ΑΡΗΣΟΣΔΛΔΗΟ ΠΑΝΔΠΗΣΖΜΗΟ ΘΔΑΛΟΝΗΚΖ
ΥΟΛΖ ΘΔΣΗΚΩΝ ΔΠΗΣΖΜΩΝ
ΣΜΖΜΑ ΜΑΘΖΜΑΣΗΚΩΝ
ΔΤΣΑΘΗΟ Ν. ΑΝΣΩΝΗΟΤ
ΑΝΑΛΤΖ ΗΓΗΑΕΟΝΣΩΝ ΓΡΑΜΜΗΚΩΝ
ΤΣΖΜΑΣΩΝ ΓΗΑΚΡΗΣΟΤ ΥΡΟΝΟΤ
Γηδαθηνξηθή Γηαηξηβή
ΘΔΑΛΟΝΗΚΖ 2000
ΑΡΗΣΟΣΔΛΔΗΟ ΠΑΝΔΠΗΣΖΜΗΟ ΘΔΑΛΟΝΗΚΖ
ΥΟΛΖ ΘΔΣΗΚΩΝ ΔΠΗΣΖΜΩΝ
ΣΜΖΜΑ ΜΑΘΖΜΑΣΗΚΩΝ
ΔΠΗΣΖΜΟΝΗΚΖ ΔΠΔΣΖΡΗΓΑ ΑΡ ..... ΣΟΤ .... ΣΟΜΟΤ
ΔΤΣΑΘΗΟ Ν. ΑΝΣΩΝΗΟΤ
ΑΝΑΛΤΖ ΗΓΗΑΕΟΝΣΩΝ ΓΡΑΜΜΗΚΩΝ
ΤΣΖΜΑΣΩΝ ΓΗΑΚΡΗΣΟΤ ΥΡΟΝΟΤ
ΓΗΓΑΚΣΟΡΗΚΖ ΓΗΑΣΡΗΒΖ
πνπ εθπνλήζεθε ζηνλ ηνκέα Αξηζκεηηθήο Αλάιπζεο θαη Δπηζηήκεο Ζ/Τ, ηνπ
ηκήκαηνο καζεκαηηθώλ ηνπ Αξηζηνηειείνπ Παλεπηζηεκίνπ Θεζζαινλίθεο κε ηελ
επνπηεία ηεο ζπκβνπιεπηηθήο επηηξνπήο πνπ νξίζηεθε από ηηο 17/21-11-1994 θαη
55/17-4-2000 απνθάζεηο ηνπ ηκήκαηνο θαη ηεο νπνίαο κέιε είλαη:
Σίηλος
Ονομαηεπώνσμο
Τπογραθή
1. Καζεγεηήο ηνπ ηκήκαηνο
Μαζεκαηηθώλ ηνπ ΑΠΘ
Ανηώνιος - Ηωάννες Βαρδοσλάκες
.....................
2. Αλαπιεξσηήο Καζεγεηήο
ηνπ ηκήκαηνο
Μαζεκαηηθώλ ηνπ ΑΠΘ
σμεών Μποδαπαλίδες
.....................
3. Αλαπιεξσηήο Καζεγεηήο
Μαζεκαηηθώλ ηνπ ΑΠΘ
Υρόνες Μωσζιάδες
.....................
ii
Ζ επηακειήο εμεηαζηηθή επηηξνπή πνπ νξθίζηεθε γηα ηελ θξίζε ηεο δηδαθηνξηθήο
δηαηξηβήο ηνπ Δπζηάζηνπ Ν. Αλησλίνπ Μαζεκαηηθνύ ζπλήιζε ζε ζπλεδξίαζε ζην
Παλεπηζηήκην ηεο Θεζζαινλίθεο 26/6/2000 όπνπ παξαθνινύζεζε ηελ ππνζηήξημε
ηεο δηαηξηβήο κε ηίηιν «ΑΝΑΛΤΖ ΗΓΗΑΕΟΝΣΩΝ ΓΡΑΜΜΗΚΩΝ ΤΣΖΜΑΣΩΝ
ΓΗΑΚΡΗΣΟΤ ΥΡΟΝΟΤ»
Ζ επηηξνπή έθξηλε νκόθσλα ( επηά (7) ςήθνη ππέξ, κεδέλ (0) θαηά) όηη ε δηαηξηβή
είλαη πξσηόηππε θαη απνηειεί νπζηαζηηθή ζπκβνιή ζηελ επηζηήκε.
