Adv Syst Sci Appl 2021; 01; 76-85 Published online at https://ijassa.ipu.ru. On Uniform Convergence Property of Solutions for Periodic Differential Inclusions with Asymptotically Stable Sets Mikhail Morozov V.A. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia E-mail: granmiguel@mail.ru Abstract: The paper considers a periodic differential inclusion with an asymptotically stable set. The uniform character of convergence of solutions to an asymptotically stable set is established. An exponential estimate is obtained for solutions of a periodic differential inclusion homogeneous in state vector. Examples of control systems leading to consideration of periodic differential 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 AC, control systems with pulse amplitude modulation, and systems used to solve problems related to investigating vibrations of milling machines. Keywords: time-invariant differential inclusion, periodic differential inclusion, homogeneous differential inclusion, asymptotically stable set, control system 1. INTRODUCTION Studying control systems has led to the use of differential inclusions theory. Consider a control system ),,,( uxtfx = (1.1) where nRx , dtdxx /= is the velocity vector, t is time, and )(tuu = is the control on which the next constraint is imposed ,)( Utu  (1.2) U is an arbitrary set, nRU  . Under fairly general assumptions control system (1.1) with constraint (1.2) is equivalent to the differential inclusion ),,( xtFx (1.3) where ),( xtF denotes a multivalued mapping, i.e., a function that assigns a set nRxtF  ),( to each time t and each point nRx in the state space. Differential inclusion (1.3) can be adopted not only for a control system with given control constraints but also for other objects. For example, such objects include systems of differential inequalities, implicit differential equations, control systems with state-space constraints, systems with variable structure and with sliding modes, and differential equations with discontinuous right-hand sides. Theory of differential inclusions is well developed and presented in systematic form in [1,4,17]. The use of differential inclusions in control systems theory is covered in the monograph [6]. The examples leading to differential inclusions are given below in Section 4. In some cases, e.g., such as the problem of absolute stability, the study of linear nonstationary systems, the matrix of the right-hand side of which satisfies interval ON UNIFORM CONVERGENCE PROPERTY OF SOLUTIONS FOR PERIODIC DIFFERENTIAL… 77 Copyright ©2021 ASSA. Adv. in Systems Science and Appl. (2021) constraints, and the stability analysis of control systems that contain elements with incomplete information linear-selectionable differential inclusions can be used. A linear- selectionable inclusion is an inclusion of the form ),,( xtFx  ,)()( ,)(:),( ttAxtAyyxtF == (1.4) where nRyx  , and )(t is a set in the space of nn matrices. An inclusion of the form (1.4) is called a linear-selectionable inclusion, because the multivalued mapping ),( xtF in (1.4) is the union of linear single-valued mappings (selectors) ,)( xtA )()( ttA  . In the case of a time-invariant linear-selectionable inclusion, the right-hand side F of this inclusion − the matrix A and the set  − are time-invariant. A linear-selectionable inclusion is said to be asymptotically stable if its trivial solution 0x is asymptotically stable. Time-invariant linear-selectionable inclusions are studied in a number of publications. For time-invariant linear-selectionable inclusions for which the set  is a compact or convex polyhedron, necessary and sufficient conditions of the zero solution