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Adv Syst Sci Appl 2024; 02; 187-191 
Published online at https://ijassa.ipu.ru. 

 

Lyapunov Functions for Periodic 
Selector-Linear Difference Inclusions  

Mikhail Morozov 

V.A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, 
Moscow, Russia  

Abstract: The paper considers periodic selector-linear difference inclusions. A class of time-
periodic quasi-quadratic Lyapunov functions is distinguished, as well as parametric classes of 
piecewise-quadratic and piecewise-linear Lyapunov functions. These functions establish 
necessary and sufficient conditions for asymptotic stability. An example leading to periodic 
selector-linear differential and difference inclusions is given. The results can find applications in 
the stability analysis of control systems with periodic parameters, in particular, servomechanisms 
whose elements operate on alternating current, control systems with amplitude-frequency 
modulation. 

Keywords: periodic selector-linear difference inclusion, asymptotic stability, Lyapunov functions. 

1. INTRODUCTION 

The problem of the stability of periodic difference inclusions arises in the study of discrete 
control systems with periodic parameters and uncertainty, in particular, tracking systems that 
operate on alternating current and systems with amplitude-frequency modulation. In some 
cases, such as the absolute stability problem, the study of linear nonstationary systems whose 
right part matrix satisfies interval constraints can be used selector-linear difference inclusions. 
In [5] periodic difference inclusions are considered. The definitions of asymptotic, uniform 
asymptotic and uniform exponential stability are given and the equivalence of these properties 
for selector-linear difference inclusions is proved. On the basis of the variational approach, a 
necessary and sufficient condition for uniform asymptotic stability in the form of some limit 
relation is obtained. In this paper we obtain asymptotic stability criteria for periodic selector-
linear difference inclusions based on the Lyapunov function method. 

The remainder of this paper is structured as follows. In Section 2, we consider periodic 
selector-linear difference inclusions and set the problem of obtaining asymptotic stability 
criteria for such inclusions on the basis of the Lyapunov function method. In section 3 we 
distinguish a class of periodic in time Lyapunov functions of quasi-quadratic form and 
parametric classes of piecewise quadratic and piecewise linear Lyapunov functions. Theorems 
1-3 are formulated and proven. They establish necessary and sufficient conditions for 
asymptotic stability of the considered inclusions. In Section 4, an example of a system leading 
to periodic differential and difference inclusions is given. In Section 5, we offer concluding 
remarks.  

2. STATEMENT OF THE PROBLEM 

Consider the dynamic systems described by periodic selector-linear difference inclusion  
𝑥(𝑠 + 1) ∈ 𝐹(𝑠, 𝑥), 𝑠 = 0, 1, …  ,   𝑥 ∈ 𝑅 , (2.1) 

where the set-valued map 𝐹: 𝑅 → 𝑅  has the form 

𝐹(𝑠, 𝑥) = {𝑦: 𝑦 = 𝐵(𝑠)𝑥,  𝐵(𝑠) ∈ Ω(𝑠)}, 



188             M. MOROZOV 

Copyright ©2024 ASSA                                                                                    Adv. in Systems Science and Appl. (2024) 

Here 𝛺(𝑠), 𝛺(𝑠 + 𝑁) = 𝛺(𝑠) (𝑠 = 0,1, . .. , 𝑁is a natural number) is a convex, compact set of 
real (𝑛 × 𝑛) – matrices 𝐵. Such set-valued maps are called selector-linear, since the right-
hand side is a union of linear maps. The sequence of vectors {𝑥(𝑠)}, satisfying for all 𝑠 =
0,1, … inclusion (2.1), is the solution of the inclusion (2.1). Let 𝑥(𝑠, 𝑠 , 𝑥 ) be the solution of 
inclusion (2.1) with initial conditions (𝑠 , 𝑥 ). Due to the periodicity of the multivalued 
function 𝐹(𝑠, 𝑥)  in 𝑠  without generality restriction we can assume that 0 ≤ 𝑠 ≤ 𝑁.  The 
equivalence of the properties of asymptotic stability, uniform asymptotic stability, and 
uniform exponential stability for inclusion (2.1) was proved in [5]. Hereinafter we will refer 
to the asymptotic stability of inclusion (2.1). 