ΣΑ ΜΔΛΖ ΣΖ ΔΠΣΑΜΔΛΟΤ ΔΞΔΣΑΣΗΚΖ ΔΠΗΣΡΟΠΖ
Σίηλος
Ονομαηεπώνσμο
Τπογραθή
1. Καζεγεηήο ηνπ ηκήκαηνο
Μαζεκαηηθώλ ηνπ ΑΠΘ
Ανηώνιος - Ηωάννες Βαρδοσλάκες
.....................
2. Καζεγεηήο ηνπ ηκήκαηνο
Μαζεκαηηθώλ ηνπ ΑΠΘ
σμεών Μποδαπαλίδες
.....................
3. Αλαπιεξσηήο Καζεγεηήο
ηνπ ηκήκαηνο
Μαζεκαηηθώλ ηνπ ΑΠΘ
Υρόνες Μωσζιάδες
.....................
4. Καζεγεηήο ηνπ ηκήκαηνο
Μαζεκαηηθώλ ηνπ ΑΠΘ
Παναγιώηες – Υρήζηος Βαζιλείοσ
.....................
5. Καζεγεηήο ηνπ ηκήκαηνο
Μαζεκαηηθώλ ηνπ ΑΠΘ
Κωνζηανηίνος Καρανίκας
.....................
6. Δπίθνπξνο θαζεγεηήο ηνπ
ηκήκαηνο Μαζεκαηηθώλ
ηνπ ΑΠΘ
Θωμάς Κσβενηίδες
.....................
7. Καζεγεηήο ηνπ ηκήκαηνο
Ζιεθηξ. Μεραληθώλ ηνπ
ΑΠΘ
Βαζίλειος Πεηρίδες
.....................
iii
ΒΗΟΓΡΑΦΗΚΟ ΖΜΔΗΩΜΑ
Ο Δπζηάζηνο Ν. Αλησλίνπ γελλήζεθε ζηηο 24/11/1972 ζηε Λάξηζα. Σνλ Ηνύλην ηνπ
1990 απνθνίηεζε από ην 2ν Γεληθό Λύθεην Λάξηζαο κε βαζκό «Λίαλ Καιώο». Σν ίδην
έηνο θαη κεηά από επηηπρία ζηηο παλειιαδηθέο εμεηάζεηο εηζήρζεθε ζην ηκήκα
καζεκαηηθώλ ηνπ Αξηζηνηειείνπ Παλεπηζηεκίνπ Θεζζαινλίθεο, από όπνπ θαη
απνθνίηεζε ηνλ Οθηώβξην ηνπ 1994 κε βαζκό «Λίαλ Καιώο».
