Microsoft Word - 1190 Article Text, Copyedited.doc Adv Syst Sci Appl 2022; 01; 167-175 Published online at https://ijassa.ipu.ru. On the Stability of Periodic Difference Inclusions Mikhail Morozov* V.A. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia E-mail: granmiguel@mail.ru Abstract: The paper considers asymptotically stable periodic difference inclusions. The uniform character of the convergence of solutions to zero is established. For selector-linear difference inclusions the equivalence of the uniform asymptotic stability and the uniform exponential stability is proved, and a necessary and sufficient condition for the uniform asymptotic stability in the form of a certain limit relation is obtained. Examples of systems leading to periodic difference inclusions are given. These 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, systems used to solve problems associated with the study of large electric power systems in the presence of forced oscillations. Keywords: periodic difference inclusion, periodic selector-linear difference inclusion, uniform asymptotic stability, uniform exponential stability 1. INTRODUCTION The problem of the stability of difference and discrete inclusions arises in various areas of mathematics: in control theory, in linear algebra, in the study of convergence of iterative processes, in problems related to discrete wavelet transforms and Markov chains. In some cases, selector-linear inclusions can be used. For example, this is the problem of absolute stability, the study of linear non-stationary systems, the matrix of the right side of which satisfies interval constraints, the study of the stability of control systems that contain elements with incomplete information. Nonlinear discrete control systems were considered in [7,5]. Their equivalence to autonomous selector-linear difference inclusions is proved. Various stability conditions were obtained using the method of Lyapunov functions. For autonomous selector-linear difference inclusions, the Lyapunov indicator was introduced and some properties of solutions were established in [1], asymptotic stability conditions in the form of constraints on the right-hand side of inclusions in [2] and those in the form of the existence of smooth and finite-step Lyapunov functions in [3,4] were obtained. The problems of absolute and robust stability of discrete control systems with periodically varying parameters were solved in [6,8]. In particular, it was established that the considered control systems with periodic parameters are equivalent, in the sense of the coincidence of the sets of solutions, to time-periodic selector-linear difference inclusions. In this paper, periodic difference inclusions are considered. It is known that the properties of the uniform asymptotic stability and the uniform exponential stability of zero solution of autonomous selector-linear difference inclusions are equivalent. For periodic selector-linear difference inclusions, the question of the equivalence of these properties has not previously been considered. The remainder of this paper is structured as follows. In Section 2, we consider periodic difference inclusions of general form and periodic selector-linear difference inclusions. We * Corresponding author: granmiguel@mail.ru 168 M. MOROZOV Copyright ©2022 ASSA. Adv. in Systems Science and Appl. (2022) give preliminary remarks and definitions. In Section 3, it is proved that under some restriction of initial conditions, the convergence of solutions of a periodic difference inclusion to zero has the uniform character. We also establish the equivalence of the uniform asymptotic stability and the uniform exponential stability of the zero solution for periodic selector-linear difference inclusions. Using the variational method for periodic selector-linear difference inclusions, a necessary and sufficient condition for uniform asymptotic stability is obtained in the form of a limit relation. In Section 4, two examples of systems leading to periodic difference inclusions are given. In Section 5, we offer concluding remarks. 