asymptotic stability were obtained in [12,13] on the base of Lyapunov functions method. The papers [7,12,13] give various algebraic criteria for the asymptotic stability of time-invariant linear-selectionable inclusions. The publications on time-periodic (in short, periodic) differential inclusions (e.g., [2,8,9]) were mostly devoted to the existence of periodic solutions. Few investigations were focused on the analysis of solutions of periodic differential inclusions and their properties. For example, the weak asymptotical and weak exponential stability of an equilibrium of a periodic differential inclusion were studied in [5,18]. In accordance with the definitions introduced therein, an equilibrium of a given periodic differential inclusion is weakly asymptotically (weakly exponentially) stable if there exists at least a single solution satisfying the standard definitions conditions of the asymptotical (exponential) stability for a differential inclusion equilibrium. The method consists in the design of a first- approximation inclusion and further analysis of the properties of its solutions. In addition, the theorems of the weak asymptotical (weak exponential) stability of an equilibrium of an original inclusion using the corresponding properties of an equilibrium of its first approximation-inclusion were established. It was demonstrated that the proposed method can be used to study the weak asymptotical (weak exponential) stability of the inclusions equivalent to control systems. The problems of absolute and robust stability of control systems with periodic variable parameters were solved in [10,11,14,15]. In particular, control systems with periodic parameters under consideration were proved to be equivalent to a periodic differential inclusion in the sense of the coincidence of the sets of absolutely continuous solutions. As was demonstrated in [16], in some cases solutions of periodic differential inclusions with the asymptotically stable trivial solution have the same properties as solutions of autonomous differential inclusions. This paper continues the research of [16]. It considers periodic differential inclusions with an asymptotically stable set. The remainder of this paper is structured as follows. In Section 1 we consider periodic differential inclusion of general form and give preliminary remarks. The definition of an asymptotically stable set is also given. In Section 2 the uniform character of convergence of solutions to an asymptotically stable set is established. For solutions of periodic differential inclusion that is homogeneous in state vector we derive an exponential estimate. In Section 4 examples of control systems leading to the periodic differential inclusions are given. In the final section we offer concluding remarks. 2. STATEMENT OF THE PROBLEM Consider the dynamic systems described by periodic differential inclusion 78 M.V. MOROZOV Copyright ©2021 ASSA. Adv. in Systems Science and Appl. (2021) 0.T const,T , 0, t),,(),( ),,( =+ nRxxTtFxtFxtFx (2.1) Everywhere below we assume that in some domain }},:{ , ,0{ 00 RxxGGxTtG RR == the multivalued function ),( xtF satisfies the main conditions [4, p. 60]; i.e., for all Gxt ),( the set nRxtF ),( is nonempty, bounded, closed, and convex, and the function ),( xtF is upper semicontinuous [4, p. 52] with respect to ).,( xt A solution of inclusion (2.1) is understood as an absolutely continuous vector function )(tx defined on an open or closed interval I and satisfying (2.1) almost everywhere on I . By virtue of periodicity of the multivalued function ),( xtF in t , when studying the properties of solutions ),,( 00 xttx of inclusion (2.1), we can