The problem is to identify the parametric classes of Lyapunov functions establishing 
necessary and sufficient conditions for asymptotic stability of inclusion (2.1) and to construct 
the stability criteria for inclusion (2.1) using a discrete analogue of the direct Lyapunov 
method.  

3. RESULTS 

Theorem 3.1:  

The following conditions are equivalent: 

1. Inclusion (2.1) is asymptotically stable. 

2. There exists the Lyapunov function 𝑣(𝑠, 𝑥) of the quasiquadratic form 
𝑣(𝑠, 𝑥) = 𝑥 ′𝐿(𝑠, 𝑥)𝑥,  𝐿(𝑠, 𝑥) = (𝑙 (𝑠, 𝑥)) ,  , 𝐿(𝑠 + 𝑁, 𝑥) = 𝐿(𝑠, 𝑥), 

𝐿′(𝑠, 𝑥) = 𝐿(𝑠, 𝑥) = 𝐿(𝑠, 𝜇𝑥), 𝑥 ≠ 0, 𝜇 ≠ 0, 𝑣(𝑠, 0) ≡ 0 (3.1) 

which is 𝑁-periodic in 𝑠, homogeneous (of second order), strictly convex in 𝑥, and it satisfies 

the following inequality:  

𝑚𝑎𝑥
∈ ( , )

𝑣(𝑠, 𝑦) ≤ 𝜃𝑣(𝑠, 𝑥), 𝑥 ∈ 𝑅 , 𝑠 = 0,1, … (3.2) 

for some 𝜃 (0 < 𝜃 < 1). 

In (3.1) the prime means the transposition operation.  
The sufficiency of the conditions of Theorem 3.1 is established (using the lemma given 

in [4]) by the reasoning used in the proof of exponential stability of discrete systems in [2]. 
The proof of the necessity follows the same scheme as the proof of the corresponding 
conditions of the theorem in [4], where exponential estimates for solutions of inclusion (2.1) 
are used.  Theorem 3.1 is an extension of the classical theorem for the discrete analogue of 
direct Lyapunov method in [2] to selector-linear periodic difference inclusions (2.1).  

The reasoning used in the proof of the lemma in [4] proves that, under Theorem 1, there 
exist constants such 𝜆 ≥ 𝜆 > 0 that for all 𝑠 ≥ 0 and  𝑥 ∈ 𝑅 , the inequalities 

𝜆 ‖𝑥‖ ≤ 𝑣(𝑠, 𝑥) ≤ 𝜆 .‖𝑥‖ . (3.3) 

(3.3) implies positive definiteness of the function 𝑣(𝑠, 𝑥). 

The problem of constructing the Lyapunov function is simplified if this function is 
selected from a certain parametric class of functions that depend on a finite number of 
parameters. The class of quasi-quadratic Lyapunov functions (3.1) is not parametric. The 
existence of a parametric class of Lyapunov functions, also defining necessary and sufficient 
conditions for the asymptotic stability of inclusion (3.1), can be formulated as a theorem. 

Theorem 3.2:  

Inclusion (2.1) is asymptotically stable iff, for some integer 𝑀 ≥ 𝑛, there exists a periodic 
in 𝑠 (of period 𝑁), piecewise-quadratic Lyapunov function  



                    LUAPUNOV FUNCTIONS FOR PERIODIC SELECTOR-LINEAR DIFFERENCE INCLUSIONS  189 

Copyright ©2024 ASSA.                                                                                    Adv. in Systems Science and Appl. (2024) 

 

𝑣 (𝑠, 𝑥) = 𝑚𝑎𝑥 𝑙 (𝑠), 𝑥⟩ , (𝑙 (𝑠 + 𝑁) = 𝑙 (𝑠), (3.4) 

for which the inequality (3.2) is satisfied for all 𝑠 ≥ 0 and 𝑥 ∈ 𝑅 , and the 𝑛-dimensional 
periodic vectors (𝑙 (𝑠)  (𝑙 (𝑠 + 𝑁) = 𝑙 (𝑠)), 𝑗 = 1, 𝑀 satisfy the condition  

𝑟𝑎𝑛𝑘𝐿(𝑠) = 𝑛 ≤ 𝑀, 𝐿(𝑠) = 𝑙 (𝑠), … , 𝑙 (𝑠) , 𝑠 = 0,1, … (3.5) 