2. STATEMENT OF THE PROBLEM Consider the dynamic systems described by periodic difference inclusion (2.1) (2.1) where , is the set of natural numbers. Everywhere below we assume that in some domain for all the set is nonempty, bounded, closed, convex, and the multivalued function is upper semicontinuous. We will call a solution of inclusion (2.1) a sequence of vectors satisfying for all inclusion (2.1). We assume that the sequence is the equilibrium position of the inclusion (2.1). Due to the multivaluedness of the function a point , generally speaking, specifies not one but a set of solutions. Due to the periodicity of the multivalued function in when studying the properties of solutions of inclusion (2.1), without loss of generality, we can assume that Definition 2.1: Inclusion (2.1) is stable if for any there exists such that, as soon as the initial conditions satisfy the condition , the solution with initial condition satisfies the inequality for all Definition 2.2: Inclusion (2.1) is asymptotically stable if the conditions of Definition 2.1 are satisfied and the limit relation holds true, that is, for any there exists such that for all the solution of inclusion (2.1) satisfies the inequality . Definition 2.3: Inclusion (2.1) is uniformly asymptotically stable if the conditions of Definitions 2.1 and 2.2 are satisfied and numbers and do not depend on solution , and Consider a periodic selector-linear difference inclusion ),,()1( xsFsx Î+ ),,(),( xMsFxsF += , ... ,1 ,0=s nRxÎ ,NM Î N }:{ , ,0{ 00 RxxGGxMsG RR £=Σ£= Gxs Î),( nRxsF Ì),( nn RRF ®+1: )(sx )},({ sx NsÎ Nsxsx Î= s 0,)( :)}({ 0)( ºsx ),( xsF ),( 00 xs ),( xsF s ),,( 00 xssx .0 0 Ms ££ 0>e 0),),,,(( 000 >ed sxssx 00 )( xsx = ),),,,(( 0000 ed sxssxx < ),,( 00 xssx 0x e<),,( 00 xssx .00 ³³ ss 0),,(lim 00 = +¥® xssx s 0>h NxsxssxS Î),,),,,(( 0000 h ),,),,,(( 00000 hxsxssxSss +³ ),,( 00 xssx h<),,( 00 xssx d S ),,( 00 xssx 00 ³s .0 nRx Î ON THE STABILITY OF PERIODIC DIFFERENCE INCLUSIONS 169 Copyright ©2022 ASSA. Adv. in Systems Science and Appl. (2022) , , (2.2) where is the equilibrium position of the difference inclusion (2.2), is a convex compact set of real - matrices. Let be a solution of inclusion (2.2) defined by a matrix . Definition 2.4: Inclusion (2.2) is uniformly exponentially stable if there exist numbers and such that, for any solution of inclusion (2.2), the inequality holds true for any matrix , any and . The problem is to establish for inclusion (2.1) uniform character of the limit relation in Definition 2.2. In the case of inclusion (2.2), the problem is to determine a condition for uniform asymptotic stability and to prove the equivalence of the properties of uniform asymptotic stability and uniform exponential stability. 