A NOTE ON PERIODIC SOLUTIONS OF RICCATI EQUATIONS

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1 A NOTE ON PERIODIC SOLUTIONS OF RICCATI EQUATIONS M. R. MOKHTARZADEH, M. R. POURNAKI, AND A. RAZANI Absrac. In his noe, we show ha under cerain assumpions he scalar Riccai differenial equaion x = a()x + b()x 2 + c() wih periodic coefficiens admis a leas one periodic soluion. Also, we give wo illusraive examples in order o indicae he validiy of he assumpions. 1. Inroducion A large class of dynamical sysems appearing hroughou he field of engineering and applied mahemaics is described by he second order differenial equaion of he form (1) y + p()y + q()y = r(), where p, q, and r are real funcions on R. In general, here is no mehod for solving nonhomogeneous linear second order differenial equaions and, herefore, a complee analysis of (1) does no exis. Neverheless, in he homogeneous case, when r = in (1), by making he change of variable x = y /y, we are led o a firs order differenial equaion of he form (2) x = p()x + x 2 + q(). Alhough he analysis of hese kinds of differenial equaions are sill in a preliminary sage, recenly various issues concerning heoreical aspecs of such differenial equaions have been successfully clarified. In he lieraure, (2) is a special case of a more general one, so-called scalar Riccai differenial equaion, namely (3) x = a()x + b()x 2 + c(), where a, b, and c are real funcions on R. The sudy of (3) has long been an imporan opic and daes from he early period of modern mahemaical analysis. I began wih examinaions of paricular cases of (1) by James Bernoulli ( ) and hen by Coun Jacopo Francesco Riccai ( ). The generalizaion of scalar Riccai differenial equaion o he marix case gives us marix Riccai differenial equaion which is one of he cenral objecs of presen day 2 Mahemaics Subjec Classificaion. Primary 34C25; Secondary 47B5. Key words and phrases. Scalar Riccai differenial equaion, Periodic soluion, Banach space, Compac operaor, Schauder s fixed poin heorem. The research of M. R. Pournaki and A. Razani was in par suppored by a gran from IPM (No and No ). 1

2 2 M. R. MOKHTARZADEH, M. R. POURNAKI, AND A. RAZANI conrol heory. In fac, in he heory of conrol sysems, he qualiaive conrol problem has received considerable research ineress. This problem is regarded as an exension of he classical resul of Kalman e al. [1] on conrollabiliy and sabiliy of linear sysems which is relevan o such differenial equaions (see [5, 16, 4, 3, 7, 8]). Marix Riccai differenial equaions also play predominan roles in oher conrol heory problems such as dynamic games, linear sysems wih Markovian jumps, and sochasic conrol. The sudy of such differenial equaions, which also appears in a number of oher areas such as biomahemaics and mulidimensional ranspor processes, is an ineresing area of curren research. There exiss a raher exensive lieraure on he marix Riccai differenial equaion, mainly developed wihin he auomaic conrol lieraure. We refer he readers o [3] as an exensive survey as well as o [5, 16, 4, 9] as fundamenal papers on his area. The analysis of periodic sysems has long been a opic of ineres. In his direcion, an imporan quesion, which has been sudied exensively by a number of auhors (see, for example [1, 12, 14, 18, 17]), is wheher Riccai differenial equaions can suppor periodic soluions or no. For example, in heoreical aspecs, knowledge of he periodic soluions is imporan for undersanding he phase porrai of he Riccai differenial equaions and, in paricular, he qualiaive behavior of soluions (see, for example [16]). On he oher hand, on he applied side, in he problem of quadraic periodic opimizaion, arising for insance in he design of solar heaing sysems where he ambien emperaure represens a periodic inpu, here occurs he need o compue he periodic soluions, if any, of a scalar or marix Riccai differenial equaion wih periodic coefficiens. Anoher applicaion is found in Kalman filering of periodic sysems such as orbiing saellies, seasonal phenomena like river flows, and economeric models, ec. We refer he readers o [2] for an overview on he srucural properies of periodic sysems, o [3] for he properies of periodic soluions o periodic Riccai differenial equaions, and o [6] for he sudy of he periodic Lyapunov differenial equaions. Also, he book by Reid [15] covers many areas in Riccai differenial equaions and is concerned wih applicaions of hese differenial equaions such as ransmission line phenomena, heory of random processes, variaional heory and opimal conrol heory, diffusion problems, and invarian imbedding. In his noe, we deal wih scalar Riccai differenial equaions. In he ligh of he above discussion, i seems reasonable o consider (3) and asks when his differenial equaion has a periodic soluion. In his direcion, we assume ha a, b, and c are -periodic coninuous real funcions on R and give cerain condiions o guaranee he exisence of a leas one periodic soluion for (3). The proof hinges on Schauder s fixed poin heorem (see Theorem 2.2) applied o inegral equaion (5) which is a reformulaion of (3). In order o indicae he validiy of he assumpions made in our resul, we also rea wo illusraive examples.