assume without loss of generality that ],0[0 Tt  . By the definition of solutions and the periodicity of the right-hand side of (2.1) in t the solutions of this inclusion have two properties as follows. If a function )(tx is a solution of inclusion (2.1) (for   t ), then 1) the function ),( kTtx + where kTtkT −−  and k denotes any integer, is also a solution of inclusion (2.1); moreover, the solutions )(tx and )( kTtx + have the same trajectory; 2) for any ttTt and , ],,0[ 10  such that ,10 ttt  the equality =))(,,( 11 txttx ),,( 00 xttx holds, where ),,()( 0011 xttxtx = . Let nRa , nRb be points (vectors) with coordinates ia and ib respectively, ni ,1= , and also let nRB  be a set. The distance  between two points or between a point and a set is interpreted as the nonnegative values ,)(),( 2/1 2 1       −=−=  = n i ii bababa ),(inf),( baBa Bb = . As is well-known, the function ),( Bx is uniformly continuous and for any points nRx and nRy )y,(),(y-),(x xBB   . A closed  -neighborhood M of a set M is a set of such points x that  M),(x . Let GM 0  , .00  Definition 2.1: A set M is asymptotically stable for inclusion (2.1) if for any 0 there exists a value 0)(  such that, for each 0x satisfying the inequality )(),( 0  Mx , there exists a solution with the initial condition 00 )( xtx = and all solutions with the above- mentioned property are extendable on the interval  tt0 and also satisfy the conditions  )),(( Mtx for  tt0 and 0)),(( →Mtx as →t . The problem is to study solutions of inclusion (2.1) with an asymptotically stable set .M 3. RESULTS Theorem 3.1: If inclusion (2.1) has an asymptotically stable set ,M then there exists a value 00  such that all solutions ),,( 00 xttx of inclusion (2.1) satisfy the condition ON UNIFORM CONVERGENCE PROPERTY OF SOLUTIONS FOR PERIODIC DIFFERENTIAL… 79 Copyright ©2021 ASSA. Adv. in Systems Science and Appl. (2021) 0)),(( →Mtx as →t (3.1) uniformly with respect to ),( 00 xt for any ],0[0 Tt  and . 0 0  Mx  Proof. First, let us show that there exists a value 00  such that all solutions of inclusion (2.1) with initial condition =)( 0tx ,0x , 0 0  Mx  satisfy condition (3.1) uniformly with respect to 0 x for any given ].,0[0 Tt  Suppose the contrary. Then for any ,0 there exists ,0)(  ],0[0 Tt  , a sequence of solutions )(txk of inclusion (2.1), a numerical sequence kTttk + 0 , and a sequence of vectors ,...,2,1 ,0 =kxk such that Mxk 0 and . tas ,...,2,1 ,0)()),,,(( k00 →= kMxttx k kk  Since the set M is asymptotically stable, it follows that for any 0 we can choose a value  small enough to ensure that all solutions with Mtx )( 0 satisfy the relations  ))),(,,(( 00 Mtxttx ),( 0  tt 0)),(( →Mtx ).(t → (3.2) Consequently, there exists a number 0 such that all solutions of inclusion (2.1) with Mtx )( 0 satisfy the inequality )())),(,,(( 00  Mtxttx ).( 0  tt (3.3) Let us show that all ),,( 00 k k xttx satisfy the inequalities , 0 Mxk  1,2,...k ,k1,m ,)),x,,(( k 000 ==+  MtmTtxk (3.4) Indeed, otherwise there would exist k ~ and ) ~ (km ( kkm ~ ) ~ (1  ) such that ,)),x,,) ~ ((( k ~ 000~  + MtTkmtx k . ~ 0 Mxk  Then the solution =+++=+= )),,) ~ ((,) ~ (,) ~ (()),,) ~ ((,,()( ~ 000~0~ ~ 000~0 k kk k k xtTkmtxTkmtTkmtxxtTkmtxttztz ),,) ~ (( ~ 00~ k k xtTkmtx += )( 0  tt of inclusion (2.1) would satisfy the relations =)),(( 0 Mtz ,)),x,,) ~ ((( k ~ 000~  + MtTkmtx k ),()),,,(())),,,) ~ ((,,)Tk ~ ((( ~ 00~~ ~ 000~0~  =+− MxttxMxtTkmtxtmtz k kk k kk which contradict (3.3). The sequence of segments of solutions )(txk contains a subsequence uniformly convergent for 10 ttt  ; in turn, the subsequence contains a new subsequence uniformly convergent for 20 ttt  , and so on. Since ),( xtF satisfies the basic assumptions, it follows that the limit function )(tx is a solution of inclusion (2.1) [4, p. 60], which, together with (3.4), implies that Mtx )( 0 and  + ))),(,,(( 000 MtxtkTtx , 1,2,...,k = and we have arrived at a contradiction with (3.2). Therefore, the assertion started at the beginning of the proof of the theorem is valid. 