Σν Ννέκβξην ηνπ 1994 έγηλε δεθηόο γηα ηελ εθπόλεζε δηδαθηνξηθήο δηαηξηβήο ζην
ηκήκα Μαζεκαηηθώλ ηνπ ΑΠΘ, ππό ηελ επίβιεςε ηνπ θαζεγεηή Α.Η.Γ. Βαξδνπιάθε,
κε ζέκα ηεο δηαηξηβήο «Αλάιπζε Ηδηαδόλησλ Γξακκηθώλ πζηεκάησλ Γηαθξηηνύ
Υξόλνπ» (απόθαζε Γ..: 20/6-2-1995). Καηά ηελ δηάξθεηα ηεο εθπόλεζεο ηεο
δηαηξηβήο ζπκκεηείρε σο λένο εξεπλεηήο ζε πιεζώξα εξεπλεηηθώλ θαη εθπαηδεπηηθώλ
πξνγξακκάησλ. Μεηαμύ απηώλ πξόγξακκα ΠΔΝΔΓ ηεο ΓΓΔΣ κε ηίηιν «Ανάπτυξη
Λογισμικού για την Ανάλυση και Σύνθεση Πολυμεταβλητών Συστημάτων Αυτομάτου
Ελέγχου» (2/96 – 2/98), πξόγξακκα κνξθσηηθώλ αληαιιαγώλ κε ηελ Σζερία
(ρξεκαηνδόηεζε ΓΓΔΣ) (5/95 – 5/96), European Robust and Adaptive Control
Network (EURACO) (9/94 – 10/96) θαη research assistant (EPRSC fellowship) ζην
Department of Applied Mathematics ηνπ Loughborough University ηεο Αγγιίαο
(3/98-6/98).
Σα ελδηαθέξνληα ηνπ θηλνύληαη ζην ρώξν ηεο Μαζεκαηηθήο Θεσξίαο πζηεκάησλ
θαη εηδηθόηεξα ζε αιγεβξηθέο κεζόδνπο γηα αλάιπζε θαη ζύλζεζε γξακκηθώλ
πνιπκεηαβιεηώλ ζπζηεκάησλ, ελώ παξάιιεια αζρνιείηαη επαγγεικαηηθά κε ηελ
αλάπηπμε software κε θύξην πξνζαλαηνιηζκό ηηο δηθηπαθέο εθαξκνγέο θαη ην
internet.
iv
«Ζ έγθξηζε ηεο παξνύζεο δηδαθηνξηθήο δηαηξηβήο ππό ηνπ ηκήκαηνο Μαζεκαηηθώλ
ηεο ρνιήο Θεηηθώλ Δπηζηεκώλ ηνπ Αξηζηνηειείνπ Παλεπηζηεκίνπ Θεζζαινλίθεο
δελ απνδεινί απνδνρή ησλ γλσκώλ ηνπ ζπγγξαθέσο»
(Ν. 5343/1932, αξζξ. 202, παξαγξ. 2)
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0RQSQ¯OG PC RQ¯OG ¬UK QK F¯Q GLKTTGKX QZTKCTUKM¦ UCZU©[QPUCK CRN¦ G©PCK ISCOO§PGX
OG CPU©TUSQHJ YSQPKM¨ HQS¦ J O©C WX RSQX UJP ¦NNJ FJN CP 1fc 1 c 1 3c 1 G©PCK
OKC CMQNQZV©C RQZ KMCPQRQKG© UJP U¬UG J YSQPKM¦ CPGTUSCO§PJ CMQNQZV©C
1 c 1 3 c c 1 c 1 f KMCPQRQKG© UJP 6Q RCSCR¦PW N¨OOC CRQFGKMP¯GK ¬UK QK TU¨NGX UQZ R©PCMC f8 a8& FGP GRCSMQ¯P IKC UQ
TYJOCUKTOQ¯ OKCX RN¨SQZX D¦TJX UQZ E $ZU¬ QHG©NGUCK TUQ IGIQP¬X ¬UK ZR¦SYQZP N¯TGKX
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PC RCSCYVQ¯P CR¬ M¦RQKQ ISCOOKM¬ TZPFZCTO¬ UWP TUJNP UQZ R©PCMC f8 a8& $ZU§X