(i.e., periodic (𝑛 × 𝑀) matrix 𝐿(𝑠)(𝐿(𝑠 + 𝑁) = 𝐿(𝑠)) has a maximum rank for all 𝑠 ≥ 0). 
We denote by ⟨⋅, ⋅⟩ a scalar product of vectors.  
The set of Lyapunov functions (3.4) forms a parametric class. The parameters defining 

this class are the components of periodic vectors 𝑙 (𝑠) (𝑗 = 1, 𝑀, 𝑠 ≥ 0 ) and the integer 𝑀 ≥
𝑛. If the rank condition (3.5) is satisfied, the function 𝑣 (𝑠, 𝑥) is positively defined in 𝑅  and 
its level surfaces at any fixed one 𝑠 ≥ 0 are centrally symmetric convex polyhedrons. The 
vectors 𝑙 (𝑠),  𝑗 = 1, 𝑀 defining the norms to their faces. 

Proof. Sufficiency. The sufficiency of the conditions of Theorem 3.2 is established 
according to the standard scheme of the proof of exponential stability in [2] using inequality 
(3.2) and estimates 𝜆 ‖𝑥‖ ≤ 𝑣 (𝑠, 𝑥) ≤ 𝜆 ‖𝑥‖ ,  𝜆 ≥ 𝜆 > 0  for function (3.4) under 
condition (3.5).  

Necessity. It follows from Theorem 3.1 that for inclusion (2.1) there exists a periodic in 𝑠 
Lyapunov function 𝑣(𝑠, 𝑥) of the quasiquadratic form satisfying the conditions of Theorem 
3.1. Consider centrally symmetric convex bodies 

𝑃 (𝑠) = {𝑥: 𝑣(𝑠, 𝑥) ≤ 1}, 𝑃 (𝑠) = {𝑥: 𝑣(𝑠, 𝑥) ≤ 𝑟}, 𝑠 = 0,1, . . . ,0 < 𝑟 < 1. 

Let 𝑖𝑛𝑡 𝐴 be the set of interior points of the set 𝐴. Since 0 < 𝑟 < 1, then 𝑃 (𝑠) ⊂ 𝑖𝑛𝑡 𝑃 (𝑠). 
It follows from Theorem 20.4 in [6] that there exists a centrally symmetric convex polyhedron 
𝐷 (𝑠), that the relations 

𝑃 (𝑠) ⊂ 𝑖𝑛𝑡 𝐷 (𝑠) ⊂ 𝐷 (𝑠) ⊂ 𝑖𝑛𝑡 𝑃 (𝑠) ⊂ 𝑃 (𝑠). (3.6) 

Let 2𝑀 be the number of faces of the polyhedron 𝐷 (𝑠),  then there are 𝑀  pairs of 
centrally symmetric faces. Let 𝑟 > 0 be the distance from a point 𝑥 = 0 to the faces of the𝑗 - 

pair, and let ±𝑛  ( 𝑛 = 1), 𝑗 = 1, 𝑀 be the unit external normals to the faces of this pair. In 
order the surface of the polyhedron 𝐷 (𝑠)  to be the surface of the level 𝛺 (𝑠) =

{𝑥: 𝑣 (𝑙(𝑠), 𝑥) = 1} for function 𝑣 (𝑙(𝑠), 𝑥)  (3.4), choose 𝑙 (𝑠) = 𝑟 𝑛 , 𝑗 = 1, 𝑀, 𝑠 =

0,1, . ..  Then the vectors 𝑙 (𝑠), 𝑗 = 1, 𝑀,  𝑠 = 0,1, . ..  satisfy rank condition (3.5), because 
otherwise the surfaces of the level function 𝑣 (𝑙(𝑠), 𝑥) will not be bounded. 

Let 𝛿(𝐾, 𝑧) = 𝑚𝑎𝑥
∈

(𝑧, 𝑥),  𝑧 ∈ 𝑅  be the reference function of compact 𝐾 ⊂ 𝑅  (see 

[6]). Since 𝛿(𝑃 (𝑠), 𝑧) = √𝑟𝛿(𝑃 (𝑠), 𝑧), it follows from (3.6) and property 9 in [1] 
√𝑟𝛿(𝑃 (𝑠), 𝑧) < 𝛿(𝐷 (𝑠),z) < 𝛿(𝑃 (𝑠),z),  z ≠ 0. (3.7) 