3 A NOTE ON PERIODIC SOLUTIONS OF RICCATI EQUATIONS 3 2. Preliminaries In his secion, we presen a brief survey of noions and resuls of funcional analysis which we shall need laer. The reader is referred o [11] for a fuller reamen of he subjec. We sar by recalling he definiion of a normed space. Le X be a vecor space over R. Then a norm on X is a funcion : X R such ha (1) x for all x X and x = if and only if x =, (2) αx = α x for all α R and for all x X, (3) x + y x + y for all x, y X (riangle inequaliy). A vecor space X over R ogeher wih a norm is called a normed space. In his case, he disance from x o y in X is defined by d(x, y) = x y. This defines a meric on X. Therefore, every normed space has a meric and so has an associaed opology. All he sandard opological noions same as open ses, closed ses, bounded ses, convergence, ec. may be applied o X. Also, we recall ha in a normed space X, a subse A of X is called convex if αx + (1 α)y A for all x, y A and for all α wih α 1. We are now going o define Banach spaces which are he mos manageable among all ypes of normed spaces due o heir meric srucure. A normed space X is called a Banach space if i is complee, i.e., if every Cauchy sequence in X is convergen. Also, for a given Banach space X, a compac operaor on X is a bounded operaor S : X X ha maps he uni ball in X o a se in X wih compac closure. I is easy o see ha he operaor S is compac on X if and only if every bounded sequence {φ n } on X has a subsequence {φ ni } such ha {S(φ ni )} is convergen on X. We also recall ha a given sequence {φ n ()} of funcions from [a, b] o R, is called equiconinuous if for every ɛ >, here exiss a δ > such ha for all n N and for all 1, 2 [a, b], 1 2 < δ implies ha φ n ( 1 ) φ n ( 2 ) < ɛ. In he sequel, we also need he following weak version of Ascoli Arzelà heorem. Theorem 2.1 (Ascoli Arzelà). Le {φ n ()} be a sequence of funcions from [a, b] o R which is uniformly bounded and equiconinuous. Then {φ n ()} has a uniformly convergen subsequence. Finally, we close his secion wih he following fixed poin heorem which is originally due o Schauder and is a key ool for proving he main resul of his noe (see Main Theorem 3.2). Theorem 2.2 (Schauder). Le X be a Banach space and Ω be a closed, bounded, and convex subse of X. If S : Ω Ω is a compac operaor, hen S has a leas one fixed poin on Ω. 3. Main resul In his secion, we sae and prove he main conribuion of his noe (see Main Theorem 3.2). In order o do his, suppose ha a, b, and c are -periodic coninuous real funcions on R such ha a(ξ)dξ. Define he funcion G : [, ] [, ]

4 4 M. R. MOKHTARZADEH, M. R. POURNAKI, AND A. RAZANI R as follows: (4) G(, s) = 1 1 exp( R a(ξ)dξ) exp ( s a(ξ)dξ) : s, exp( R a(ξ)dξ) 1 exp( R exp ( a(ξ)dξ) : s. a(ξ)dξ) s The following lemma is useful for proving he Main Theorem 3.2. Lemma 3.1. Le a, b, and c be -periodic coninuous real funcions on R such ha a(ξ)dξ. Suppose ha x is a coninuous real funcion on R. If x is a soluion of he inegral equaion (5) x() = G(, s) ( b(s)x 2 (s) + c(s) ) ds, hen x is a soluion of (3). Proof. By assumpion, x is a soluion of inegral equaion (5). Therefore, by using expression (4), we may wrie x() = = G(, s) ( b(s)x 2 (s) + c(s) ) ds G(, s) ( b(s)x 2 (s) + c(s) ) ds + G(, s) ( b(s)x 2 (s) + c(s) ) ds (6) = 1 1 exp( R a(ξ)dξ) exp(r a(ξ)dξ) 1 exp( R a(ξ)dξ) exp ( s a(ξ)dξ)( b(s)x 2 (s) + c(s) ) ds exp ( s a(ξ)dξ)( b(s)x 2 (s) + c(s) ) ds = exp(r a(ξ)dξ) 1 exp( R 1 a(ξ)dξ) exp( R s exp(r a(ξ)dξ) exp(r a(ξ)dξ) 1 exp( R 1 a(ξ)dξ) exp( R s a(ξ)dξ) ( b(s)x 2 (s) + c(s) ) ds a(ξ)dξ) ( b(s)x 2 (s) + c(s) ) ds.