FGP G©PCK ¦NNGX CR¬ UKX N¯TGKX RQZ CPUKTQKYQ¯P TG KFKQUKO§X UQZ R©PCMC Ej TUQ j ' f
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UQZ Ej CPUKTUQKYQ¯P TG RGRGSCTO§PC HCTOCUKM¦ [G¯IJ UQZ Ej RQZ CPUQKTUQKYQ¯P
TUJP KFKQUKO¨ b3f c UC QRQ©C RGSK§YQPUCK TUQ 5UJP RGS©RUWTJ RQZ Q R©PCMCX
Ej §YGK KFKQUKO§X TUQ j ' f UQ RGRGSCTO§PQ KFKQ[G¯IQX RQZ CPUKTUQKYG© TUJP OJFGPKM¨
KFKQUKO¨ G©PCK UQ ¦RGKSQ HCTOCUKM¬ [G¯IQX UQZ Ej UQ QRQ©Q RSQHCPX FGP §YGK NJHVG©
ZR¬EJP TUJP *KC PC TYJOCU©TQZOG OKC RN¨SJ D¦TJ UQZ FKCPZTOCUKMQ¯ YSQZ E
RS§RGK PC TZORGSKN¦DQZOG T
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$ZUQRCNN©PFSQOGX RCSCTU¦TGKX FKCMSKUQ¯ YS¬PQZ
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GRKNZQZTP RKP¦MWP FKCUZRPQZOG UQ GR¬OGPQ VGSJOC
8HYUKPD =?
+ V\PSHULIRU¢ WR\ V\VW¤PDWRZ SR\ SHULJU¢IHWDL DS¨ WKQ S¢QY DS¨ WR SHSHUDVP£QR V«QROR
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GL©TWTJX G©PCK Q RZS¨PCX UQZ - nE + TZORGSKHQS¦ UJX G©PCK RSQHCPX
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¬RQZ ? ' _i} mEjmG©PCK Q TZPQNKM¬X CSKVO¬X RGRGSCTO§PWP TUQKYGKWFP FKCKSGUP,
> G©PCK Q TZPQNKM¬X CSKVO¬X TUQKYGKWFP FKCKSGUP TUQ 4 ¬RQZ QK RQNNCRN¬UJUGX UWP
FKCKSGUP NCOD¦PQPUCK ZR¬EKP MCK TUKX F¯Q RGSKRUTGKX MCK GRKRN§QP KTY¯GK ? n > ' o^
DN§RG =? (RKRN§QP G©PCK IPWTU¬ =? ¬UK QK TU¨NGX UQZ R©PCMC SJ,Ef8 a8 c f"a" 3'f
G©PCK ISCOOKM¦ CPGL¦SUJUGX (RKRN§QP CR¬ UJP ZR¬VGTJ ¬UK Q R©PCMCX Ej G©PCK
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
/¯TJ UJX $ZUQRCNN©PFSQOJX GL©TWTJX
MCPQPKM¬X §YQZOG
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ZRQYSWP GM UWP QRQ©WP Q RSUQX CPUKTUQKYG© TUJP RGRGSCTO§PJ HCTOCUKM¨ F¬OJ UQZ
EjMCK Q FG¯UGSQX TUJP HCTOCUKM¨ FQO¨ UQZ R©PCMC TUQ ¦RGKSQ FJN
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CSYKM¦ UWP N§LGWP Forward MCK Backward CPU©TUQKYC 2 RSUQX ZRQYSQX E8 RGSK§YGK
N¯TGKX RQZ MKPQ¯PUCK RSQX UC GORS¬X TUQP YS¬PQ CZU¬ FKMCKQNQIG© UQP YCSCMUJSTO¬ UQZ
YSQZ WX Forward MCK CPUCPCMN¦ UJP FK¦FQTJ UWP CSYKMP TZPVJMP 1 fc 1 c c 1 ^3