Let us show that there exists a number 𝑟   (𝑟 < 𝑟 < 1), such that the inequality 

√𝑟𝛿(𝑃 (𝑠), 𝑧) < 𝑟 𝛿(𝐷 (𝑠),z),  z ≠ 0. (3.8) 

Suppose that inequality (3.8) is not satisfied, i.e., for any 𝑟   (𝑟 < 𝑟 < 1)  

√𝑟𝛿(𝑃 (𝑠), 𝑧) ≥ 𝑟 𝛿(𝐷 (𝑠),z) or
𝑟

𝑟
𝛿(𝑃 (𝑠), 𝑧) ≥ 𝛿(𝐷 (𝑠),z),  z ≠ 0,  

that contradicts inequality (3.7). 
Function √𝑟 𝛿(𝐷 (𝑠),z) = 𝛿(𝐷 ,z) is the reference function of the polyhedron 𝐷 (𝑠) =

{𝑥: 𝑣 (𝑙(𝑠), 𝑥) ≤ 𝑟 },  which is similar to polyhedron 𝐷 (𝑠) = {𝑥: 𝑣 (𝑙(𝑠), 𝑥) ≤ 1}. 
Therefore (3.8) is equivalent to the relation 

𝑃 (𝑠) ⊂ 𝑖𝑛𝑡 𝐷 (𝑠) ⊂ 𝐷 (𝑠). (3.9) 



190             M. MOROZOV 

Copyright ©2024 ASSA                                                                                    Adv. in Systems Science and Appl. (2024) 

It follows from inequality (3.2) for the function 𝑣(𝑠, 𝑥) that 𝐹(𝑃 (𝑠)) ⊂ 𝑃 (𝑠), where  

𝐹(𝑃 (𝑠)) = ∪
∈ ( )

𝐹(𝑠, 𝑥). (3.6) implies that 𝐷 (𝑠) ⊂ 𝑃 (𝑠).  Therefore 

𝐹 𝐷 (𝑠) ⊂ 𝐹 𝑃 (𝑠) ⊂ 𝑃 (𝑠). (3.10) 

It follows from (3.9) and (3.10) 𝐹(𝐷 (𝑠)) ⊂ 𝑖𝑛𝑡 𝐷 (𝑠) and therefore the condition 

𝑚𝑎𝑥
∈ ( , )

𝑣 (𝑙(𝑠), 𝑦) < 𝑟 , 𝑥 ∈ 𝐷 (𝑠), 𝑠 = 0,1, … (3.11) 

Since  

𝑣 (𝑙(𝑠), 𝜇𝑥) = 𝜇 𝑣 (𝑙(𝑠), 𝑥) and 𝐹(𝑠, 𝜇𝑥) = 𝜇𝐹(𝑠, 𝑥) 

for all 𝑠 ≥ 0, 𝑥 ∈ 𝑅  and 𝜇 ∈ 𝑅 , then, for the above-constructed function 𝑣 (𝑙(𝑠), 𝑥) from 

(3.11), the inequality 

𝑚𝑎𝑥
∈ ( , )

𝑣 (𝑙(𝑠), 𝑦) ≤ 𝑟 𝑣 (𝑙(𝑠), 𝑥), 0 < 𝑟 < 1, 𝑥 ∈ 𝑅 , 𝑠 = 0,1, … (3.12) 

of the form (3.2). This completes the proof of Theorem 3.2.  
Consider piecewise linear Lyapunov functions 

𝑉 (𝑠, 𝑥) = 𝑚𝑎𝑥 𝑙 (𝑠), 𝑥⟩ . (3.13) 

A corollary of Theorem 3.2 is 

Theorem 3.3: Inclusion (2.1) is asymptotically stable iff, for some integer 𝑀 ≥ 𝑛 there 
exists a periodic on 𝑠  (period 𝑁 ), piecewise-linear Lyapunov function 𝑉 (𝑠, 𝑥)  (3.13), 
satisfying condition (3.5) and inequality (3.2) for all 𝑥 ∈ 𝑅 , 𝑠 ≥ 0. 

