Optimal control of infinite dimensional bilinear systems: application to the heat and wave equations

In this paper we consider second order optimality conditions for a bilinear optimal control problem governed by a strongly continuous semigroup operator, the control entering linearly in the cost function. We derive first and second order optimality conditions, taking advantage of the Goh transform....

Full description

Saved in:

Bibliographic Details
Published in:	Mathematical programming Vol. 168; no. 1-2; pp. 717 - 757
Main Authors:	Aronna, M. Soledad, Bonnans, J. Frédéric, Kröner, Axel
Format:	Journal Article
Language:	English
Published:	Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2018 Springer Nature B.V Springer Verlag
Subjects:	Calculus of Variations and Optimal Control; Optimization Combinatorics Full Length Paper Mathematical analysis Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Mathematics of Computing Nonlinear programming Numerical Analysis Operators (mathematics) Optimal control Optimization Optimization and Control Theoretical Wave equations Wave equation 49K20 Partial differential equations Heat equation Optimal control 35L05 Second-order optimality conditions 35K05 Semigroup theory 90C48 Bilinear control systems Goh transform second-order optimality conditions heat equation bilinear control systems wave equation partial differential equations semigroup theory
ISSN:	0025-5610, 1436-4646
Online Access:	Get full text
Tags:	Add Tag No Tags, Be the first to tag this record!

Abstract	In this paper we consider second order optimality conditions for a bilinear optimal control problem governed by a strongly continuous semigroup operator, the control entering linearly in the cost function. We derive first and second order optimality conditions, taking advantage of the Goh transform. We then apply the results to the heat and wave equations.
AbstractList	In this paper we consider second order optimality conditions for a bilinear optimal control problem governed by a strongly continuous semigroup operator, the control entering linearly in the cost function. We derive first and second order optimality conditions, taking advantage of the Goh transform. We then apply the results to the heat and wave equations.
Author	Kröner, Axel Bonnans, J. Frédéric Aronna, M. Soledad
Author_xml	– sequence: 1 givenname: M. Soledad surname: Aronna fullname: Aronna, M. Soledad organization: EMAp/FGV – sequence: 2 givenname: J. Frédéric orcidid: 0000-0003-0988-9865 surname: Bonnans fullname: Bonnans, J. Frédéric email: Frederic.Bonnans@inria.fr organization: Centre de Mathématiques Appliquées, Ecole Polytechnique, INRIA-Saclay, Université Paris-Saclay – sequence: 3 givenname: Axel surname: Kröner fullname: Kröner, Axel organization: Centre de Mathématiques Appliquées, Ecole Polytechnique, INRIA-Saclay, Université Paris-Saclay
BackLink	https://inria.hal.science/hal-01273496$$DView record in HAL
BookMark	eNp9kEtLAzEUhYMoWB8_wF3AlYvRm0kmmXFXii8ouNF1SDMZG5kmY5JW-u9NHUUQdBVuzncu95wjtO-8MwidEbgkAOIqEiAgCiC8INDQgu2hCWGUF4wzvo8mAGVVVJzAITqK8RUACK3rCWofh2RXqsfauxR8j32Hreuss8ng1q6Mi9a7rC9sb51RAcdtTGYVr7Eaht5qlbKOk8dpafDSqISVa_G72hhs3tafajxBB53qozn9eo_R8-3N0-y-mD_ePcym80LTClKxYKBqbQjXSjUVq0qmheCsLTvSaOBMd7rMA-cUWmALRjVlwoCGVqu67gw9Rhfj3qXq5RByrrCVXll5P53L3R-QUlDW8E2Z2fORHYJ_W5uY5Ktfh5w0yjIXKpqq5iJTYqR08DEG00lt02eoFJTtJQG5q1-O9ef9XO7qlyw7yS_n90H_ecrREzPrXkz4uelv0wftWpmH