80 M.V. MOROZOV Copyright ©2021 ASSA. Adv. in Systems Science and Appl. (2021) Let us now prove the assertion of Theorem 2.1. Suppose that is fails. Then for any ,0 there exist a number ,0)(  a sequence )(txk of solutions of inclusion (2.1), a sequence of numbers kt0 ( ],0[0 Tt k  and kTtt k k + 0 ), and a sequence of vectors ,...,2,1 ,0 =kxk such that Mxk 0 , . t,...;2,1 ,0)()),,,(( k00 →= kMxttx kk kk  (3.5) Without loss of generality, we can assume that ,0  where 0 is a number for which the above-proved assertion holds. Since , 0 Mxk  ],0[0 Tt k  , ,...,2,1 =k it follows that the sequence ,...2,1 =k contains a subsequence ),( }{ → kk such that the limits 00lim xxk k =  → ( M0x ) and 00lim tt k k =  → ( ],0[0 Tt  ) simultaneously exist. To simplify the notation, we assume that kk tt 00 =  , kk xx 00 =  ,...,2,1 k == k and ,lim 00 xx k k = → .lim 00 tt k k = → (3.6) Let .0M),( 00 =−  x By (3.6), there exist a 1K such that 2/)x,( 00  kx (3.7) for all 1Kk  Since all solutions of inclusion (2.1) are equicontinuous [2, p. 61] in any closed bounded domain lying in ,G it follows that they are equicontinuous in the domain , { 0Mx ]},0[ Tt as well. Therefore, for the point 0t , there exist a neighbourhood ]},,0[ ,2/:{)( 00 TtttttS −=  ],[)( 0 batS = ( ta 0= if 00 =t and 0tb = if Tt =0 ), such that .2/))t ~ x(,)) ~ (, ~ ,(( 000  txtbx (3.8) for all solutions )) ~ (, ~ ,( 00 txttx of inclusion (2.1) with ],[ ~ 0 bat  and .) ~ ( 0 Mtx  Then, taking into account relations (3.7), (3.8) and uniform continuity of the function ),()( Mxx  = we obtain two relations 2/)x,(),(x-),(x),(),( 000 k 000  − kk xMMMxMx , 2/)x),,,((),(x-)),x,t(b,(x),()),,,(( k 000 k 0 k 0 k 0k000  − kk k kkk k xtbxMMMxMxtbx for all solutions )(txk with .1Kk  Adding the last two inequalities, we obtain 0000 ),()),,,((  =+ MxMxtbx kk k , .1Kk  This, together with (3.5) and (3.6), means that there exist a 2K such that the relations ,0 Tbt k  000 )),,,((  Mxtbx kk k , )()),,,(( 00  Mxttx kk kk (3.9) are simultaneously valid for all .2Kk  Taking into account relations (3.9), from the sequence of solutions )(txk of inclusion (2.1) satisfying inequalities (3.5) for 2Kk  , we pass to the sequence of solutions )(tyk ),,( 00 kk k xttx= defined by the relations ),()( 0 bxyby k k k == ,),( 00  Myk ),()),y,,(( k 0  Mbty kk .k →t (3.10) ON UNIFORM CONVERGENCE PROPERTY OF SOLUTIONS FOR PERIODIC DIFFERENTIAL… 81 Copyright ©2021 ASSA. Adv. in Systems Science and Appl. (2021) The existence of a sequence ),(tyk ,2Kk = ,12 +K ,...22 +K ,(  tb ),0 Tb  satisfying conditions (3.10) contradicts the already known fact that relation (3.1) is valid uniformly with respect to 0x (for 0 0  Mx  and ],0[0 Tt  ) for all solutions of inclusion (2.1). The proof of Theorem 3.1 is complete.  In what follows, we consider differential inclusions homogeneous with respect to .nRx If B is a set in nR and c is a number, then cB stands for a set of points of the form cx for all .Bx Definition 3.2: A multivalued function ),( xtF is homogeneous (of degree one) in x if ),( cxtF ),( xtcF for all 0c . Definition 3.3: A differential inclusion )0 ),,(),(( ),(  cxtcFcxtFxtFx (3.11) is homogeneous in .x Homogeneous differential inclusion (3.11) is preserved under the substitution 1cxx = with arbitrary .0c It means that if a function )(tx = is a solution of inclusion (3.11), then