Proof. Since 

𝑣 (𝑠, 𝑥) = 𝑚𝑎𝑥 ⟨𝑙 (𝑠), 𝑥⟩ = 𝑚𝑎𝑥 𝑙 (𝑠), 𝑥⟩ = 𝑉 (𝑠, 𝑥), 

then each function 𝑣 (𝑠, 𝑥) can be matched with function 𝑉 (𝑠, 𝑥) =  𝑣 (s,x). In the same 
way each function 𝑉 (𝑠, 𝑥)  (3.13) can be associated with function 𝑣 (𝑠, 𝑥) = 𝑉 (𝑠, 𝑥). 
From this and Theorem 3.2 follows the validity of Theorem 3.3.  

4. EXAMPLE 

Consider the torsional vibrations of the crankshafts of a single-cylinder engine with a flywheel 
with allowance for the inertia of connecting rods and pistons. Let us assume that the mass of 
the flywheel is sufficiently large, so that the rotation of the shaft can be considered uniform. 
Let 𝜔 - angular velocity of the flywheel, 𝑐 - torsional stiffness coefficient of the shaft. In [3] 
it is shown that vibrations are described by second order differential equation 

𝑑 𝑞

𝑑𝑡
+

𝑐

𝜔
𝑝(𝑡)𝑞 = 0, (4.14) 

where 𝑞  – generalized coordinate associated with the angle of rotation of the crank, and 

periodic function 𝑝(𝑡) of the period 2𝜋can be determined by the kinetic energy of the crank 

together with the associated moving masses (connecting rod and piston). 

Equation (4.1) is a special case of an equation of more general form 

𝑑 𝑧

𝑑𝑡
+ 𝑏𝑓(𝑡)𝑧 = 0, (4.2) 



                    LUAPUNOV FUNCTIONS FOR PERIODIC SELECTOR-LINEAR DIFFERENCE INCLUSIONS  191 

Copyright ©2024 ASSA.                                                                                    Adv. in Systems Science and Appl. (2024) 

 

where 𝑓(𝑡) is a periodic function of time (with period 𝑇 > 0), and  𝑏 ∈ 𝐼, 𝐼 = [𝑏 , 𝑏 ] is a 
certain parameter. Let us introduce notation 

 𝑥 = 𝑧,  𝑥 =  = ,  𝑥 =
𝑥
𝑥

,  𝐴(𝑡, 𝑎, 𝑏) =
  0           1
−𝑏𝑓(𝑡)    0 

. 

Equation (4.2) will take the form of second order system 
𝑑𝑥

𝑑𝑡
= 𝐴(𝑡, 𝑏)𝑥,  𝐴(𝑡 + 𝑇) ≡ 𝐴(𝑡),

𝑏 ∈ 𝐼,  𝑡 ≥ 0,  𝑇 > 0,  𝑥 ∈ 𝑅 .

 (4.3) 

Consider discrete analogue of system (4.3)  
𝑥(𝑠 + 1) = 𝐴(𝑠, 𝑏)𝑥(𝑠),  𝐴(𝑠 + 𝑀) ≡ 𝐴(𝑠),

𝑏 ∈ 𝐼, 𝑀 ∈ 𝑁,  𝑥 ∈ 𝑅 ,
 (4.4) 

where 𝑠 = 0,1, . .. is discrete time. 
System (4.4) is equivalent to periodic selector-linear difference inclusion (2.1), where the 

multivalued function 𝐹(𝑠, 𝑥)  (𝐹(𝑠 + 𝑀, 𝑥) ≡ 𝐹(𝑠, 𝑥)) is defined at each point (𝑠, 𝑥),  𝑥 ∈
𝑅 by the relation 

𝐹(𝑠, 𝑥) = {𝑦: 𝑦 = 𝐴(𝑠, 𝑏)𝑥,  𝑏 ∈ 𝐼}.    
5. CONCLUSION 

For periodic selector-linear difference inclusion (2.1) asymptotic stability criteria were 
obtained. They use quasi-quadratic, piecewise quadratic and piecewise linear Lyapunov 
functions. The example of technical problem leading to consideration of periodic differential 
and difference inclusions is given. 

The obtained results can be used in the study of the stability of control systems with 
periodic parameters. In particular, such systems are the tracking systems, which elements 
operate on alternating current and the control systems with pulse-amplitude modulation.  The 
extracted in Theorems 3.2, 3.3 piecewise-quadratic and piecewise-linear Lyapunov functions 
of the form (3.4), (3.13) establish necessary and sufficient conditions of asymptotic stability 
for inclusion (2.1). These functions can be used when developing numerical methods of 
stability analysis for systems equivalent to difference inclusion (2.1).  

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