5 Thus, we obain x () = A NOTE ON PERIODIC SOLUTIONS OF RICCATI EQUATIONS 5 a() exp(r a(ξ)dξ) 1 exp( R 1 a(ξ)dξ) exp( R s exp( R a(ξ)dξ) ( b()x 2 () + c() ) a(ξ)dξ) ( b(s)x 2 (s) + c(s) ) ds (7) a() exp(r a(ξ)dξ) exp(r a(ξ)dξ) 1 exp( R 1 a(ξ)dξ) exp( R s exp(r a(ξ)dξ) 1 exp( R a(ξ)dξ) ( b()x 2 () + c() ) a(ξ)dξ) ( b(s)x 2 (s) + c(s) ) ds = a()x() + b()x 2 () + c(), which shows ha x is a soluion of (3) as requesed. Noe ha expression (4) is, in fac, he Green s funcion of (3). Therefore, by using mehods for finding Green s funcion, we may find he kernel G(, s) appeared in expression (4). However, our approach is differen, bu for going hrough he deails of finding Green s funcion we refer he reader o [13]. We now sae and prove he Main Theorem 3.2 which is he main conribuion of his noe. Main Theorem 3.2. Le a, b, and c be -periodic coninuous real funcions on R such ha a(ξ)dξ. Consider (8) M = sup G(, s),,s (9) N = sup and suppose ha (1) G(, s)c(s)ds, b(ξ) dξ 1 4MN. Then (3) admis a leas one -periodic soluion. Proof. Le (11) X = {φ φ is a -periodic coninuous real funcion on R} and for φ X define φ = sup φ(). I is easy o see ha X is a Banach space. Define he funcion ψ : [, ] R as (12) ψ() = G(, s)c(s)ds

6 6 M. R. MOKHTARZADEH, M. R. POURNAKI, AND A. RAZANI and consider (13) Ω = {φ X φ ψ N}. I is easy o see ha Ω is closed, bounded, and convex subse of X. operaor S : Ω X by sending φ o S(φ), where S(φ) defined as (14) S(φ)() = G(, s) ( b(s)φ 2 (s) + c(s) ) ds. Define he Firs, we claim ha S maps Ω ino Ω. In order o show his, by using (13) and (12), we obain φ() N + ψ() 2N holds for all φ Ω and for all [, ]. Therefore, (14), (12), (8), and (1) imply ha for all φ Ω and for all [, ], (15) S(φ)() ψ() = G(, s)b(s)φ2 (s)ds 4MN 2 b(s) ds N. Thus, for all φ Ω, we have S(φ) ψ N and so S(φ) Ω. This shows ha S is an operaor from Ω ino Ω. Nex, we show ha S is compac. In order o do his, suppose {φ n } is a sequence on Ω which is, by (13), bounded. Thus, here exiss L > such ha for all n N and for all [, ], we have φ n () L. We should show ha {φ n } has a subsequence, say {φ ni }, such ha {S(φ ni )} is convergen on Ω. Noe ha, by Lemma 3.1, for all n N, he funcion S(φ n ) is, in fac, differeniable and for all [, ] we have (16) S(φ n ) () = a()φ n () + b()φ n 2 () + c(). Therefore, for all n N and for all [, ], we have S(φ n ) () AL + BL 2 + C, where A, B, and C are he maximum values of a, b, and c on [, ], respecively. Therefore, for given ε >, if we consider δ = ε/(al + BL 2 + C), hen for all n N and for all 1, 2 [, ], 1 2 < δ implies ha (17) S(φ n )( 1 ) S(φ n )( 2 ) (AL + BL 2 + C) 1 2 < ε. Thus, {S(φ n ())} as a sequence of funcions on [, ] is equiconinuous and Theorem 2.1 hen implies ha here exiss a subsequence of {S(φ n ())}, say {S(φ ni ())}, which is uniformly convergen on [, ]. This means ha {S(φ ni )} is convergen on Ω and so S is compac. Therefore, Theorem 2.2 implies ha here exiss x Ω such ha S(x) = x, i.e., for all [, ], (18) x() = G(, s) ( b(s)x 2 (s) + c(s) ) ds. Since x Ω, x is a -periodic coninuous real funcion on R and so wha remains o be proved is ha x is indeed a soluion o (3). Bu his already proven by Lemma 3.1.

