Microsoft Word - Morozov+MV Adv Syst Sci Appl 2025; 02; 75-80 Published online at https://ijassa.ipu.ru. Stability Criteria 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. Lyapunov functions from the parametric class of homogeneous forms of even degree are constructed. These functions establish necessary and sufficient conditions of asymptotic stability and can be used in the development of numerical methods for investigating the stability of systems equivalent to the considered difference inclusions. Using piecewise linear Lyapunov functions, an algebraic criterion of asymptotic stability is obtained. An example of a mechanical system leading to periodic selector-linear difference inclusion is considered. Keywords: periodic selector-linear difference inclusions, asymptotic stability, Lyapunov functions, algebraic criterion of asymptotic stability 1. INTRODUCTION. STATEMENT OF THE PROBLEM The study of discrete control systems in a number of cases leads to difference inclusions. In [9] a brief review of publications on this topic is given, see for example [1-3,5]. In [9] for periodic difference inclusions a necessary and sufficient condition of uniform asymptotic stability in the form of some limit relation is obtained on the basis of the variational approach. In [10], for periodic selector-linear difference inclusions, a class of time-periodic Lyapunov functions of quasi-quadratic form, as well as parametric classes of piecewise quadratic and piecewise linear Lyapunov functions were distinguished. With the help of these functions the necessary and sufficient conditions for asymptotic stability were obtained. This paper is a continuation of [9,10] and is devoted to obtaining new asymptotic stability criteria for periodic selector-linear difference inclusions based on the method of Lyapunov functions. Consider periodic selector-linear difference inclusion ),,()1( xsFsx  , ,...,1,0 nRxs  (1.1) where the set-valued map nn RRF 1: has the form  ,)()( ,)(:),( ssBxsByyxsF  here ),(s )()( sNs  )number natural a is ,...,1,0 ( Ns  is a convex, compact set of real )( nn - matrices .B The sequence of vectors  ,)(sx satisfying for all ,...1,0s inclusion (1.1), is the solution of inclusion (1.1). The definitions of asymptotic stability, uniform asymptotic stability and uniform exponential stability of inclusion (1.1) are given in [9]. The equivalence of these properties for inclusion (1.1) is proved there. Keeping this in mind, further we will speak about asymptotic stability of inclusion (1.1). The problem is to construct stability criteria for inclusion (1.1) using the discrete analogue of the direct Lyapunov method [4]. 76 M. MOROZOV Copyright ©2025 ASSA Adv. in Systems Science and Appl. (2025) 2. RESULTS In [10] parametric classes of piecewise quadratic ,),(max),( 2 1 xslxsV j Mj M   )()( slNsl jj  (2.1) and piecewise linear ,),(max),( 1 xslxsV j Mj M   )()( slNsl jj  (2.2) Lyapunov functions were considered. In (2.1) and further we denote by , a scalar product of vectors. In [10] it was proved that for asymptotic stability of inclusion (1.1) it is necessary and sufficient that for some integer nM  there exists periodic on s (period N ) Lyapunov function (2.1) or (2.2) satisfying the condition ,)( MnsrankL  ))(),...,(()( 1 slslsL M (2.3) and the inequality ),(),1(max ),( xsvysv MM xsFy   (2.4) for all ,nRx 0s and some ).10(  Consider periodic on s Lyapunov functions from the class of homogeneous forms of degree r2 ( hereafter r is natural) ,),(),( 1 2 ,    M j rj rM xslxsV ),()( slNsl jj  ,...1,0s (2.5) The condition of positive definiteness of function (2.5), as well as for functions (2.1), (2.2), is condition (2.3). If this condition is fulfilled, the function ),( , xsV rM will be strictly convex on nRx for every .0s Theorem 2.1: Inclusion (1.1) is asymptotically stable iff there exists Lyapunov function (2.5) periodic in s (of period N ), its vectors )),()(( )( slNslsl jjj  Mj ,1 satisfy for all 0s condition (2.3), for the function