϖ(t)))223(ϖ(t))τζD3(t,1,2,3)+τ2ζ(D3(t,1,2,3))2][(30)2]}=lim supτ0+1τζ{[tt0τ]ι=0(1)ι(ζCι)[(30)2]}+lim supτ0+1τζ{[tt0τ]ι=0(1)ι(ζCι)[(3(ϖ(t)))2]}lim supτ0+{[tt0τ]ι=1(1)ι(ζCι)[23(ϖ(t))τζD3(t,1,2,3)]}.

    From (2.12) and (2.14) in [23], we obtain

    CΔζ+V3(tt0)ζΓ(1ζ)[(3(ϖ(t)))2][21(ϖ(t))D3(t,1,2,3)].

    As

    t,   (tt0)ζΓ(1ζ)[(3(ϖ(t)))2]0,

    then

    CΔζ+V32[3(ϖ(t))D3(t,1,2,3)].

    Using

    (ϖ(t))τCΔζ(t)+(t),

    we obtain

    CΔζ+V3=2[τ(t)D23(t,3,2,3)+υ3(t)D3(t,1,2,3)]=2[τ(t)(213+33)2+3(213+33)]=2τ(t)[(213+33)2]23[213+33]. (4.11)

    If T is continuous, then τ=0, and (4.11) reduces to:

    CΔζ+V3(1,2,3)=23[213+33]=221623=(2  0  6)(V1  V2  V3)T. (4.12)

    Otherwise, for discrete T τ=1, (4.11) reduces to:

    CΔζ+V3=2[(213+33)2]21[213+33]21[213+33],

    then we also obtain (4.12).

    From (4.8), (4.10), and (4.12), we obtain

    CΔζ+V(826681206)(V1V2V3)=Θ(t,V). (4.13)

    Next, we choose a comparison system of the form

    CΔζ+κ=Θ(t,κ)=Mκ

    with

    M=(826681206).

    The vector inequality (4.13), along with all the conditions outlined in Theorems 3.1 and 3.2, hold true if the matrix M has eigenvalues with negative real parts. Given that the eigenvalues of M are

    λ1=12.433,λ2=7.199,λ3=2.369,

    it follows that (4.6) is uniformly practically stable. Thus, we conclude that system (4.6) exhibits UPS.

    In conclusion, this paper develops a unified approach for UPS analysis of CFrDyT, utilizing vector LFs and a two-measure framework. By applying the CFrΔD and the CFrΔDiD, this study extends stability analysis methods across both continuous and discrete domains, making it versatile for hybrid systems exhibiting both gradual and abrupt changes. The use of vector LFs over scalar LFs enables a broader and more robust stability analysis, accommodating complex system dynamics with improved precision and adaptability. Illustrative examples, Eqs (4.1) and (4.6), demonstrate the practical application of the proposed framework, underscoring its effectiveness and relevance in capturing the essential stability characteristics of fractional dynamic systems. This work offers significant contributions to stability theory by enhancing the tools available for fractional dynamic systems, with implications for fields such as engineering, control theory, and applied sciences.

    Michael Ineh: conceptualization, methodology, software, formal analysis, writing—original draft preparation, writing—review and editing; Umar Ishtiaq: conceptualization, validation, investigation, resources, visualization, funding acquisition; Jackson Ante: conceptualization, methodology, supervision; Mubariz Garayev: validation, investigation, resources, visualization, funding acquisition; Ioan-Lucian Popa: conceptualization, software, validation, investigation, project administration. All authors have read and agreed to the published version of the manuscript.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The authors extend their appreciation to King Saud University for funding this work through Researchers Supporting Project number (RSPD2025R1056), King Saud University, Riyadh, Saudi Arabia.

    The authors declare that they have no conflicts of interest.



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