7 A NOTE ON PERIODIC SOLUTIONS OF RICCATI EQUATIONS 7 4. Two illusraive examples In his secion, we rea he following wo illusraive examples. The firs example is generaed by rial and error process, using compuer codes in Mahemaica 5.2 wih symbolic operaions. Therefore, i seems o be quie far from a real pracical problem. However, i shows ha he novely of our approach, since he previous resuls in he lieraure are inapplicable for proving he exisence of periodic soluions of i. Also, his example indicaes he validiy of he assumpions made in he Main Theorem 3.2. These are he only jusificaion for puing his example. Insead, he second example comes from mahemaical modeling of naural phenomenons. Example 4.1. Suppose ha a, b, and c are 1-periodic coninuous real funcions on R which are defined as follows: (19) a() = π(447 cos π 288 cos 2π+cos 3π sin π+288 sin 2π 4 sin 3π+48) 2(cos π+3)(136 cos π+4 cos 2π 312 sin π+sin 2π 863), (2) b() = 48π(3 cos π+53 sin π+1) 1536 cos π+16 cos 2π+4 cos 3π 1871 sin π 36 sin 2π+sin 3π 48242, (21) c() = π(33 cos π 288 cos 2π+cos 3π sin π+288 sin 2π 4 sin 3π+432) 1536 cos π 16 cos 2π 4 cos 3π+1871 sin π+36 sin 2π sin 3π Here, we have 1 a(ξ)dξ = , and so he funcion G is as follows: exp ( a(ξ)dξ) : s 1, s (22) G(, s) = exp ( a(ξ)dξ) : s 1. s Since (23) M = sup G(, s) = ,s 1 and (24) N = sup we have (25) G(, s)c(s)ds = , b(ξ) dξ = < = 1 4MN. Therefore, he Main Theorem 3.2 implies ha (3) admis a leas one 1-periodic soluion. This 1-periodic soluion may be given by expression (29), where k is an arbirary consan. Noe ha expression (29) is he general soluion of (3) defined by (19), (2), and (21). The general soluion is generaed by knowing hree paricular soluions (26), (27), and (28) of he equaion. Here, all numerical resuls are correc o 5 digis, using arbirary precision faciliies devised in his sofware, and hen all

8 8 M. R. MOKHTARZADEH, M. R. POURNAKI, AND A. RAZANI resuls runcaed o 6 decimal places. For sacking more accurae resuls, we also normalized variable and resriced ourselves o he inerval [, 1). (26) x 1 () = 1, (27) x 2 () = cos π, 3 (28) x 3 () = (cos π + sin π), 8 (29) x() = k cos 2π+12k sin π+k sin 2π+673k+2(62k+5) cos π 6 sin π 288 2(3k+5) cos π+6(8(7k 6)+(k 1) sin π). Example 4.2. Consider he Riccai differenial equaion (3) x = x + b()x 2 + sin, where b is a 2π-periodic coninuous real funcion on R. This equaion comes from mahemaical modeling of naural phenomenons. Here, we have 2π dξ = 2π, and so he funcion G is as follows: 1 exp( s) : s 2π, 1 exp(2π) (31) G(, s) = exp(2π) exp( s) : s 2π. 1 exp(2π) We have (32) M = sup G(, s) = 1.187,,s 2π (33) N = sup and so if (34) 2π 2π 2π G(, s)(sin s)ds = , b(ξ) dξ 1 4MN = , hen he Main Theorem 3.2 implies ha (3) admis a leas one 2π-periodic soluion. For insance, if we consider 28(784 + cos + 29 sin ) (35) b() =, (784 + cos ) 2 hen since (36) 2π b(ξ) dξ =.2244 < , we obain ha (3) admis a leas one 2π-periodic soluion, ha is, (37) x() = 1 (784 + cos ). 28