inequality (2.4) is satisfied for some 1r . Proof. Sufficiency. The sufficiency of the conditions of Theorem 2.1 is proved by using inequality (2.4) and estimates 0 ,),( 12 2 2 , 2 1   r rM r xxsvx for the positively homogeneous strictly convex function (2.5), which were obtained in Lemma [8]. Necessity. To prove the necessity, we will use the statement of Theorem 3.2 in [10], according to which for asymptotically stable inclusion (1.1) there exists a piecewise quadratic Lyapunov function (2.1) satisfying (2.3) and the inequality ),(),1(max 1 ),( xsVysV MM xsFy   (2.6) for some 1 ).10( 1   We construct the function ),( , xsV rM by choosing as vectors ),(sl j Mj ,1 in (2.5) the vectors defining the Lyapunov function ).,( xsVM Since the inequalities are true     M j rjrjrj Mj xslMxslxsl 1 2 Mj1 22 1 ,),( max ),(),(max then the estimates ),,(),(),( , xsMVxsVxsV r MrM r M  ,nRx ,...1,0s (2.7) The inequality follows from (2.6) and (2.7) STABILITY CRITERIA FOR PERIODIC SELECTOR-LINEAR DIFFERENCE INCLUSIONS 77 Copyright ©2025 ASSA. Adv. in Systems Science and Appl. (2025) ),(),1(max ,1, ),( xsVMysV rM r rM xsFy   . Choosing a positive integer ,0rr  where ,1)]ln/([ln 10  Mr we obtain that the inequality ),(),1(max ,2, ),( xsvysv rMrM xsFy   is satisfied for the constructed function ),( , xsv rM , since for 0rr  the number 2 12  M satisfies the condition .10 2   The periodicity of the function ),( xsvM implies the fulfillment of the equality ).,(),( , , xsVxNsV rMrM  Theorem 2.1 is proved.  Lyapunov function (2.5) can be represented in the form ,)()(),( 1 ,    rN i iirM xsxsV  ),()( sNs ii   ,,1 ,...,1,0 rNis  (2.8) where  ,1 ),( ri Nix all possible elementary monomials of degree r2    n j r rnrji m n m i CNrmxxx nii 1 2 121 )2 ,...)(( 1 – the total number of such monomials. In accordance with the discrete analog of the direct Lyapunov method [4], the construction of Lyapunov function (2.8) for difference inclusion (1.1) is reduced to the search of the parameter vector ,,1 )),(( ri Nix   defining the solution of the set of inequalities ,0 ,0),())(,1(),,( ,,  xxsVxsBsVxs rMrMB  ).()( ssB  (2.9) Equation (2.9) specifies the conditions of strict monotonic decreasing of the function ),(, xsV rM on the solutions of inclusion (1.1). Considering (2.8) and (2.9) we obtain the inequalities ,0 ,0))()())(()1((),,( 1   xxsxsBsxs pN i iiiiB  ).()( ssB  (2.10) Let i ni xx   1 max be cubic norm of vector .x Consider the problem of mathematical programming ),,,(max max max min )()(110 xsB ssBxNsG      .1: 1 2          rN i iG  (2.11) Using the scheme of the proof of Theorem 1 in [11], it can be shown that for existence of Lyapunov function (2.8), satisfying condition (2.10), it is necessary and sufficient that the solution of the problem (2.11) satisfies the inequality .0 To solve the minimax problem (2.11), known methods of numerical solution can be used. The periodic components of the vector , ,,1 ),( ri Nix  can be searched, for example, in the form of Fourier series segments [11]. Through     M j ij Mi dsD 1 1 max)( denote the cubic norm of the square matrix M jiijdD 1,)(  of order .M With Lyapunov functions (2.2) we obtain the criterion of asymptotic stability of inclusion (1) in algebraic form. Theorem 2.2: For asymptotic stability of inclusion (1.1) it is necessary and sufficient that the following conditions are satisfied: 1. For some 1p there exist periodic -)( nn matrices )(sBi ( ),()( sBNsB ii  pi ,1 ), satisfying the condition )),(),...,(()( 1 sBsBcos p ,...1,0s 78 M. MOROZOV Copyright ©2025 ASSA Adv. in Systems Science and Appl. (2025) 2. There exist a number ,nM  periodic -)( Mn matrix )(sL ))()(( sLNsL  of rank ,n and periodic -)( MM  matrices )(sDi )),()(( sDNsD ii  ,,1 pi  satisfying the conditions 1)(   sDi ( ,,1 pi  ,...1,0s ), such that the matrix relations are satisfied ),()()1()( sDsLsLsB ii  ,,1 pi  ,...1,0s (2.12) In (2.12) and further the dash denotes the transpose operation. Proof. Necessity. By Theorem 3.3 [10], for asymptotically stable