3. RESULTS Theorem 3.1: If inclusion (2.1) is asymptotically stable, then there exists such that all solutions of inclusion (2.1) satisfy the condition uniformly with respect to for any and The proof of Theorem 3.1 is carried out mutatis mutandis by the scheme of Theorem 1 in [9]. Let be a solution of inclusion (2.2) with the initial conditions corresponding to a matrix To obtain a condition for uniform asymptotic stability of inclusion (2.2), we introduce into consideration the functions and (3.1) By the Weierstrass theorem, taking into account the form of inclusion (2.2), the function is defined for any and since the functional is a continuous function of the variables and . Using the function we can formulate the following criterion for asymptotic stability. ),,()1( xsFsx Î+ { })()( ,)(:),( ssBxsByyxsF WÎ== )()( sMs W=+W , ... ,1 ,0=s ,NM Î 0)( ºsx )(sW )( nn´ ),,( 00 xssxB )()( ssB WÎ 0>a 1³b ),,( 00 xssxB ))(exp(),,( 0000 ssxxssxB --£ ab )()( ssB WÎ 00 ³³ ss nRx Î0 00 >d ),,( 00 xssx 0),,(lim 00 = +¥® xssx s ),( 00 xs Ms ££ 00 .: 000 d£xx ),,( 00 xssxB ),,( 00 xs ).()( ssB WÎ 2 00)()(00 ),,(max),,( xssxxstW BssB WÎ = .0 ),,,(max),( 00010 0 ³³= = ssxssWss x r ),( 0ssr 0x NsÎ 2 00 ),,( xssxB 0x )(sB ),( 0ssr 170 M. MOROZOV Copyright ©2022 ASSA. Adv. in Systems Science and Appl. (2022) Theorem 3.2: Inclusion (2.2) is uniformly asymptotically stable if and only if the limit relation (3.2) holds true uniformly with respect to . Proof. Necessity. Since inclusion (2.2) is uniformly asymptotically stable, then for any there exists such that for all From the last inequality and (3.1) follows that for all and , which proves the necessity of condition (3.2). Sufficiency. Due to the uniform boundedness of the elements of the matrix the solution of inclusion (2.2) with will be uniformly bounded for all Therefore, for any there exists , such that for all It follows from the last inequality and relation (3.2) that there exists, that is independent of such that for all . For any , , , . (3.3) From (3.3), linearity and periodicity of inclusion (2.2) for any , any , and follows . We set . Then it follows from (3.3) that solutions of inclusion (2.2) satisfy the inequality for all and , if only . It follows from (3.2) that for any and there exists such that is independent of and , such that for all the inequality holds true and hence the inequality due to the linearity and periodicity of inclusion (2.2). Therefore, for all Thus, the solution of inclusion (2.2) satisfies all the conditions of Definition 2.2. Theorem 3.2 is proved. � 0),(lim 0 = +¥® ss s r 00 ³s 0>h ,)( NS Îh h<),,( 00 xssxB ,00 ³s ,0 Sss +³ ),()( ssB WÎ .1: 00 =xx 2 0 ),( hr s 1)( * ³sg )(),( * 0 sss gr £ . ..., ,1 , 000 *++= sssss 10 ³r 0s 00 ),( rr e 00 ³s ,1: 00 =xx 0ss ³ )()( ssB WÎ ere <)/,,( 000 xssxB 0/)( reed = e<),,( 00 xssxB 00 ³³ ss )()( ssB WÎ )(0 edh ,0>R ,),( NRS Îh )()( ssB WÎ 00 ³s ,0{ 0 ³s ),,(0 RSss h+³ ,10 =x )}()( ssB WÎ h<),,( 00 RxssxB h<),,( 00 xssxB ,0{ 0 ³s ),,(0 RSss h+³ ,10 =x )}.()( ssB WÎ ,/),(),,( 22 0 2 00 RssxssxB hr <£ ON THE STABILITY OF PERIODIC DIFFERENCE INCLUSIONS 171 Copyright ©2022 ASSA. Adv. in Systems Science and Appl. (2022) Consider the system of difference equations (3.4) where The transition matrix of system (3.4) is the matrix connecting the solution and i.e., satisfying the equalities and , where is the unit matrix of order From the equality it follows Corollary 3.1: Inclusion (2.2) is uniformly asymptotically stable if and only if the transition matrix of system (3.4) satisfies the condition , uniformly in and The equivalence of the uniform asymptotic stability and the uniform exponential stability for inclusion (2.2) is established by the following theorem. Theorem 3.3: Inclusion (2.2) is uniformly asymptotically stable if and only if inclusion (2.2) is uniformly exponentially stable. Proof. The sufficiency follows directly from Definitions 2.3 and 2.4, and only necessity needs to be proved. To prove the necessity we use the statement of Theorem 3.2. Let us show that the uniform exponential stability of inclusion (2.2) follows from condition (3.2). For solutions of inclusion (2.2) we have (3.5) for all where the function is defined in (3.1). Note that solutions of inclusion (2.2) satisfy the equality for all Let where ( denotes the integer part of ). Using (3.5), we obtain ,)()1( xsBsx =+ ),()( ),()( sMsssB W=+WWÎ , ... ,1 ,0=s ,nRxÎ .NM Î ),( 0ssBP ),,( 00 xssxB ,0x 0000 ),(),,( xssxssx BB P= ,),( 00 nB Ess =P 0ss ³ nE .n ,),(max),( 2 0)()(0 ssss BssB P= WÎ r ,0ss ³ ),( 0ssBP 0),(lim 0 =P +¥® ssBs 00 ³s ).