CitedBy_id	crossref_primary_10_1007_s00245_021_09803_6 crossref_primary_10_1016_j_ejcon_2023_100943 crossref_primary_10_1002_asjc_3125 crossref_primary_10_1137_19M1286906 crossref_primary_10_1080_00207179_2022_2079005 crossref_primary_10_1080_02331934_2024_2421384 crossref_primary_10_3934_naco_2025004 crossref_primary_10_1007_s00028_024_00968_5 crossref_primary_10_1137_17M1117653 crossref_primary_10_1080_00036811_2023_2299714 crossref_primary_10_1007_s10957_022_02143_7 crossref_primary_10_1016_j_jmaa_2023_127024
Cites_doi	10.1080/10556788.2013.830220 10.1007/s10107-015-0976-0 10.1137/S1052623403426519 10.1007/978-1-4612-5561-1 10.1137/0329049 10.1137/0304026 10.3934/naco.2012.2.511 10.2140/pjm.1951.1.525 10.1137/S0363012994261987 10.1109/CDC.2013.6759937 10.1137/140978855 10.1365/s13291-014-0109-3 10.1137/120862892 10.1007/BF02551380 10.2514/3.2562 10.1137/0331045
ContentType	Journal Article
Copyright	Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2016 Mathematical Programming is a copyright of Springer, (2016). All Rights Reserved. Distributed under a Creative Commons Attribution 4.0 International License
Copyright_xml	– notice: Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2016 – notice: Mathematical Programming is a copyright of Springer, (2016). All Rights Reserved. – notice: Distributed under a Creative Commons Attribution 4.0 International License
DBID	AAYXX CITATION 7SC 8FD JQ2 L7M L~C L~D 1XC VOOES
DOI	10.1007/s10107-016-1093-4
DatabaseName	CrossRef Computer and Information Systems Abstracts Technology Research Database ProQuest Computer Science Collection Advanced Technologies Database with Aerospace Computer and Information Systems Abstracts Academic Computer and Information Systems Abstracts Professional Hyper Article en Ligne (HAL) Hyper Article en Ligne (HAL) (Open Access)
DatabaseTitle	CrossRef Computer and Information Systems Abstracts Technology Research Database Computer and Information Systems Abstracts – Academic Advanced Technologies Database with Aerospace ProQuest Computer Science Collection Computer and Information Systems Abstracts Professional
DatabaseTitleList	Computer and Information Systems Abstracts
DeliveryMethod	fulltext_linktorsrc
Discipline	Engineering Mathematics
EISSN	1436-4646
EndPage	757
ExternalDocumentID	oai:HAL:hal-01273496v2 10_1007_s10107_016_1093_4
GrantInformation_xml	– fundername: Foundation Hadamard/Gaspard Monge Program for Optimization and Operations Research (PGMO)
GroupedDBID	--K --Z -52 -5D -5G -BR -EM -Y2 -~C -~X .4S .86 .DC .VR 06D 0R~ 0VY 199 1B1 1N0 1OL 1SB 203 28- 29M 2J2 2JN 2JY 2KG 2KM 2LR 2P1 2VQ 2~H 30V 3V. 4.4 406 408 409 40D 40E 5GY 5QI 5VS 67Z 6NX 6TJ 78A 7WY 88I 8AO 8FE 8FG 8FL 8TC 8UJ 8VB 95- 95. 95~ 96X AAAVM AABHQ AACDK AAHNG AAIAL AAJBT AAJKR AANZL AARHV AARTL AASML AATNV AATVU AAUYE AAWCG AAYIU AAYQN AAYTO AAYZH ABAKF ABBBX ABBXA ABDBF ABDZT ABECU ABFTV ABHLI ABHQN ABJCF ABJNI ABJOX ABKCH ABKTR ABMNI ABMQK ABNWP ABQBU ABQSL ABSXP ABTEG ABTHY ABTKH ABTMW ABULA ABUWG ABWNU ABXPI ACAOD ACBXY ACDTI ACGFS ACGOD ACHSB ACHXU ACIWK ACKNC ACMDZ ACMLO ACNCT ACOKC ACOMO ACPIV ACUHS ACZOJ ADHHG ADHIR ADIMF ADINQ ADKNI ADKPE ADRFC ADTPH ADURQ ADYFF ADZKW AEBTG AEFIE AEFQL AEGAL AEGNC AEJHL AEJRE AEKMD AEMOZ AEMSY AENEX AEOHA AEPYU AESKC AETLH AEVLU AEXYK AFBBN AFEXP AFFNX AFGCZ AFKRA AFLOW AFQWF AFWTZ AFZKB AGAYW AGDGC AGGDS AGJBK AGMZJ AGQEE AGQMX AGRTI AGWIL AGWZB AGYKE AHAVH AHBYD AHKAY AHQJS AHSBF AHYZX AIAKS AIGIU AIIXL AILAN AITGF AJBLW AJRNO AJZVZ AKVCP ALMA_UNASSIGNED_HOLDINGS ALWAN AMKLP AMXSW AMYLF AMYQR AOCGG ARAPS ARCSS ARMRJ ASPBG