Relative controllability of nonlinear systems with delays in control

Transcription

1 Relave conrollably o nonlnear sysems wh delays n conrol Jerzy Klamka Insue o Conrol Engneerng, Slesan Techncal Unversy, 44- Glwce, Poland. phone/ax : , {jklamka}@a.polsl.glwce.pl Keywor: Conrollably. Nonlnear conrol sysems. Conrol sysems wh delays. Schauder s xed-pon heorem. Absrac In he paper relave conrollably o ceran nonlnear dynamcal conrol sysem wh rbued and me varyng mulple lumped delays n conrol s consdered. Usng Schauder s xed pon heorem sucen condon or relave conrollably n a gven ne me nerval s ormulaed and proved. Some specal cases are also consdered. Inroducon Conrollably problems or deren ypes o nonlnear dynamcal sysems wh delays n conrol has been suded by several auhors wh he help o deren xed pon heorems [-5]. In parcular n he paper [7] he noon o measure o noncompacness o a se and Darbo s xed pon heorem are used or he sudy o he relave conrollably or nonlnear sysems wh lumped mulple me varyng delays n conrol and wh mplc dervave. The papers [] and [2] conan sucen condons or relave conrollably n a gven nerval or nonlnear dynamcal sysems wh lumped mulple me varyng delays n conrol. The same problems bu or nonlnear dynamcal sysems wh rbued delays n conrol have been cussed and solved n he papers [3] and [4]. Nonlnear dynamcal sysems wh delays n conrol are he mahemacal models o several processes. The man purpose o hs paper s o sudy he relave conrollably o nonlnear perurbed dynamcal sysems wh rbued and mulple lumped me varyng delays n conrol. Usng Schauder s xed pon heorem sucen condons or relave conrollably n a gven me nerval are ormulaed and proved. Some specal cases are also consdered. The resuls generalze or nonlnear dynamcal sysems wh rbued delays n conrol relave conrollably condons presened n he papers [-3], [5], [7], [-5]. 2 Sysem descrpon Le us consder he nonlnear dynamcal sysem wh rbued and mulple me-varyng delays n conrol, descrbed by he ollowng ordnary derenal equaon dened or [, ]. x'( = g(, x(, u( d η B(, η, x(, u( u( η (, x(, u( v(, u( v(,..., u( v (,..., u( v ( ( where x R n, u s an p-dmensonal conrol uncon wh u C p [ -h, ], B(,η,x,u s an n p-dmensonal marx, connuous n (,x,u or xed η and o bounded varaon n η on [-h,] or each (,x,u [, ] R np. Assume he n- dmensonal vecor uncon s connuous n s argumens. The negral s n he Lebesque-Seljes sense whch s denoed by he symbol d η. The connuous srcly ncreasng uncons v (:[, ] R, =,,2,...,, represen devang argumens n he conrol.e., v (=h (, where h ( are lumped me varyng delays or =,,2,... Le h> be gven. For a uncon u:[ -h, ] R p and [, ], he symbol u denoes he uncon on [-h, dened by u (s=u(s or s [-h,. Le us assume ha he uncon g(,x,u s connuous wh respec o s argumens and connuously derenable wh respec o x. Thereore, he equaon x (=g(,x(,u( has a unque soluon x( = G(,,x(,u(

2 or each connuous conrol u( on [, ] and each nal condon G(,,x(,u(=x( R n. Le he soluon G(,,y,u( be derenable wh respec o y. Le us nroduce he ollowng n n-dmensonal marces A(,x,u=g x (,x,u and F(,,y,u=G y (,,y,u. I s easy o show ha F (,s,y,u=a(,g(,s,y,u,uf(,s,y,u, and F(,,y,u=I oreover, we have [5] F (,, x(, u = I A( s, G( s,, x(, u, u F( s,, x(, u (2 Thus, he varaon o parameer ormula or he equaon ( s gven by [5] x( = G(,, x(, u F(, s, x, u dη B( s, η, x, u u( s η h F(, s, x, u ( s, x( s, u( v ( s, u( v ( s,..., u( v ( s (3 Usng he unsymmerc Fubn heorem equaon (2 can be wren as [5], [3], [4] x( = G (,, x(, u d B F (, s η, x, u B ( s η, η, x, u u ( s η η F (, s η, x, u dη B ( s η, η, x, u u( s F (, s, x, u ( s, x( s, u( v ( s,..., u( v ( s,..., u( v ( s (4 where d Bη denoes ha he negraon s n he Lebesque- Seljes sense wh respec o he varable ηn B and B( s, η, x, u, or s B ( s, η, x, u =, or s > Dene q(, x, u = (5 = d F (, s η, x, u B( s η, η, x, u u ( s Bη h η (6 S(, s, x, u = F (, s η, x, u d η B ( s η, η, x, u (7 and he n n-dmensonal conrollably marx [5], [3], [4] W (,, x, u = S(, s, x, u S (, s, x, u where he sar denoes he marx ransposon. (8 Now, le us assume ha he uncons v (:[, ] R, =,,2,..., are wce connuously derenable and srcly ncreasng uncons n he me nerval [, ]. oreover, v ( or [, ], and =,,2,3,...,. Le us nroduce he me-lead uncons r (:[v (,v ( ] [, ], =,,2,3,...,, such ha r (v (= or [, ]. Furhermore, only or smplcy and compacness o noaon le us assume ha v (= and or = he uncons v ( sasy he ollowng nequales [], [2] h = v ( v - (... v m ( = v m ( < v m- (... v ( v ( = (9 In he sequel, we shall also consder a specal case o he dynamcal sysem (.e., nonlnear dynamcal sysem wh only mulple me varyng lumped delays n conrol sasyng he relaons (9. Ths sysem s descrbed by he ollowng ordnary derenal equaon. = x' ( = g (, x(, u( B (, x (, u( u( v ( = (, x(, u( v (, u( v (,..., u( v (,..., u( v ( (

