State-Constrained Control-Affine Parabolic Problems I: First and Second Order Necessary Optimality Conditions
In this paper we consider an optimal control problem governed by a semilinear heat equation with bilinear control-state terms and subject to control and state constraints. The state constraints are of integral type, the integral being with respect to the space variable. The control is multidimension...
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| Published in | Set-valued and variational analysis Vol. 29; no. 2; pp. 383 - 408 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
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Springer Netherlands
01.06.2021
Springer Nature B.V |
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| ISSN | 1877-0533 1877-0541 |
| DOI | 10.1007/s11228-020-00560-2 |
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| Abstract | In this paper we consider an optimal control problem governed by a semilinear heat equation with bilinear control-state terms and subject to control and state constraints. The state constraints are of integral type, the integral being with respect to the space variable. The control is multidimensional. The cost functional is of a tracking type and contains a linear term in the control variables. We derive second order necessary conditions relying on the concept of alternative costates and quasi-radial critical directions. The appendix provides an example illustrating the applicability of our results. |
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| AbstractList | In this paper we consider an optimal control problem governed by a semilinear heat equation with bilinear control-state terms and subject to control and state constraints. The state constraints are of integral type, the integral being with respect to the space variable. The control is multidimensional. The cost functional is of a tracking type and contains a linear term in the control variables. We derive second order necessary conditions relying on the concept of alternative costates and quasi-radial critical directions. The appendix provides an example illustrating the applicability of our results. |
| Author | Kröner, Axel Bonnans, J. Frédéric Aronna, M. Soledad |
| Author_xml | – sequence: 1 givenname: M. Soledad orcidid: 0000-0002-0640-4722 surname: Aronna fullname: Aronna, M. Soledad email: soledad.aronna@fgv.br organization: Escola de Matemática Aplicada FGV EMAp – sequence: 2 givenname: J. Frédéric surname: Bonnans fullname: Bonnans, J. Frédéric organization: Inria Saclay and CMAP, Ecole Polytechnique, CNRS, Université Paris Saclay – sequence: 3 givenname: Axel surname: Kröner fullname: Kröner, Axel organization: Weierstrass Institute for Applied Analysis and Stochastics |
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| Keywords | Optimal control of partial differential equations Second order analysis Control-affine problems Semilinear parabolic equations State constraints |
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Vol. 11968ParisDunod0165.10801 CasasEde Los ReyesJCTröltzschFSufficient second-order optimality conditions for semilinear control problems with pointwise state constraintsSIAM J. Optim.2008192616643242503210.1137/07068240X CasasETröltzschFRecent advances in the analysis of pointwise state-constrained elliptic optimal control problemsESAIM Control Optim. Calc. Var.2010163581600267462710.1051/cocv/20090101201.49004 LiebermanGMSecond order parabolic differential equations1996River Edge, NJWorld Scientific Publishing Co., Inc.10.1142/3302https://doi.org/10.1142/3302 de Los ReyesJCMerinoPRehbergJTröltzschFOptimality conditions for state-constrained PDE control problems with time-dependent controlsControl. Cybern.200837153824407221153.49006 BonnansJFSecond-order analysis for control constrained optimal control problems of semilinear elliptic systemsAppl. Math. Optim.1998383303325164670310.1007/s0024599000930917.49020 AubinJ-PUn théorème de compacitéC. R. Acad. Sci. Paris1963256504250441528600195.13002 BonnansJFJaissonPOptimal control of a parabolic equation with time-dependent state constraintsSIAM J. Control Optim.201048745504571268389810.1137/0807446081208.49024 CasasETröeltzschFUngerASecond order sufficient optimality conditions for a nonlinear elliptic control problemJ. for Analysis and its Applications (ZAA)19961568770714060830879.49020 DmitrukAVJacobi type conditions for singular extremalsControl & Cybernetics200837228530624728781235.49043 BonnansJFHermantASecond-order analysis for optimal control problems with pure state constraints and mixed control-state constraintsAnn. Inst. H. Poincaré Anal. Non Linéaire2009262561598250404410.1016/j.anihpc.2007.12.0021158.49023 BonnansJFShapiroAPerturbation analysis of optimization problems2000New YorkSpringer-Verlag10.1007/978-1-4612-1394-9 Aronna, M.S., Bonnans, J.F., Kröner, A: State-constrained control-affine parabolic problems II: Second-order sufficient optimality conditions. 