AVWKF AXYYD AYJHY AZFZN AZQEC B-. B0M BA0 BAPOH BBWZM BDATZ BENPR BEZIV BGLVJ BGNMA BPHCQ BSONS CAG CCPQU COF CS3 CSCUP DDRTE DL5 DNIVK DPUIP DU5 DWQXO EAD EAP EBA EBLON EBR EBS EBU ECS EDO EIOEI EJD EMI EMK EPL ESBYG EST ESX FEDTE FERAY FFXSO FIGPU FINBP FNLPD FRNLG FRRFC FSGXE FWDCC GGCAI GGRSB GJIRD GNUQQ GNWQR GQ6 GQ7 GQ8 GROUPED_ABI_INFORM_COMPLETE GXS H13 HCIFZ HF~ HG5 HG6 HMJXF HQYDN HRMNR HVGLF HZ~ H~9 I-F I09 IAO IHE IJ- IKXTQ ITM IWAJR IXC IZIGR IZQ I~X I~Z J-C J0Z JBSCW JCJTX JZLTJ K1G K60 K6V K6~ K7- KDC KOV KOW L6V LAS LLZTM M0C M0N M2P M4Y M7S MA- N2Q N9A NB0 NDZJH NPVJJ NQ- NQJWS NU0 O9- O93 O9G O9I O9J OAM P19 P2P P62 P9R PF0 PQBIZ PQBZA PQQKQ PROAC PT4 PT5 PTHSS Q2X QOK QOS QWB R4E R89 R9I RHV RIG RNI RNS ROL RPX RPZ RSV RZK S16 S1Z S26 S27 S28 S3B SAP SCLPG SDD SDH SDM SHX SISQX SJYHP SMT SNE SNPRN SNX SOHCF SOJ SPISZ SRMVM SSLCW STPWE SZN T13 T16 TH9 TN5 TSG TSK TSV TUC TUS U2A UG4 UOJIU UTJUX UZXMN VC2 VFIZW W23 W48 WH7 WK8 XPP YLTOR Z45 Z5O Z7R Z7S Z7X Z7Y Z7Z Z81 Z83 Z86 Z88 Z8M Z8N Z8R Z8T Z8W Z92 ZL0 ZMTXR ZWQNP ~02 ~8M ~EX AAPKM AAYXX ABBRH ABDBE ABFSG ABRTQ ACSTC ADHKG ADXHL AEZWR AFDZB AFFHD AFHIU AFOHR AGQPQ AHPBZ AHWEU AIXLP AMVHM ATHPR AYFIA CITATION PHGZM PHGZT PQGLB 7SC 8FD JQ2 L7M L~C L~D PUEGO 1XC VOOES
ID	FETCH-LOGICAL-c350t-b40a8ce16caa954524c7764d2f19c064cfc2d2f6630d04b43c347e0c0dca88fe3
IEDL.DBID	RSV
ISICitedReferencesCount	17
ISICitedReferencesURI	http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000426071000029&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D
ISSN	0025-5610
IngestDate	Tue Oct 14 21:01:15 EDT 2025 Thu Sep 25 00:57:15 EDT 2025 Sat Nov 29 03:33:59 EST 2025 Tue Nov 18 21:04:25 EST 2025 Fri Feb 21 02:32:44 EST 2025
IsDoiOpenAccess	true
IsOpenAccess	true
IsPeerReviewed	true
IsScholarly	true
Issue	1-2
Keywords	Wave equation 49K20 Partial differential equations Heat equation Optimal control 35L05 Second-order optimality conditions 35K05 Semigroup theory 90C48 Bilinear control systems Goh transform second-order optimality conditions heat equation bilinear control systems wave equation partial differential equations semigroup theory
Language	English
License	Distributed under a Creative Commons Attribution 4.0 International License: http://creativecommons.org/licenses/by/4.0
LinkModel	DirectLink
MergedId	FETCHMERGED-LOGICAL-c350t-b40a8ce16caa954524c7764d2f19c064cfc2d2f6630d04b43c347e0c0dca88fe3
Notes	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14
ORCID	0000-0003-0988-9865
OpenAccessLink	https://inria.hal.science/hal-01273496
PQID	2007795867
PQPubID	25307
PageCount	41
ParticipantIDs	hal_primary_oai_HAL_hal_01273496v2 proquest_journals_2007795867 crossref_citationtrail_10_1007_s10107_016_1093_4 crossref_primary_10_1007_s10107_016_1093_4 springer_journals_10_1007_s10107_016_1093_4
PublicationCentury	2000
PublicationDate	2018-03-01
PublicationDateYYYYMMDD	2018-03-01
PublicationDate_xml	– month: 03 year: 2018 text: 2018-03-01 day: 01
PublicationDecade	2010
PublicationPlace	Berlin/Heidelberg
PublicationPlace_xml	– name: Berlin/Heidelberg – name: Heidelberg
PublicationSubtitle	A Publication of the Mathematical Optimization Society
PublicationTitle	Mathematical programming
PublicationTitleAbbrev	Math. Program
PublicationYear	2018
Publisher	Springer Berlin Heidelberg Springer Nature B.V Springer Verlag
Publisher_xml	– name: Springer Berlin Heidelberg – name: Springer Nature B.V – name: Springer Verlag