()( ssB WÎ =200 ),,( xssxB ( ) £ 2 000 2 0 /,, xxssxx B ,),( 2 00 xssr ,0{ 0 ³s ,0: 00 ¹xx )},()( ssB WÎ ),( 0ssr ),,()),,(,,( 0020012 xssxxssxssx BBB = .012 sss ³³ ,0 mkSss ++= ,0]/)[( 0 ³-= Sssk Sm <£0 ][z z =++ 2 000 ),,( xsmkSsxB £+-++-+++ 2 00000 )),,)1((,)1(,( xsmSksxmSksmkSsx BB ×+-+++£ ))1(,( 00 mSksmkSsr £+-+ 2 000 )),,)1(( xsmSksxB 172 M. MOROZOV Copyright ©2022 ASSA. Adv. in Systems Science and Appl. (2022) … (3.6) Since inclusion (2.2) is uniformly asymptotically stable, relation (3.2) holds. Therefore, for any there exists such that for each of the factors in (3.6). Therefore, for any (3.7) Taking into account that from (3.5) we obtain the estimate (3.8) where From (3.7) and (3.8) it follows Since we have for solutions of inclusion (2.2), where and the numbers and do not depend on , and . Consequently, under Theorem 3.3, inclusion (2.2) is uniformly exponentially stable. Theorem 3.3 is proved. � It follows from Theorem 3.2 that for uniformly asymptotically stable inclusion (2.2) there exists a number independent of , such that Using this property in the proof of Theorem 3.3 allows us to establish the uniform exponential stability of inclusion (2.2). Therefore, the following statement is true. Corollary 3.2: For inclusion (2.2) to be uniformly asymptotically stable, it is necessary and sufficient that there exists the number such that 4. EXAMPLES Example 4.1: The problem of absolute stability was solved in [6] for nonlinear nonstationary discrete control systems with a periodic linear part described by the equations , (4.1) ),( Sss -£ r ×-- )2,( SsSsr ×+--× ),)1(( 0 msSksr .),,( 2 000 xsmsxB + 0>h ),(hS 1)exp(),)1(( <-=<--- chjSsSjsr ).0( >c kjjSsSjs ,1 ),,)1(( =---r ...2,1,0=k £200 ),,( xssxB .),,()exp( 2 000 xsmsxk B +-c ,1),( 0 =sss £+ 2 000 ),,( xsmsxB £+ 2 000 ),( xsmsr ,200 xb .1),(max 0000 ³+= ££ smsr Sm b £200 ),,( xssxB ,...2,1,0 ),exp(2 00 =- kkx cb ],/)[( 0 Sssk -= £),,( 00 xssxB )),(exp( 00 ssx --ab ),2/(,10 Se cabb c =³= 0>a 1³b 00 ³s 0x )()( ssB WÎ ,NSÎ ),( 00 xs .1),( 0 ³º+ÎÎ TttfTtfIbIa )(tf ].,[ ],,0[ 21211 bbIaI == )(tf p , 0 )(-- 1 0 ),,( , , , 2 2 11 21 ÷÷ ø ö çç è æ =÷÷ ø ö çç è æ ==== tbfa batA x x x dt dz dt dxxzx 174 M. MOROZOV Copyright ©2022 ASSA. Adv. in Systems Science and Appl. (2022) (4.4) Consider the discrete analogue of system (4.4) (4.5) where is discrete time. System (4.5) is equivalent to periodic selector-linear difference inclusion (2.2), where the multivalued function is defined in each point by the relation 5. CONCLUSION Periodic difference inclusions are considered. It is proved that under some restriction on the initial conditions, the convergence of solutions to zero is uniform. For periodic selector- linear difference inclusions the equivalence of the properties of uniform asymptotic stability and uniform exponential stability of the zero solution is established. Using the variational method the necessary and sufficient condition for the uniform asymptotic stability in the form of the limit relation has been obtained. Examples of systems leading to the consideration of periodic difference inclusions are presented. 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