3 Conrollably marx or he dynamcal sysem ( has he ollowng orm [], [2] W (,, x, u = S (, s, x, u S (, s, x, u m m m where = m ( S (, s, x, u = F (, r ( s, x, u B ( r ( s, x, u r '( s m = (2 oreover, or dynamcal sysem ( we have [], [2] q(, x, u = F (,, x, u x x( = m = v ( = = m v ( F (, r ( s, x, u B ( r ( s, x, u r ' ( s u ( s v ( F (, r ( s, x, u B ( r ( s, x, u r ' ( s u ( s (3 Remark 2.. I should be menoned ha he dynamcal sysems ( or ( wh uncon g(,x(,u( = A(,x(,u(x( B(,x(,u(u( where A(,x,u and B(,x,u are n n and n p dmensonal marces respecvely, are he so called quaslnear dynamcal sysems [8]. I s well known, ha or dynamcal sysems wh delays n conrol ( we may nroduce wo deren conceps o sae a me, namely nsananeous sae x( and he so called complee sae z(={x(,u } R n C p (,. Denon 2.. The dynamcal sysem ( s sad o be relavely conrollable on [, ], or every nal complee sae z( and every vecor x R n, here exss a conrol u C p [, ] such ha he soluon o he equaon ( sases x( =x. Remark 2.2. I should be sressed, ha or he dynamcal sysems wh delays n conrol or n he sae varables we can also consder he so called absolue or unconal conrollably, whch s he sronger noon han he relave conrollably [-4]. Roughly speakng, absolue conrollably means ha s possble o seer he dynamcal sysem rom an arbrary nal complee sae z( o an arbrary nal complee sae z( usng admssble conrols. 3 Relave conrollably condons In hs secon usng Schauder s xed pon heorem sucen condon or relave conrollably n a gven me nerval or he nonlnear dynamcal sysem ( wll be ormulaed and proved. For smplcy o noaon le us denoe d = (x,u,u,...,u,...,u R n(p and d = x u u... u... u, where denoes he sandard norm n he ne dmensonal Eucldean space. Theorem 3.. Le he connuous uncon sasy he so called growh condon d d lm (, = (4 d and suppose ha he uncon g s connuous n s all argumens and connuously derenable wh respec o argumen x. oreover, le us assume ha here exss a posve consan c such ha or each par o uncons {x,u} C n [, ] C p [, ] dew(,,x,u c (5 Then he nonlnear dynamcal sysem ( s relavely conrollable on [, ]. Proo. The proo o he Theorem 3. s based on he Schauder s xed pon heorem. Le H=C n [, ] C p [, ] and dene he nonlnear connuous operaor T:H H, T{x,u}={y,w} as ollows: w ( = S (,, x, u W (,, x, u [ x G(,, x(, u q(, x, u F (, s, x, u ( s, x( s, u( v ( s,..., u( v ( s,..., u( v ( s ] (6

4 y ( = G(,, x(, u q(, x, u S (, s, x, u w( s F (, s, x, u ( s, x( s, u( v ( s,..., u( v ( s,..., u( v ( s Le us nroduce he ollowng noaons: c ( = sup{ S (,, x, u : S { x, u} H, [, ]} c ( = sup{ W (,, x, u : W { x, u} H, [, ]} c ( = sup{ G (,, x(, u q (, x, u : q { x, u} H, [, ]} x c ( = sup{ G (,, x (, u : u C (, } G p (7 c ( = sup{ (, x(, u( v (,..., u ( v (,..., u( v ( : { x, u} H, [, ]} c = max{( - c S, } c 2 = 6c c S c W c G ( - c 3 = 6c G ( - c 4 = ( - c S c 5 = 6c S c W c q c c 6 = 6c q c 7 = max{c 2, c 3 } c 8 = max{c 5,c 6 } Thus, by (6 and (7 he ollowng nequales hold: y( c ( c ( w q S C p (, ( cg ( c ( ( c c c 8 7 6c w( c c ( c c c ( S W q G c w ( c c c C p (, (8 I ollows rom he growh condon (4 ha here exss a posve consan k such ha, d k, hen [5], [7] c 8 c 7 (,d k, or all [, ] (9 Le us dene he se Q as ollows: Q = {{ x, u} C (, C (, : x k ( 2 and x Cn (, C n (, n k ( 2 } Thereore, we ake x p k ( 2, u k n (, C p (, ( 2 C and moreover, u C p (, k ( 2 hen = d = x( u( v ( k or all [, ] = (2 Thus, akng no accoun equales (6, (7 and nequales (8, (9, (2 we have proved ha he nonlnear operaor T maps Q no sel. Snce uncon s connuous, mples ha he operaor T s connuous and hence s compleely connuous by he applcaon o Arzela-Ascol heorem. Snce he se Q s bounded, closed and convex, hen by Schauder s xed pon heorem here exss a leas one xed pon {x*,u*} Q such ha T{x*,u*}={x*,u*}. Hence, or {y,w}={x*,u*} we have * * * x ( = G (,, x (, u q (, x, u * * S (, s, x, u u ( s * * * * * * F (, s, x, u ( s, x ( s, u ( v ( s,..., u ( v ( s,..., u ( v ( s Thus x*( s he soluon o he nonlnear ordnary derenal equaon (. oreover, s easy o very ha hs soluon sases he nal condon x*( = x. Thereore, by Denon 2. he nonlnear dynamcal sysem s relavely conrollable on me nerval [, ]. Hence Theorem 3. ollows. Now, le us ormulae sucen condon or relave conrollably or he specal case o he dynamcal sysem (.e., or he nonlnear dynamcal sysem wh only mulple me varyng lumped delays n conrol descrbed by he ordnary derenal equaon (.