References	GohBSThe second variation for the singular Bolza problemJ. SIAM Control19664230932520354510.1137/03040260146.11906 BonnansJFTibaDControl problems with mixed constraints and application to an optimal investment problemMath. Rep. (Bucur.)200911(61)429330626561661212.49037 AronnaMSBonnansJFDmitrukAVLotitoPAQuadratic order conditions for bang-singular extremalsNumer. Algebra Control Optim.201223511546297090110.3934/naco.2012.2.5111252.49028(AIMS Journal, special issue dedicated to Professor Helmut Maurer on the occasion of his 65th birthday) CasasERyllCTröltzschFSecond order and stability analysis for optimal sparse control of the FitzHugh–Nagumo equationSIAM J. Control Optim.201553421682202337463610.1137/1409788551326.49032 DmitrukAVQuadratic conditions for a weak minimum for singular regimes in optimal control problemsSoviet Math. Dokl.19771824184220385.49008 LiXYongJNecessary conditions for optimal control of distributed parameter systemsSIAM J. Control Optim.1991294895908111166610.1137/03290490733.49025 AubinJ-PUn théorème de compacitéC. R. Acad. Sci. Paris1963256504250441528600195.13002 PoggioliniLStefaniGSufficient optimality conditions for a bang-singular extremal in the minimum time problemControl Cybern.200837246949024728861235.49054 HestenesMRApplications of the theory of quadratic forms in Hilbert space to the calculus of variationsPac. J. Math.195115255814659010.2140/pjm.1951.1.5250045.20806 Milyutin, A.A., Osmolovskiim, N.P.: Calculus of variations and optimal control, Translations of Mathematical Monographs, vol. 180, American Mathematical Society, Providence, RI (1998), Translated from the Russian manuscript by Dimitrii Chibisov. MR1641590 BallJMStrongly continuous semigroups, weak solutions, and the variation of constants formulaProc. Am. Math. Soc.19776323703734427480353.47017 BensoussanADa PratoGDelfourMCMitterSKRepresentation and Control of Infinite Dimensional Systems: Systems and Control: Foundations and Applications2007BostonBirkhäuser Boston, Inc.1117.93002 PazyASemigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences1983New YorkSpringer10.1007/978-1-4612-5561-10516.47023 Frankowska, H., Tonon, D.: The Goh necessary optimality conditions for the Mayer problem with control constraints. In: Decision and Control, pp. 538–543 (2013) BonnansJFOptimal control of a semilinear parabolic equation with singular arcsOptim. Methods Softw.2014295964978322501410.1080/10556788.2013.8302201301.49053 BergouniouxMTibaDGeneral optimality conditions for constrained convex control problemsSIAM J. Control Optim.1996342698711137771910.1137/S03630129942619870859.49026 CasasESecond order analysis for bang-bang control problems of PDEsSIAM J. Control Optim.201250423552372297474210.1137/1208628921257.49021 CasasETröltzschFSecond order optimality conditions and their role in PDE controlJahresber. Dtsch. Math. Ver.20151171344331194810.1365/s13291-014-0109-31311.49002 DmitrukAVQuadratic conditions for the Pontryagin minimum in an optimal control problem linear with respect to control. II. Theorems on the relaxing of constraints on the equalityIzv. Akad. Nauk SSSR Ser. Mat.1987514812832911 KelleyHJA second variation test for singular extremalsAIAA J.196421380138216601810.2514/3.25620136.10101 LionsJ-LMagenesEProblèmes aux limites non homogènes et applications1968ParisDunod0165.10801 FattoriniHOFrankowskaHNecessary conditions for infinite-dimensional control problemsMath. Control Signals Syst.1991414167108285510.1007/BF025513800737.49017 AronnaMSBonnansJFGohBSSecond order analysis of control-affine