the function )(tcx = with arbitrary 0c is also a solution. Let us consider the differential inclusion ), 0,( ),,( nRxtxtFx  0),T const,(T ),,(),( =+ xTtFxtF ),0( ),(),(  cxtcFcxtF (3.12) periodic in t and homogeneous in .x Since inclusion (3.12) is homogeneous in ,x we have the following assertion. Corollary 3.1: If a bounded set M is asymptotically stable for inclusion (3.12), then all solutions ))(,,( 00 txttx of inclusion (3.12) with the initial conditions ],0[0 Tt  and RGx 0 (where R is an arbitrary positive number) satisfy condition (3.1) uniformly with respect to ).,( 00 xt For solutions of periodic homogeneous differential inclusion (3.12) with an asymptotically stable set M an exponential estimate is valid. Theorem 3.2: If a bounded set M is asymptotically stable for inclusion (3.12), then there exist numbers 0 ,0 10  cc such that any solution ),,( 00 xttx of inclusion (3.12) satisfies the estimate ).(t )exp()),x,,(( 010000 − ttcxcMttx (3.13) for any 0t and .0tt  Proof. By virtue of corollary 3.1, for some ,0 there exists a 0 (independent of ),( 00 xt ) such that Tk ~ = ( k ~ is some positive integer) and ,2/)),x,,(( 00  Mttx + t0t , for all solutions ),,( 00 xttx with the initial condition .0 x By virtue of Theorem 3 in [4, p. 62] the set of solutions of inclusion (3.12) is compact on the closed interval ]t,[ 00 +t in the metric of ]t,[ 00 +tC , whence it follows that  200 )),x,,(( cMttx  )0( 2 c for .t00 + tt If ),,( 00 xttx is a solution with an arbitrary 00 x and ,/ 0xc = then the function ),,(),,( 0000 xttcxytty = is also a solution and ,0 =y whence it follows that 82 M.V. MOROZOV Copyright ©2021 ASSA. Adv. in Systems Science and Appl. (2021) 2/)),y,,(( 00  Mtty )t( 0 + t and  200 )),y,,(( cMtty  ).t( 00 + tt Returning from ),,( 00 ytty to ),,( 00 xttx , we obtain 0200 )),x,,(( xcMttx  ),t( 00 + tt 2/)),x,,(( 000 xMttx  ).t( 0 + t (3.14) for any 0t and .0x Since after replacement of t by kTt + ( k is an arbitrary integer), a solution of inclusion (3.12) remains a solution, it follows from (3.14) that a solution ),,( 00 xttx with an arbitrary 0x satisfies the relations 000 2)),x,,(( xMttx i− ),t( i  t (3.15) where ,...2,1 ,t 0i =+= iit  For any ,0tt  we choose an i such that .t 1-i itt  Then itt + 0 and ./)( 0 tti − Therefore, the inequality )/2lnexp()/2lnexp()/)(2lnexp()2lnexp(2 00  ttttii −=−−−=− is fulfilled. The last inequality, together with (3.15), implies (3.13) with )/2lnexp( 00 tc = and ./2ln1 =c The proof of Theorem 3.2 is complete.  4. EXAPMLES Example 4.1: In the paper [13] the nonlinear control system is considered ,),( 1  = += m j jj j tbAxx   = == n i i j i j j xcxc 1 ,, ,,1,0),,0( mjtj = (4.1) where nn ii Rxxx = = ,)( 1 is n -dimensional vector, characterizing the deviation of the system from the mode prescribed by control objective (the zero solution otx )( of system (4.1) corresponds to this mode), A is a constant square matrix of order ,n and jb and ),1( mjc j = are constant n -dimensional vectors. The brackets .,. denote the scalar product. It is also assumed that nonlinear functions ,,1),,( mjtjj = that define the characteristics of nonlinear elements, satisfy conditions for the existence of an absolutely continuous solution of system (4.1) for any initial conditions and inequalities ( )mjkkktk jjjjjjjjj ,1, ),( 21 2 2 2 1 =+−  for all j and .0t System (4.1) is equivalent, see the paper [13], to time-invariant linear-selectionable inclusion (1.4), where  is a compact set in the 2n -dimensional matrix space. Here and in the following examples, equivalence is understood in the sense of solutions sets identity for the system under consideration and the corresponding inclusion