5 Corollary 3.. Le he uncons and g sasy he assumpons saed n Theorem 3.. oreover, le us assume ha here exss a posve consan c such ha or each par o uncons {x,u} C n [, ] C p [, ] dew m (,,x,u c Then he nonlnear dynamcal sysem s relavely conrollable on [, ]. Proo. Snce dynamcal sysem ( s a specal case o he dynamcal sysem ( hen, Corollary 3. ollows drecly rom Theorem 3. and equales (, (2 and (3. 4 Conclusons The paper conans sucen condons or relave conrollably n a gven ne me nerval or general nonlnear ne-dmensonal dynamcal sysems wh boh rbued and lumped mulple me varyng delays n he conrol. In he proo o he man resul well known Schauder s xed pon heorem has been used. oreover, some specal cases are also cussed. The resul gven n Theorem 3. exen or more general nonlnear dynamcal conrol sysems wh deren kn o delays n conrol, relave conrollably condons presened n he papers [-3], [5], [7], and [-5]. Reerences [] Balachandran K. and Somasundaram D. Conrollably o a class o nonlnear sysems wh rbued delays n conrol, Kyberneka, Vol.2, No.6, pp , 983. [2] Balachandran K. and Somasundaram D. Relave conrollably o nonlnear sysems wh me varyng delays n conrol, Kyberneka, Vol.2, No., pp.65-72, 985. [3] Balachandran K. and Dauer J.P. Relave conrollably o perurbaons o nonlnear sysems, Journal o Opmzaon Theory and Applcaons, Vol.63, No., pp.5-56, 989. No.2, pp.9-22, 993. [6] Balachandran K. and Dauer J.P. Null conrollably o nonlnear nne delay sysems wh me varyng mulple delays n conrol, Appled ahemacs Leers, Vol.9, No.3, pp.5-2, 996. [7] Dacka C. Relave conrollably o perurbed nonlnear sysems wh delay n conrol, IEEE Transacons on Auomac Conrol, Vol.AC-27, No., pp , 982. [8] Dauer J.P. Nonlnear perurbaons o quaslnear conrol sysems, Journal ahemacal Analyss and Applcaons, Vol.54, No.3, pp , 976. [9] Klamka J. On he global conrollably o perurbed nonlnear sysems, IEEE Transacons on Auomac Conrol, Vol.AC-2, No., pp.7-72, 975. [] Klamka J. On he local conrollably o perurbed nonlnear sysems, IEEE Transacons on Auomac Conrol, Vol.AC-2, No.2, pp , 975. [] Klamka J. Conrollably o nonlnear sysems wh delay n conrol, IEEE Transacons on Auomac Conrol, Vol.AC-2, No.5, pp.72-74, 975. [2] Klamka J. Relave conrollably o nonlnear sysems wh delays n conrol, Auomaca, Vol.2, No.6, pp , 976. [3] Klamka J. Relave conrollably o nonlnear sysems wh rbued delays n conrol, Inernaonal Journal o Conrol, Vol.28, No.2, pp.37-32, 978. [4] Klamka J. Conrollably o nonlnear sysems wh rbued delays n conrol, Inernaonal Journal o Conrol, Vol.3, No.5, pp.8-89, 98. [5] Onwuau J.U. On conrollably o nonlnear sysems wh rbued delays n he conrol, Indan Journal o Pure and Appled ahemacs, Vol.2, No.2, pp.23-28, 989. [4] Balachandran K. and Dauer J.P. Null conrollably o nonlnear nne delay sysems wh rbued delays n conrol, Journal ahemacal Analyss and Applcaons, Vol.45, No., pp , 99. [5] Balachandran K. and Dauer J.P. Relave conrollably o general nonlnear sysems, Derenal Equaons and Dynamcal Sysems, Vol.,