problems with scalar state constraintMath. Progr.20161601–2115147355538410.1007/s10107-015-0976-01352.49020 Li, X., Yao, Y.: Maximum principle of distributed parameter systems aith time lages, Die Grundlehren der mathematischen Wissenschaften, vol. 181. Springer, New York-Heidelberg, Lecture Notes in Control and Information Science (1985) GoldbergHTröltzschFSecond-order sufficient optimality conditions for a class of nonlinear parabolic boundary control problemsSIAM J. Control Optim.199331410071025122754410.1137/03310450787.49011 FattoriniHOInfinite Dimensional Linear Control Systems, North-Holland Mathematics Studies2005AmsterdamElsevier Tröltzsch, F.: Regular Lagrange multipliers for control problems with mixed pointwise control-state constraints. SIAM J. Optim. 15(2), 616–634 (electronic) (2004/2005) Aronna, M.S., Bonnans, J.F., Kröner, A.: Optimal control of PDEs in a complex space setting; application to the Schrödinger equation. Research report, INRIA (2016) BS Goh (1093_CR18) 1966; 4 MS Aronna (1093_CR3) 2016; 160 J-P Aubin (1093_CR4) 1963; 256 1093_CR22 MR Hestenes (1093_CR20) 1951; 1 L Poggiolini (1093_CR27) 2008; 37 1093_CR28 1093_CR25 MS Aronna (1093_CR2) 2012; 2 E Casas (1093_CR11) 2015; 53 1093_CR1 A Pazy (1093_CR26) 1983 J-L Lions (1093_CR23) 1968 AV Dmitruk (1093_CR13) 1977; 18 H Goldberg (1093_CR19) 1993; 31 A Bensoussan (1093_CR6) 2007 JM Ball (1093_CR5) 1977; 63 E Casas (1093_CR12) 2015; 117 E Casas (1093_CR10) 2012; 50 1093_CR17 JF Bonnans (1093_CR8) 2014; 29 HO Fattorini (1093_CR15) 2005 M Bergounioux (1093_CR7) 1996; 34 AV Dmitruk (1093_CR14) 1987; 51 HO Fattorini (1093_CR16) 1991; 4 X Li (1093_CR24) 1991; 29 JF Bonnans (1093_CR9) 2009; 11(61) HJ Kelley (1093_CR21) 1964; 2
References_xml	– reference: BallJMStrongly continuous semigroups, weak solutions, and the variation of constants formulaProc. Am. Math. Soc.19776323703734427480353.47017 – reference: DmitrukAVQuadratic conditions for the Pontryagin minimum in an optimal control problem linear with respect to control. II. Theorems on the relaxing of constraints on the equalityIzv. Akad. Nauk SSSR Ser. Mat.1987514812832911 – reference: Milyutin, A.A., Osmolovskiim, N.P.: Calculus of variations and optimal control, Translations of Mathematical Monographs, vol. 180, American Mathematical Society, Providence, RI (1998), Translated from the Russian manuscript by Dimitrii Chibisov. MR1641590 – reference: HestenesMRApplications of the theory of quadratic forms in Hilbert space to the calculus of variationsPac. J. Math.195115255814659010.2140/pjm.1951.1.5250045.20806 – reference: BonnansJFOptimal control of a semilinear parabolic equation with singular arcsOptim. Methods Softw.2014295964978322501410.1080/10556788.2013.8302201301.49053 – reference: AronnaMSBonnansJFGohBSSecond order analysis of control-affine problems with scalar state constraintMath. Progr.20161601–2115147355538410.1007/s10107-015-0976-01352.49020 – reference: LionsJ-LMagenesEProblèmes aux limites non homogènes et applications1968ParisDunod0165.10801 – reference: PoggioliniLStefaniGSufficient optimality conditions for a bang-singular extremal in the minimum time problemControl Cybern.200837246949024728861235.49054 – reference: AubinJ-PUn théorème de compacitéC. R. Acad. Sci. Paris1963256504250441528600195.13002 – reference: FattoriniHOInfinite Dimensional Linear Control Systems, North-Holland Mathematics Studies2005AmsterdamElsevier – reference: CasasETröltzschFSecond order optimality conditions and their