A New Generalization of Lemma Gronwall-Bellman
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1 Applied Mthemticl Sciences, Vol. 6, 212, no. 13, A New Generliztion of Lemm Gronwll-Bellmn Younes Lourtssi LA2I, Deprtment of Electricl Engineering, Mohmmdi School Engineering Agdl, Rbt, Morocco y lourtssi@yhoo.fr El Houssine El Mzoudi LREEER, Deprtment of Economy, Cddi Ayd University Mrrkech, Morocco h mzoudi@yhoo.fr Noureddine Ellmi LA2I, Deprtment of Electricl Engineering, Mohmmdi School Engineering Agdl, Rbt, Morocco ellmi@emi.c.m Abstrct In this pper, we present new generliztion of the Gronwll- Bellmn lemm constitutes key element in the stbiliztion of nonliner systems considered. This new generliztion cn develop simple commnd to exponentilly stbilize lrge clss of nonliner systems. Mthemtics Subject Clssifiction: 26D15, 26D2, 34A4, 34H15 Keywords: Gronwll-Bellmn ineulity, nonnegtive continuous functions, stbility, output feedbck 1 Introduction Gronwll-Bellmn ineulity, which is usully proved in elementry differentil eutions using continuity rguments see [6], [7], [9], is n importnt tool in the study of boundedness, uniueness nd other spects of ulittive behvior of solutions of differentil nd stbility.
2 622 Y. Lourtssi, El H. El Mzoudi nd N. Ellmi In 1918, T. Gronwll gve the Gronwll-Bellmn ineulity see [5]. After tht, mny uthors gve number of generliztions of this ineulity nd these generliztions hd significnt pplictions in differentil nd integrl eutions. In this pper, we give new generliztion of the Gronwll Bellmn lemm nd n ppliction for type of nonliner dynmic systems. 2 Sttement of results Integrl ineulities ply n importnt role in the study of differentil eutions. In prticulr, there hs been n incresed interest in the following Gronwll-Bellmn ineulity. Lemm 2.1 see [1], [5] Let zt nd ft be nonnegtive continuous functions on t T, for which the ineulity zt c + fszsds, t [,T] 1 holds, where c is constnt. Then zt c exp fsds, t [,T] 2 With different motivtions, mny generliztions nd pplictions of this lemm hs been obtined nd widely used, such s the study of the stbility of solutions of nonliner differentil eutions. The purpose of this work is to estblish generliztions of Lemm Grnwll-Bellmn nd some conseuences of our results re lso given. Our min results re given in the following theorems: Theorem 2.2 see [2], [3], [4] Let zt be positive differentible function stisfying the ineulity: zt c + fszs+gsz n sds, t I =[, b] 3 where c, the functions ft nd gt re continuous in I nd n>1 is constnt. Then: c exp zt 1 n 1c gs exp fsds s 1 n 1 fτdτ ds Under the ssumption, for t, s [, b] s 1 n 1c gs exp n 1 fτdτ ds > 5 4
3 A new generliztion of lemm Gronwll-Bellmn 623 Proof 2.3 Expression 3 cn be written: for ll t [, b] zt c + using 2, we obtin zt c exp the lst ineulity is euivlent to z t c exp n 1 fs+gsz szsds, 6 fs+gsz s ds, 7 fs+gsz s ds, 8 multiply by n 1gt exp n 1 gsz sds, we obtin n 1gtzt exp n 1 gsz sds t n 1c gt exp n 1 fsds, which implies tht d exp n 1 dt gsz sds c gs exp n 1 nd integrting to t, we obtin exp n 1 gsz sds 1 c under the ssumption 5, we get exp n 1 gsz 1 sds 1 n 1c gs exp Hence the ineulity 8 becomes s: c exp n 1 z t 1 n 1c gs exp 9 fsds, 1 s gs exp n 1 fτdτ ds, 11 fsds n 1 s s, n 1 fτdτ ds 12 fτdτ ds Therefore, tking into ccount tht n>1, we will c exp fsds zt s 1 1 n 1c gs exp n 1 fτdτ ds This completes the proof of the theorem
4 624 Y. Lourtssi, El H. El Mzoudi nd N. Ellmi Remrk 2.4 If the ssumption 5 is not verified, there is subdivision t i m 1 i= defined by: t =, t m = b nd for 1 i m 1: i exp n 1 z sgsds c ti+1 s gs exp n 1fτdτ ds = t i 15 then in this cse zt exp fsds i+1 n 1 gs exp n 1 t i s 1 fτdτ ds 16 Remrk 2.5 In cse < n < 1, the hypothesis 5 is lwys stisfied. Which leds to: s 1 zt c exp 1 n 1c gs exp n 1 fτdτ ds 17 Theorem 2.6 Let zt be positive differentible function stisfying the ineulity: zt c + f i sz i sds, t I =[, b] 18 where c is constnt, the functions f i t for i =1,...,n re continuous in I nd n>1 is constnt. Then: zt c exp f 1 sds s 1 n 1 c i 1 f i s exp n 1f 1 σdσ ds Under the ssumption, for t, s [, b] 1 n s c i 1 f i s exp n 1f 1 σdσ ds > 2 Proof 2.7 Ineulity 18 is written s: t zt c + f 1 s+ f i sz i 1 s zsds, 21