inclusion (1.1), there exists piecewise linear Lyapunov function (2.2) satisfying condition (2.3) and inequality (2.4) for some ).10(  It follows from Theorem 20.4 [12] that any non-empty closed bounded set can be approximated by a polyhedral convex set. Thus, for the set )(s there exists a natural number 1p and such periodic -)( nn matrices )(sBi ( ),()( sBNsB ii  pi ,1 ), that )),(),...,(()( 1 sBsBcos p and the sets )(s and ))(),...,(( 1 sBsBco p will differ from each other as little as possible. Also, the vectors ,)()( xsBsy ii  ,,1 pi  0s will be as close as desired to the elements of the set ),( xsF in (1.1). Therefore, from (2.2) and (2.4) follow the inequalities ,),(max)(,),1(max 11 xslxsBxsl j Mj i f Mf    ,,1 pi  ,...,1,0s ,nRx hence the inequalities ,),(max)(,),1()( 1 xsqlxsBxslsB j Mj j f i   ,nRx (2.13) for all sif and , ,,1( Mf  ,,1 pi  ,...).1,0s Applying Lemma [7] to (2.13) we obtain that for each i and f ,,1( Mf  ),1 pi  there exist periodic (period )N functions ),(si fj ,,1 Mj  ,...,1,0s such that ,)()()1()( 1    M j ji fj f i slsqslsB     M j i fj s 1 .1)( Introducing the periodic functions ),()( sqsd i fj i fj  we come to the following equations ,)()()1()( 1    M j ji fj f i slsdslsB ,,1 Mf  ,,1 pi  ,...1,0s (2.14) Since    M j i fj qsd 1 )( for any sif и , ,,1( Mf  ,,1 pi  ,...),1,0s then each of the -)( MM  matrices ,))(()( , M jf i fji sdsD  ,,1 pi  satisfies the condition 1)(   qsDi for all ,...1,0s In matrix notation, vector equations (2.14) are equivalent to (2.12). The periodicity of the matrices )(sL and )(sDi in (2.12) follows from the periodicity of the vectors ),(sl f Mf ,1 in (2.2). The necessity is proved. Sufficiency. According to the conditions of the theorem )),(),...,(()( 1 sBsBcos p therefore, any vector ),( xsFy in (1.1) can be represented as ,)()( 1    p i ii xsBsy  ,0)( si ,,1 pi  .1)( 1    p i i s Determining by the matrix )(sL in (2.12), satisfying condition (2.3), Lyapunov function (2.2), we obtain ))()(,1(max),1(max 1),( sxsBsVysv iM pi M xsFy   = (2.15) STABILITY CRITERIA FOR PERIODIC SELECTOR-LINEAR DIFFERENCE INCLUSIONS 79 Copyright ©2025 ASSA. Adv. in Systems Science and Appl. (2025) .)(),1()(maxmax 11 sxslsB f i Mfpi   Since matrix equalities (2.12) are equivalent to vector equalities (2.14), for any ,f ,i s ( ,,1 Mf  ,,1 pi  ,...1,0s ) the following relations are true      M j M j i fjM ji fj M j ji fj f i sdxsVxslsdxslsdxslsB 1 11 .)(),(),()(),()(),1()( Hence .,1 ),,()(),1()(max 1 pixsVsDxslsB Mi f i Mf   Therefore, the equality is valid ),,(),1()(maxmax 11 xsVxslsB M f i Mfpi   (2.16) where .1)(maxmax 110   sDi piNs  It follows from (2.15) and (2.16) that the function ),( xsVM satisfies inequality (2.4). Therefore, by Theorem 3.3 [9] inclusion (1.1) will be asymptotically stable. Theorem 2.2 is proved.  4. EXAMPLE Consider a pendulum of length l , whose suspension axis makes vertical harmonic oscillations with small amplitude  and frequency . The differential equation of motion of the pendulum is [6] ,0sin1 2 22 2        zt gl g dt zd   (3.1) where g is acceleration of free fall. As for a pendulum with a fixed pendulum axis, the vertical is the equilibrium position. But in contrast to the case of a pendulum with a fixed axis, this equilibrium position can be either stable or unstable, depending on the value of . Equation (3.1) is a special case of an equation of the more general form [6] ,0)( 2 2  ztbf dt zd (3.2) where )(tf is a periodic function of time (with period 0T ), and , Ib ],[ 21 bbI  is a certain parameter. Just as in the examples in [9,10], it can be shown that equation (3.1) can be represented as a second-order system of differential equations with periodic coefficients, depending on the parameter .Ib The discrete analog of this system is equivalent to inclusion (1.1). 4. CONCLUSION For periodic selector-linear difference inclusion (1.1) the asymptotic stability criterion is obtained using Lyapunov functions (2.5). 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