role in PDE controlJahresber. Dtsch. Math. Ver.20151171344331194810.1365/s13291-014-0109-31311.49002 – reference: DmitrukAVQuadratic conditions for a weak minimum for singular regimes in optimal control problemsSoviet Math. Dokl.19771824184220385.49008 – reference: AronnaMSBonnansJFDmitrukAVLotitoPAQuadratic order conditions for bang-singular extremalsNumer. Algebra Control Optim.201223511546297090110.3934/naco.2012.2.5111252.49028(AIMS Journal, special issue dedicated to Professor Helmut Maurer on the occasion of his 65th birthday) – reference: KelleyHJA second variation test for singular extremalsAIAA J.196421380138216601810.2514/3.25620136.10101 – reference: BergouniouxMTibaDGeneral optimality conditions for constrained convex control problemsSIAM J. Control Optim.1996342698711137771910.1137/S03630129942619870859.49026 – reference: GoldbergHTröltzschFSecond-order sufficient optimality conditions for a class of nonlinear parabolic boundary control problemsSIAM J. Control Optim.199331410071025122754410.1137/03310450787.49011 – reference: CasasESecond order analysis for bang-bang control problems of PDEsSIAM J. Control Optim.201250423552372297474210.1137/1208628921257.49021 – reference: GohBSThe second variation for the singular Bolza problemJ. SIAM Control19664230932520354510.1137/03040260146.11906 – reference: BonnansJFTibaDControl problems with mixed constraints and application to an optimal investment problemMath. Rep. (Bucur.)200911(61)429330626561661212.49037 – reference: BensoussanADa PratoGDelfourMCMitterSKRepresentation and Control of Infinite Dimensional Systems: Systems and Control: Foundations and Applications2007BostonBirkhäuser Boston, Inc.1117.93002 – reference: FattoriniHOFrankowskaHNecessary conditions for infinite-dimensional control problemsMath. Control Signals Syst.1991414167108285510.1007/BF025513800737.49017 – reference: Tröltzsch, F.: Regular Lagrange multipliers for control problems with mixed pointwise control-state constraints. SIAM J. Optim. 15(2), 616–634 (electronic) (2004/2005) – reference: CasasERyllCTröltzschFSecond order and stability analysis for optimal sparse control of the FitzHugh–Nagumo equationSIAM J. Control Optim.201553421682202337463610.1137/1409788551326.49032 – reference: Aronna, M.S., Bonnans, J.F., Kröner, A.: Optimal control of PDEs in a complex space setting; application to the Schrödinger equation. Research report, INRIA (2016) – reference: Li, X., Yao, Y.: Maximum principle of distributed parameter systems aith time lages, Die Grundlehren der mathematischen Wissenschaften, vol. 181. Springer, New York-Heidelberg, Lecture Notes in Control and Information Science (1985) – reference: LiXYongJNecessary conditions for optimal control of distributed parameter systemsSIAM J. Control Optim.1991294895908111166610.1137/03290490733.49025 – reference: PazyASemigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences1983New YorkSpringer10.1007/978-1-4612-5561-10516.47023 – reference: Frankowska, H., Tonon, D.: The Goh necessary optimality conditions for the Mayer problem with control constraints. In: Decision and Control, pp. 538–543 (2013) – volume: 29 start-page: 964 issue: 5 year: 2014 ident: 1093_CR8 publication-title: Optim. Methods Softw. doi: 10.1080/10556788.2013.830220 – volume: 160 start-page: 115 issue: 1–2 year: 2016 ident: 1093_CR3 publication-title: Math. Progr. doi: 10.1007/s10107-015-0976-0 – volume-title: Infinite Dimensional Linear Control Systems, North-Holland Mathematics