under the same initial conditions. Example 4.2: In the paper [10] the linear nonstationary control systems ,)()( 1 xtAtx k m k k = =  0)( tk , 1)( 1 = = t m k k (4.2) ON UNIFORM CONVERGENCE PROPERTY OF SOLUTIONS FOR PERIODIC DIFFERENTIAL… 83 Copyright ©2021 ASSA. Adv. in Systems Science and Appl. (2021) are considered, where nт ii Rxx = =1)( is n -dimensional state vector of the system, =)(tAk (t))(a n 1ji, k ij = are given continuous periodic matrices of the period ,0T =+ )( TtAk )(tAk , mk ,1= , and )(tk ( mk ,1= ) are bounded measurable functions. System (4.2) is equivalent to the periodic differential inclusion ),,(),( ),,( xTtFxtFxtFx +  == === m kk m k k tttAtyyxtF 1k k 1 }.1)( ,0)( ),()(:{),(  (4.3) Example 4.3: In the paper [14] the linear nonstationary control system nRxxtAx = ,)( (4.4) is considered, where n jiij tatA 1,))(()( == is an arbitrary matrix ( )(taij , nji ,1, = , are measurable functions, generally speaking, not periodic). System (4.4) almost everywhere satisfies the inequalities ))(()( ,))(((t) (t),(t) (t) 1,1j, n jiij n nij tatAtaAAAA == == (4.5) on any finite interval of the semi-axis ).[0, Matrix inequalities (4.5) are understood elementwise, that is ,,1, (t),)((t) ijij njiataa ij = , where ,)(a tij , ,1, (t), (t), ijij njiaa = are arbitrary measurable functions. It is assumed that the given bounded "extreme" matrices (t) A and (t)A are periodic with a period 0T  , i.e. conditions are valid (t) )( ATtA + , ).()( tATtA + Thus, by virtue of (4.5), not one fixed system (4.4) is considered, but a set of linear nonstationary systems (4.4) with periodic interval constraints (4.5). A series of examples leading to systems (4.2) and (4.4) with constraints (4.5) were considered in [19]. Among such systems, note tracking systems with AC elements, control systems with pulse amplitude modulation and also the systems arising in the vibration analysis of milling machines. If the condition )()( tATtA =+ is additionally satisfied, the set of linear nonstationary systems (4.4) with periodic interval constraints (4.5) is equivalent to the linear-selectionable periodic inclusion ),,( xtFx  )()( ,)(:),( ttAxtAyyxtF == ,  )()()()()( :)()( 21 tAttAttAtAt  +== , )()( tTt =+ , where arbitrary bounded and measurable functions 2,1),( =ktk satisfy the conditions ,1)( ,0)( 2 1 =  =k kk tt  ),()( tTt kk  =+ .0tt  5. CONCLUSION The paper considers solutions of periodic differential inclusions with asymptotically stable sets. Theorem 3.1 establishes the uniform character of convergence to an asymptotically stable set. Homogeneous in state vector periodic differential inclusions are also considered. Corollary 3.1 is obtained for such inclusions. Theorem 3.2 gives the exponential estimate for solutions. Theorem 3.1 is a generalization of lemma 2, proved for autonomous differential inclusions [3]. Theorem 3.2 generalizes the estimate well known for solutions of an autonomous homogeneous (first degree) differential inclusion [4]. The examples of control systems leading to consideration of periodic differential inclusions are given. The results obtained can find applications in the stability analysis of control systems with periodic parameters, in particular, servomechanisms whose elements operate on AC, control 84 M.V. MOROZOV Copyright ©2021 ASSA. Adv. in Systems Science and Appl. (2021) systems with pulse amplitude modulation, and systems used to solve problems related to investigating vibrations of milling machines. Further research into the inclusions considered in the present paper can be related to producing weak asymptotic and weak exponential stability conditions. 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