5 A new generliztion of lemm Gronwll-Bellmn 625 by pplying the Gronwll-Bellmn Lemm 2.1, we will t zt c exp f 1 s+ f i sz i 1 s ds, 22 then, for ll i =2,...,n t z i 1 t c i 1 exp i 1 f 1 s+ f i sz i 1 s ds, t c i 1 exp n 1 f 1 s+ f i sz i 1 s ds, 23 multiply the lst ineulity by negtive term n 1f i t, then: t n 1f i tz i 1 t exp n 1 f i sz i 1 sds t 24 n 1f i tc i 1 exp n 1 f 1 sds by summing the ineulities i =2to n, we obtin: t n 1 f i tz i 1 t exp n 1 f i sz i 1 sds 25 n 1 f i tc i 1 exp n 1 f 1 sds by integrting to t, we find t exp n 1 f i sz i 1 sds 1 n 1 f i sc i 1 exp n 1 f 1 σdσ ds then t exp f i sz i 1 sds 1 1 n 1 f i sc i 1 exp 1 n 1 f 1 σdσ ds where ineulity 22 becomes c exp f 1 sds zt s 1 n 1 c i 1 f i s exp n 1f 1 σdσ ds
6 626 Y. Lourtssi, El H. El Mzoudi nd N. Ellmi 3 Applictions There re mny pplictions of the ineulities obtined in the previous section. in this section we study the output feedbck stbiliztion of nonliner dynmicl systems tht model mny phenomen from vrious disciplines. Now consider the following nonliner system: m ẋt = Axt+ g i xt u i t+but 29 yt = Cxt x = x where: xt R + n is the vector of stte, x R + n is the initil condition, yt R m is output, ut R p is the vector controls nd A, B, C re constnt mtrices of pproprite size such tht A, B stbilizble nd A, C is detectble. The nonliner function g i xt is mesurble with g i = nd stisfies the following hypothesis: for ll i =1,...,m, there exists n integer >1 s: where α i re positive constnts. g i xt α i xt 3 Remrk 3.1 We ssume tht the gin K exists becuse the fct tht A, B stbilizble nd A, C detectble provide necessry conditions but not sufficient for the existence of the gin K such tht ll eigenvlues of the mtrix A BKC either negtive rel prt. This implies tht there exists M> nd ω< such tht: e A BKCt Me ωt for ll t> 31 In the literture, mny uthors hve proposed sufficient conditions for problem of stbilizing sttic output feedbck for liner systems. Unfortuntely for the nonliner cse the problem remins open. Exponentil stbiliztion by output feedbck sttic system 29 is extension of Theorem 2.2, it is given by the following theorem. 1 m ω Theorem 3.2 Let α = α i, R = αm +1 KC M nd M = 1+ αm +1 KC x 1 ω For x <R, the system 29 controlled by the liner stte output feedbck
7 A new generliztion of lemm Gronwll-Bellmn 627 ut = Kyt stisfies: xt M x e ωt nd ut M KC x e ωt. And then is exponentilly stble. Proof 3.3 If C I n nd p<n, the solution of system 29 is controlled by the stte output feedbck ut = Kyt is given by: xt =e A BKCt x + e A BKCt s m where KCxs i is the i ieme component of vector KCxs. then g i xs KCxs i ds 32 m while tking into ccount 3 nd 31 with α = α i, it will: xt M x e ωt + e ωt αm KC xs +1 ds 33 xte ωt M x + where: c = M x, fs =, gs = αm KC e ωs ppliction of Theorem 2.2 gives: xte ωt αm KC e ωs xse ωs +1 ds 34 M x 1 αm +1 KC x nd n = +1 the 1 e ωs ds M x 1 αm +1 KC x ω e ωt this proves tht: for ll t M x 1+ αm +1 KC x ω 1 xt M x e ωt Under the ssumption of Theorem 2.2: 1 αm +1 KC x e ωs ds > 36
8 628 Y. Lourtssi, El H. El Mzoudi nd N. Ellmi We obtin: then: 1+ αm +1 KC x ω x < This completes the proof of the theorem. ω αm +1 KC > 1 38 References [1] R. Bellmn, Stbility Theory of Differentil Eultions, McGrw-Hill, New York, [2] N. Ellmi, A generlistion of Grnwll s lemm, 2ieme conférence interntionle sur les éutions différentielles, Mrrkech, [3] N. Ellmi, J. Ellmi, Stbilistion robuste pr retour de sorties des systèmes bilineires non homogénes pr l pproche de Gronwll-Bellmn, Congrès ntionl de mthémtiues ppliuées et industrielles, SMAI 21. [4] N. Ellmi, Une nouvelle générlistion du lemme de Gronwll-Bellmn, Congrès ntionl de mthémtiues ppliuées et industrielles, SMAI 21. [5] T. H. Gronwll, Note on the derivtives with respect to prmeter of the solutions of system of differentil eutions, Ann. of Mth., [6] M. W. Hirsch, S. Smlf, Differentil Eutions, Dynmicl Systems, nd Liner Algebr, Acdemic Press, New York/London, [7] B. G. Pchptte, A note on Gronwll-Bellmn ineulity, J. Mth. Anl. Appl., , [8] B. G. Pchptte. On some generliztions of Bellmn s lemm. J. of Mthemticl Anlysis nd Applictions, , [9] J. A. Oguntuse, Remrk on Gronwll type ineulities, An. Stiint. Univ. Al. I. Cuz Isi. Mt. N.S , no. 2, Received: July, 211
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