Studies year: 2005 ident: 1093_CR15 – ident: 1093_CR25 – volume: 256 start-page: 5042 year: 1963 ident: 1093_CR4 publication-title: C. R. Acad. Sci. Paris – ident: 1093_CR28 doi: 10.1137/S1052623403426519 – volume-title: Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences year: 1983 ident: 1093_CR26 doi: 10.1007/978-1-4612-5561-1 – volume: 18 start-page: 418 issue: 2 year: 1977 ident: 1093_CR13 publication-title: Soviet Math. Dokl. – volume: 29 start-page: 895 issue: 4 year: 1991 ident: 1093_CR24 publication-title: SIAM J. Control Optim. doi: 10.1137/0329049 – ident: 1093_CR22 – volume: 4 start-page: 309 issue: 2 year: 1966 ident: 1093_CR18 publication-title: J. SIAM Control doi: 10.1137/0304026 – ident: 1093_CR1 – volume-title: Problèmes aux limites non homogènes et applications year: 1968 ident: 1093_CR23 – volume: 51 start-page: 812 issue: 4 year: 1987 ident: 1093_CR14 publication-title: Izv. Akad. Nauk SSSR Ser. Mat. – volume-title: Representation and Control of Infinite Dimensional Systems: Systems and Control: Foundations and Applications year: 2007 ident: 1093_CR6 – volume: 2 start-page: 511 issue: 3 year: 2012 ident: 1093_CR2 publication-title: Numer. Algebra Control Optim. doi: 10.3934/naco.2012.2.511 – volume: 63 start-page: 370 issue: 2 year: 1977 ident: 1093_CR5 publication-title: Proc. Am. Math. Soc. – volume: 1 start-page: 525 year: 1951 ident: 1093_CR20 publication-title: Pac. J. Math. doi: 10.2140/pjm.1951.1.525 – volume: 34 start-page: 698 issue: 2 year: 1996 ident: 1093_CR7 publication-title: SIAM J. Control Optim. doi: 10.1137/S0363012994261987 – ident: 1093_CR17 doi: 10.1109/CDC.2013.6759937 – volume: 11(61) start-page: 293 issue: 4 year: 2009 ident: 1093_CR9 publication-title: Math. Rep. (Bucur.) – volume: 53 start-page: 2168 issue: 4 year: 2015 ident: 1093_CR11 publication-title: SIAM J. Control Optim. doi: 10.1137/140978855 – volume: 117 start-page: 3 issue: 1 year: 2015 ident: 1093_CR12 publication-title: Jahresber. Dtsch. Math. Ver. doi: 10.1365/s13291-014-0109-3 – volume: 50 start-page: 2355 issue: 4 year: 2012 ident: 1093_CR10 publication-title: SIAM J. Control Optim. doi: 10.1137/120862892 – volume: 4 start-page: 41 issue: 1 year: 1991 ident: 1093_CR16 publication-title: Math. Control Signals Syst. doi: 10.1007/BF02551380 – volume: 2 start-page: 1380 year: 1964 ident: 1093_CR21 publication-title: AIAA J. doi: 10.2514/3.2562 – volume: 37 start-page: 469 issue: 2 year: 2008 ident: 1093_CR27 publication-title: Control Cybern. – volume: 31 start-page: 1007 issue: 4 year: 1993 ident: 1093_CR19 publication-title: SIAM J. Control Optim. doi: 10.1137/0331045
SSID	ssj0001388
Score	2.3188426
Snippet	In this paper we consider second order optimality conditions for a bilinear optimal control problem governed by a strongly continuous semigroup operator, the...
SourceID	hal proquest crossref springer
SourceType	Open Access Repository Aggregation Database Enrichment Source Index Database Publisher
StartPage	717
SubjectTerms	Calculus of Variations and Optimal Control; Optimization Combinatorics Full Length Paper Mathematical analysis Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Mathematics of Computing Nonlinear programming Numerical Analysis Operators (mathematics) Optimal control Optimization Optimization and Control Theoretical Wave equations
Title	Optimal control of infinite dimensional bilinear systems: application to the heat and wave equations