9 A NOTE ON PERIODIC SOLUTIONS OF RICCATI EQUATIONS 9 5. Conclusion In his noe, we invesigae he exisence of periodic soluions for a class of scalar Riccai differenial equaions. Suppose ha a, b, and c are -periodic coninuous real funcions on R such ha a(ξ)dξ. Consider (38) M = sup,s G(, s), (39) N = sup G(, s)c(s)ds, and suppose ha (4) b(ξ) dξ 1 4MN, where he kernel G(, s) is defined as follows: 1 1 exp( R exp ( a(ξ)dξ) : s, a(ξ)dξ) s (41) G(, s) = exp( R a(ξ)dξ) 1 exp( R exp ( a(ξ)dξ) : s. a(ξ)dξ) s Then he scalar Riccai differenial equaion x = a()x + b()x 2 + c() admis a leas one -periodic soluion. In order o indicae he validiy of he assumpions made in our resul, we also rea wo illusraive examples. Acknowledgmens The auhors would like o hank he referees for helpful remarks which have conribued o improve he presenaion of he noe. The auhors also express heir graiude o Professor Ali H. Nayfeh for his kindness and suppor. References [1] R. E. Agahanjanc, On periodic soluions of he Riccai equaion, Vesnik Leningrad. Univ. 16 (1961), no. 19, [2] S. Biani, Deerminisic and sochasic linear periodic sysems, Time series and linear sysems, ix, , Lecure Noes in Conrol and Inform. Sci., 86, Springer, Berlin, [3] S. Biani, P. Colaneri, G. De Nicolao, The periodic Riccai equaion, The Riccai Equaion, , Comm. Conrol Engrg. Ser., Springer-Verlag, Berlin, [4] S. Biani, P. Colaneri, G. De Nicolao, G. O. Guardabassi, Periodic Riccai equaion: exisence of a periodic posiive semidefinie soluion, Proceedings of he IEEE Conference on Decision and Conrol Including he Symposium on Adapive Pro, 1987, [5] S. Biani, P. Colaneri, G. O. Guardabassi, Periodic soluions of periodic Riccai equaions, IEEE Trans. Auoma. Conrol 29 (1984), no. 7, [6] P. Bolzern, P. Colaneri, The periodic Lyapunov equaion, SIAM J. Marix Anal. Appl. 9 (1988), no. 4, [7] P. Colaneri, Coninuous-ime periodic sysems in H 2 and H, I. Theoreical aspecs, Kyberneika 36 (2), no. 2,

10 1 M. R. MOKHTARZADEH, M. R. POURNAKI, AND A. RAZANI [8] P. Colaneri, Coninuous-ime periodic sysems in H 2 and H, II. Sae feedback problems, Kyberneika 36 (2), no. 3, [9] C. E. de Souza, Exisence condiions and properies for he maximal periodic soluion of periodic Riccai difference equaions, Inerna. J. Conrol 5 (1989), no. 3, [1] R. E. Kalman, Y. C. Ho, K. S. Narendra, Conrollabiliy of linear dynamical sysems, Conribuions o Differenial Equaions 1 (1963), [11] E. Kreyszig, Inroducory Funcional Analysis wih Applicaions, John Wiley & Sons, Inc., New York, [12] V. S. Loščinin, Periodic soluions of Riccai s equaion, Balašov. Gos. Ped. Ins. Učen. Zap. 1 (1963), [13] Y. A. Meĺnikov, Green s Funcions in Applied Mechanics, Topics in Engineering, 27, Compuaional Mechanics Publicaions, Souhampon, [14] Y. S. Qin, Periodic soluions of Riccai s equaion wih periodic coefficiens, Kexue Tongbao 24 (1979), no. 23, [15] W. T. Reid, Riccai Differenial Equaions, Mahemaics in Science and Engineering, vol. 86. Academic Press, New York-London, [16] M. A. Shayman, On he phase porrai of he marix Riccai equaion arising from he periodic conrol problem, SIAM J. Conrol Opim. 23 (1985), no. 5, [17] F. Tang, The periodic soluions of Riccai equaion wih periodic coefficiens, Ann. Differenial Equaions 13 (1997), no. 2, [18] H. Z. Zhao, The periodic soluions of Riccai equaion wih periodic coefficiens, Ann. Differenial Equaions 7 (1991), no. 4, M. R. Mokharzadeh, School of Mahemaics, Insiue for Research in Fundamenal Sciences (IPM), P.O. Box , Tehran, Iran. address: mrmokharzadeh@ipm.ir M. R. Pournaki, Deparmen of Mahemaical Sciences, Sharif Universiy of Technology, P.O. Box , Tehran, Iran, and School of Mahemaics, Insiue for Research in Fundamenal Sciences (IPM), P.O. Box , Tehran, Iran. address: pournaki@ipm.ir URL: hp://mah.ipm.ac.ir/pournaki/ A. Razani, Deparmen of Mahemaics, Faculy of Science, Imam Khomeini Inernaional Universiy, P.O. Box , Qazvin, Iran, and School of Mahemaics, Insiue for Research in Fundamenal Sciences (IPM), P.O. Box , Tehran, Iran. address: razani@ikiu.ac.ir URL: hp://mah.ipm.ac.ir/razani/