1909.05056 (2019) AronnaMSBonnansJFGohBSSecond order analysis of control-affine problems with scalar state constraintMath. Program.20161601-2, Ser. A115147355538410.1007/s10107-015-0976-01352.49020 Bonnans, J., Giorgi, D., Grélard, V, Heymann, B., Maindrault, S., Martinon, P., Tissot, O., Liu, J.: Bocop – A collection of examples. INRIA. http://www.bocop.org (2017) KrumbiegelKRehbergJSecond order sufficient optimality conditions for parabolic optimal control problems with pointwise state constraintsSIAM J. Control Optim.2013511304331303287710.1137/1208716871272.35060 Lions, J.-L.: Contrôle des systèmes distribués singuliers. Méthodes Mathématiques de l’Informatique, vol. 13. Gauthier-Villars, Montrouge (1983) MaurerHOn optimal control problems with bounded state variables and control appearing linearlySIAM J. Control Optimization197715334536246400710.1137/0315023 KelleyHJA second variation test for singular extremalsAIAA J.196421380138216601810.2514/3.2562 RaymondJ-PTröltzschFSecond order sufficient optimality conditions for nonlinear parabolic control problems with state constraintsDiscrete Contin. Dynam. Systems200062431450173937510.3934/dcds.2000.6.431 Maurer, H.: On the minimum principle for optimal control problems with state constraints. Schriftenreihe des Rechenzentrum 41, Universität Münster (1979) AronnaMSBonnansJFDmitrukAVLotitoPAQuadratic order conditions for bang-singular extremalsNumerical Algebra, Control and Optimization, AIMS Journal201223511546297090110.3934/naco.2012.2.5111252.49028 SchättlerHLocal fields of extremals for optimal control problems with state constraints of relative degree 1J. Dyn. Control Syst.2006124563599225336110.1007/s10883-006-0005-y1123.49031 LionsJ-LQuelques méthodes de résolution des problèmes aux limites non linéaires1969ParisDunod0189.40603 GohBSNecessary conditions for singular extremals involving multiple control variablesSIAM J. Control1966471673120571910.1137/0304052 AV Dmitruk (560_CR4) 1977; 18 H Maurer (560_CR8) 2005 BS Goh (560_CR2) 1966; 4 JF Bonnans (560_CR12) 1998; 38 JC de Los Reyes (560_CR18) 2008; 37 J-L Lions (560_CR23) 1969 MS Aronna (560_CR10) 2016; 160 E Casas (560_CR14) 2010; 16 H Maurer (560_CR6) 1977; 15 E Casas (560_CR17) 2008; 19 560_CR13 K Krumbiegel (560_CR16) 2013; 51 J-P Raymond (560_CR15) 2000; 6 JF Bonnans (560_CR25) 2009; 26 LC Evans (560_CR27) 1998 H Schättler (560_CR9) 2006; 12 J-P Aubin (560_CR22) 1963; 256 E Casas (560_CR11) 1996; 15 AV Dmitruk (560_CR28) 2008; 37 JF Bonnans (560_CR20) 2000 GM Lieberman (560_CR29) 1996 HJ Kelley (560_CR3) 1964; 2 560_CR30 560_CR24 J-L Lions (560_CR21) 1968 560_CR1 560_CR26 JF Bonnans (560_CR19) 2010; 48 MS Aronna (560_CR5) 2012; 2 JP McDanell (560_CR7) 1971; 9 |
| References_xml | – reference: GohBSNecessary conditions for singular extremals involving multiple control variablesSIAM J. Control1966471673120571910.1137/0304052 – reference: LiebermanGMSecond order parabolic differential equations1996River Edge, NJWorld Scientific Publishing Co., Inc.10.1142/3302https://doi.org/10.1142/3302 – reference: EvansLCPartial differential equations1998Providence, RIAmer. Math Soc.0902.35002Graduate Studies in Mathematics 19 – reference: MaurerHOn optimal control problems with bounded state variables and control appearing linearlySIAM J. Control Optimization197715334536246400710.1137/0315023 – reference: Aronna, M.S., Bonnans, J.F., Kröner, A: State-constrained control-affine parabolic problems II: Second-order sufficient optimality conditions. 1909.05056 (2019) – reference: CasasETröeltzschFUngerASecond order sufficient optimality conditions for a nonlinear elliptic control problemJ. for Analysis and its Applications (ZAA)19961568770714060830879.49020 – reference: AubinJ-PUn théorème de compacitéC. R. Acad. Sci. Paris1963256504250441528600195.13002 – reference: AronnaMSBonnansJFGohBSSecond order analysis of control-affine problems with scalar state constraintMath. Program.20161601-2, Ser. A115147355538410.1007/s10107-015-0976-01352.49020 – reference: Casas, E., Mateos, M., Tröltzsch, F.: Necessary and sufficient optimality conditions for optimization problems in function spaces and applications to control theory. In: Proceedings of 2003 MODE-SMAI Conference, ESAIM Proceedings, vol 13, pp 18–30. EDP Sciences (2003) – reference: CasasETröltzschFRecent advances in the analysis of pointwise state-constrained elliptic optimal control problemsESAIM Control Optim. Calc. Var.2010163581600267462710.1051/cocv/20090101201.49004 – reference: Maurer, H.: On the minimum principle for optimal control problems with state constraints. Schriftenreihe des Rechenzentrum 41, Universität Münster (1979) – reference: DmitrukAVJacobi type conditions for singular extremalsControl & Cybernetics200837228530624728781235.49043 – reference: AronnaMSBonnansJFDmitrukAVLotitoPAQuadratic order conditions for bang-singular extremalsNumerical Algebra, Control and Optimization, AIMS Journal201223511546297090110.3934/naco.2012.2.5111252.49028 – reference: KrumbiegelKRehbergJSecond order sufficient optimality conditions for parabolic optimal control problems with pointwise state constraintsSIAM J. Control Optim.2013511304331303287710.1137/1208716871272.35060 – reference: Lions, J.