URI	https://link.springer.com/article/10.1007/s10107-016-1093-4 https://www.proquest.com/docview/2007795867 https://inria.hal.science/hal-01273496
Volume	168
WOSCitedRecordID	wos000426071000029&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D
hasFullText	1
inHoldings	1
isFullTextHit
isPrint
journalDatabaseRights	– providerCode: PRVAVX databaseName: Springer LINK customDbUrl: eissn: 1436-4646 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0001388 issn: 0025-5610 databaseCode: RSV dateStart: 19970101 isFulltext: true titleUrlDefault: https://link.springer.com/search?facet-content-type=%22Journal%22 providerName: Springer Nature
link	http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV1JS8UwEB7cDnpwF58bQTwpgbRN09SbiOLBDTe8lTQLCtqn9vn8-2a6PJ-igh7TJmnJZCYfzOT7ALZi7b0vFCENROgoz4OIpooramUcG8kcM85UYhPJ6am8vU3Pm3vcZVvt3qYkq0g9dNktqMokBUUKJMpHYdyfdhL1Gi4ubwbhN4ikbHVaERy0qczvpvh0GI3eYSnkEM78khqtTpzDmX_96yxMNwCT7NU7Yg5GbDEPU0O0g751MuBqLRfAnPmo8eiHNFXrpOuI33b3CEaJQfL_mriDYBlt4R2D1OzP5S4Zyn6TXpf4SQnGdqIKQ95U3xL7XDOJl4twfXhwtX9EG-0FqqOY9WjOmZLaBkIrlaIOOddJIrgJXZBqD2O006FveLzCDOM5j3TEE8s0M1pJ6Wy0BGNFt7DLQISzqXWhReJ4nqswVUJboVzuwaj2Tt8B1hoh0w0xOepjPGQflMq4nBkWo-FyZrwD24MhTzUrx2-dN71lB_2QT_to7zjDZ5h3R8b8ftiBtdbwWePFJUp0JkkaS5F0YKc19MfrH7-48qfeqzDpUZisC9vWYKz38mrXYUL3e_fly0a1ud8BBCfyqQ
linkProvider	Springer Nature
linkToHtml	http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV1ZT9wwEB5xVAIeoOUQy9FaVZ-oLDmJ4zi8IQRa1GVbtRTxFjk-xEqQBbIsfx9PjmWLoBI8OrGdyOMZf9J8_gbgW6y994UipIEIHeV5ENFUcUWtjGMjmWPGmarYRNLvy4uL9Fdzj7ts2e5tSrKK1FOX3YKKJikoSiBRPgvz3B9YKJj_-8_5JPwGkZRtnVYEB20q86Up_jmMZi-RCjmFM5-lRqsT53jlXf_6EZYbgEkO6h3xCWZssQpLU7KDvnU60Wot18D89FHj2g9pWOtk6IjfdgMEo8Sg-H8t3EGQRlt4xyC1-nO5T6ay32Q0JH5SgrGdqMKQBzW2xN7WSuLlOvw9Pjo77NKm9gLVUcxGNOdMSW0DoZVKsQ4510kiuAldkGoPY7TToW94vMIM4zmPdMQTyzQzWknpbLQBc8WwsJtAhLOpdaFF4XieqzBVQluhXO7BqPZO3wHWGiHTjTA51se4yp4klXE5MySj4XJmvAN7kyE3tSrH_zp_9Zad9EM97e5BL8NnmHdHxfxx2IGd1vBZ48UlluhMkjSWIunA99bQT69f_eLWm3p_gYXu2Wkv6530f2zDokdksia57cDc6O7e7sIHPR4NyrvP1UZ_BGtL9Y0
linkToPdf	http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwpV3LTtwwFL0qFCG6oOUlplBqIVYga5zEcZzuUNsRCBiQeIhd5PghkCADJJ3-fn0nyTAgqIRYJrGdyL62T3SPzwHYirWffaEIaSBCR3keRDRVXFEr49hI5phxZmQ2kfT78vIyPWl8TsuW7d6mJOszDajSVFTdO-O6EwffghFlUlCUQ6J8Cj5y5NHj7_rpxXgpDiIpW89WBAptWvOlJp5sTFNXSIucwJzP0qSj3af3-d3f_QXmG-BJdutIWYAPtliETxNyhP7qaKzhWi6BOfarya2v0rDZycARH47XCFKJQVOAWtCDIL228BOG1KrQ5Q8ykRUn1YD4Rgmu-UQVhvxVQ0vsfa0wXi7Dee_32c892ngyUB3FrKI5Z0pqGwitVIr-5FwnieAmdEGqPbzRTof-wuMYZhjPeaQjnlimmdFKSmejFZguBoVdBSKcTa0LLQrK81yFqRLaCuVyD1K1Xww6wNoByXQjWI6-GTfZo9QydmeGJDXszox3YHtc5a5W6_hf4U0_yuNyqLO9t3uY4T3Mx6OS_jDswHobBFkzu0u07kySNJYi6cBOO-iPj19949c3lf4Osye_etnhfv9gDeY8UJM1920dpquHP_YbzOhhdV0-bIxi_h8LU_5x
openUrl	ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Optimal+Control+of+Infinite+Dimensional+Bilinear+Systems%3A+Application+to+the+Heat+and+Wave+Equations&rft.jtitle=Mathematical+programming&rft.au=Aronna%2C+Maria+Soledad&rft.au=Bonnans%2C+Joseph+Fr%C3%A9deric&rft.au=Kr%C3%B6ner%2C+Axel&rft.date=2018-03-01&rft.pub=Springer+Verlag&rft.issn=0025-5610&rft.eissn=1436-4646&rft.volume=168&rft.issue=1-2&rft.spage=717&rft.epage=757&rft_id=info:doi/10.1007%2Fs10107-016-1093-4&rft.externalDBID=HAS_PDF_LINK&rft.externalDocID=oai%3AHAL%3Ahal-01273496v2
thumbnail_l	http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0025-5610&client=summon
thumbnail_m	http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0025-5610&client=summon
thumbnail_s	http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0025-5610&client=summon