RSQX UC GORS¬X GP Q FG¯UGSQX ZRQYSQX E F©PGK N¯TGKX RQZ MKPQ¯PUCK CPU©TUSQHC
FJNCF¨ CR¬ UQP UGNKM¨ YSQPKM¨ TUKIO¨ RSQX UQ f
+ RCSCR¦PW CP¦NZTJ ORQSG© PC TZIMSKVG© OG UC CRQUGN§TOCUC RQZ GOHCP©[QPUCK TUKX
=? =? =? 5JOGKPQZOG ¬UK J TZIMGMSKO§PJ CP¦NZTJ UQZ YSQZ E FGP G©PCK
OQPCFKM¨ MCK CPUKTUQKYG© CRN¦ TG OKC
(Maximal Forward
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TUJP =? P£JLVWK SURZ WD HPSU¨Z DQ¢O\VK
/«VK WKZ $\WRSDOO¥QGURPKZ HM¥VYVKZ
PC RQN¯ GPFKCH§SQP RS¬DNJOC G©PCK Q RSQTFKQSKTO¬X OKCX MNGKTU¨X H¬SOQZNCX
N¯TJX UJX FGFQO§PWP UWP TZPQSKCMP UKOP UQZ RSQDN¨OCUQX 2 N¬IQX RQZ
YSGKC[¬OCTUG TZPQSKCM§X TZPV¨MGX MCK ¬YK CRN¦ CSYKM§X TZPV¨MGX G©PCK RSQHCP¨X CP
N¦DQZOG ZR¬EKP UJP CP¦NZTJ UQZ YSQZ N¯TGWP TUJP RSQJIQ¯OGPJ RCS¦ISCHQ
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
$ZUQRCNN©PFSQOGX RCSCTU¦TGKX FKCMSKUQ¯ YS¬PQZ
8HYUKPD =?
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FK¦PZTOC l ' d l A8 l A" oA c ¬RQZ l 8 MCK l " §YQZP MCU¦NNJNC GRKNGIO§PGX FKCTU¦TGKX
6SC CR¬ UJP TY§TJ MCK YSJTKOQRQKPUCX UQP QSKTO¬ UWP 8 c " CR¬ UJP
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
/¯TJ UJX $ZUQRCNN©PFSQOJX GL©TWTJX
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l 8 ' 8 %f
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5JOGKPQZOG ¬UK J TY§TJ RSQFKQS©[GK UQ l 8 CNN¦ ¬YK UQ l " :SJTKOQRQKPUCX
RCS¬OQKC OGVQFQNQI©C ORQSQ¯OG PC IS¦EQZOG
l " ' " % 3^n
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$PU©TUQKYC J TY§TJ RSQTFKQS©[GK UQ l " CNN¦ ¬YK UQ l 8 6SC YSJTKOQRQKPUCX
UKX TY§TGKX MCK §YQZOG
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boundary mapping equation TUJP =? MCK ORQSG© PC VGWSJVG© ¦OGTJ IGP©MGZTJ UJX +
GL©TWTJ CRGKM¬PKTJX UWP TZPQSKCMP TZPVJMP RGSK§YGK ¬NQZX UQZX RGSKQSKTOQ¯X RQZ
V§UGK UQ T¯TUJOC TUC F¯Q CMSC©C TJOG©C UQZ TZP¬NQZ & ' fc c 2c c $P QK TZPV¨MGX
CZU§X KMCPQRQKQ¯PUCK U¬UG YSJTKOQRQKPUCX UJP ORQSQ¯OG PC RSQTFKQS©TQZOG
OQPCFKM¦ ¬NGX UKX GPFK¦OGTGX UKO§X UQZ 1& *KC CRNQ¯TUGZTJ UQZ TZODQNKTOQ¯ MCK CMQNQZV¬PUCX RCS¬OQKQ TZODQNKTO¬ OG UJP =?
V§UQZOG
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DSHLN¨QLVKZ WYQ V\QRULDN©Q V\QXKN©Q
8HYUKPD =?
o@?& ~ Efc ' o^
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$ZUQRCNN©PFSQOGX RCSCTU¦TGKX FKCMSKUQ¯ YS¬PQZ
$S¨GHLMK
V§UQZOG
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G©PCK CPUKTUS§EKOQX CR¬ UJP §YQZOG
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3SQHCPX Q R©PCMCX TUQ FGLK¬ O§NQX UJX RCSCR¦PW TY§TJX §YGK RN¨SJ U¦LJ ISCOOP
SC
o@?&~ Efc ' o^
6Q RCSCR¦PW CRQU§NGTOC ORQSG© PC TZIMSKVG© OG UQ VGSJOC TUJP =? ¬RQZ
CRQFGKMP¯GUCK ¬UK §PCX R©PCMCX CRGKM¬PKTJX UWP TZPQSKCMP TZPVJMP OG RN¨SJ U¦LJ
ZR¦SYGK CP MCK O¬PQ CP UQ CPU©TUQKYQ T¯TUJOC IGPKMGZO§PQZ YSQZ MCUCTU¦TGWP
G©PCK
Solvable MCK
Conditionable 5UJP RGS©RUWTJ
OCX UQ T¯TUJOC G©PCK T©IQZSC GRKN¯TKOQ CHQ¯ §YQZOG ¨FJ RSQTFKQS©TGK OKC N¯TJ
MCK TZPVJMQNQI¨TKOQ CHQ¯ CRQFG©LCOG ¬UK TUJP RGS©RUWTJ RQZ QK TZPQSKCM§X UKO§X
KMCPQRQKQ¯P UJP J N¯TJ CZU¨ G©PCK OQPCFKM¨ (RKRN§QP J GRKNZTKO¬UJUC MCK J
TZPVJMQNQIJTKO¬UJUC UJX ORQSG© PC GRKDGDCKWVG© YSJTKOQRQKPUCX GN§IYQZX
U¦LJX UWP CPU©TUQKYWP RKP¦MWP ¬RWX TUJP =? 5UJP RGS©RUWTJ OCX UQ IGIQP¬X ¬UK
Q R©PCMCX Ej G©PCK MCPQPKM¬X GLCTHCN©[GK UJP KTY¯ MCK UWP F¯Q RCSCR¦PW KFKQU¨UWP
(©PCK GR©TJX TJOCPUKM¬ PC TJOGKTQZOG ¬UK J TY§TJ TZPGR¦IGUCK
HSLO«VLPR
V\QXKNRORJ¤VLPR
_4 !ih ~ Efc ' o^ ' _4 E
RQZ TJOC©PGK ¬UK UC FKCP¯TOCUC TZODKDCTUP CSYKMPUGNKMP TZPVJMP GRKN§IQPUCK
CR¬ §PC FKCPZTOCUKM¬ YSQ FK¦TUCTGWX o^ SC o^ G©PCK Q TZPQNKM¬X CSKVO¬X
TZPQSKCMP UKOP RQZ MCUCP§OQPUCK TUC F¯Q ¦MSC UQZ TZP¬NQZ & ' fc c 2c c MCK
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
/¯TJ UJX $ZUQRCNN©PFSQOJX GL©TWTJX
ORQSQ¯P PC GRKNGIQ¯P CZVC©SGUC + TY§TJ OGUCL¯ UWP _4 !ih ~ Efc MCK _4 E G©PCK
RSQHCP¨X
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3SQHCPX Q R©PCMCX Ej FGP §YGK RGRGSCT§PQZX TUQKYGKFGKX FKCKS§UGX CHQ¯ G¯MQNC
FKCRKTUPGK MCPG©X ¬UK _i| Ej ' 7GWSQ¯OG USC UQ T¯TUJOC RQZ RGSKIS¦HGUCK CR¬
UJP CZUQRCNN©PFSQOJ GL©TWTJ
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IKC & ' fc c 2c ^ OG : e 5¯OHWPC OG UQP TZODQNKTO¬ RQZ §YQZOG CMQNQZV¨TGK
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