-L.: Contrôle des systèmes distribués singuliers. Méthodes Mathématiques de l’Informatique, vol. 13. Gauthier-Villars, Montrouge (1983) – reference: BonnansJFJaissonPOptimal control of a parabolic equation with time-dependent state constraintsSIAM J. Control Optim.201048745504571268389810.1137/0807446081208.49024 – reference: LionsJ-LMagenesEProblèmes aux limites non homogènes et applications. Vol. 11968ParisDunod0165.10801 – reference: DmitrukAVQuadratic conditions for a weak minimum for singular regimes in optimal control problemsSoviet Math. Doklady19771824184220385.49008 – reference: SchättlerHLocal fields of extremals for optimal control problems with state constraints of relative degree 1J. Dyn. Control Syst.2006124563599225336110.1007/s10883-006-0005-y1123.49031 – reference: BonnansJFShapiroAPerturbation analysis of optimization problems2000New YorkSpringer-Verlag10.1007/978-1-4612-1394-9 – reference: LionsJ-LQuelques méthodes de résolution des problèmes aux limites non linéaires1969ParisDunod0189.40603 – reference: RaymondJ-PTröltzschFSecond order sufficient optimality conditions for nonlinear parabolic control problems with state constraintsDiscrete Contin. Dynam. Systems200062431450173937510.3934/dcds.2000.6.431 – reference: Bonnans, J., Giorgi, D., Grélard, V, Heymann, B., Maindrault, S., Martinon, P., Tissot, O., Liu, J.: Bocop – A collection of examples. INRIA. http://www.bocop.org (2017) – reference: KelleyHJA second variation test for singular extremalsAIAA J.196421380138216601810.2514/3.2562 – reference: BonnansJFSecond-order analysis for control constrained optimal control problems of semilinear elliptic systemsAppl. Math. Optim.1998383303325164670310.1007/s0024599000930917.49020 – reference: BonnansJFHermantASecond-order analysis for optimal control problems with pure state constraints and mixed control-state constraintsAnn. Inst. H. Poincaré Anal. Non Linéaire2009262561598250404410.1016/j.anihpc.2007.12.0021158.49023 – reference: de Los ReyesJCMerinoPRehbergJTröltzschFOptimality conditions for state-constrained PDE control problems with time-dependent controlsControl. Cybern.200837153824407221153.49006 – reference: McDanellJPPowersWFNecessary conditions for joining optimal singular and nonsingular subarcsSIAM J. Control1971916117329403610.1137/0309014 – reference: MaurerHKimJ-HRVossenGOn a state-constrained control problem in optimal production and maintenance2005Boston, MASpringer US2893081151.49009https://doi.org/10.1007/0-387-25805-1_17 – reference: CasasEde Los ReyesJCTröltzschFSufficient second-order optimality conditions for semilinear control problems with pointwise state constraintsSIAM J. Optim.2008192616643242503210.1137/07068240X – volume: 48 start-page: 4550 issue: 7 year: 2010 ident: 560_CR19 publication-title: SIAM J. Control Optim. doi: 10.1137/080744608 – ident: 560_CR24 – volume: 160 start-page: 115 issue: 1-2, Ser. A year: 2016 ident: 560_CR10 publication-title: Math. Program. doi: 10.1007/s10107-015-0976-0 – ident: 560_CR1 – volume: 19 start-page: 616 issue: 2 year: 2008 ident: 560_CR17 publication-title: SIAM J. Optim. doi: 10.1137/07068240X – ident: 560_CR26 – start-page: 289 volume-title: On a state-constrained control problem in optimal production and maintenance year: 2005 ident: 560_CR8 doi: 10.1007/0-387-25805-1_17 – volume: 51 start-page: 304 issue: 1 year: 2013 ident: 560_CR16 publication-title: SIAM J. Control Optim. doi: 10.1137/120871687 – volume-title: Problèmes aux limites non homogènes et applications. Vol. 1 year: 1968 ident: 560_CR21 – volume: 9 start-page: 161 year: 1971 ident: 560_CR7 publication-title: SIAM J. 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Systems doi: 10.3934/dcds.2000.6.431 – volume-title: Partial differential equations year: 1998 ident: 560_CR27 – volume: 16 start-page: 581 issue: 3 year: 2010 ident: 560_CR14 publication-title: ESAIM Control Optim. Calc. Var. doi: 10.1051/cocv/2009010 – volume: 2 start-page: 511 issue: 3 year: 2012 ident: 560_CR5 publication-title: Numerical Algebra, Control and Optimization, AIMS Journal doi: 10.3934/naco.2012.2.511 – volume: 256 start-page: 5042 year: 1963 ident: 560_CR22 publication-title: C. R. Acad. Sci. Paris – volume: 37 start-page: 5 issue: 1 year: 2008 ident: 560_CR18 publication-title: Control. Cybern. – volume: 15 start-page: 345 issue: 3 year: 1977 ident: 560_CR6 publication-title: SIAM J. Control Optimization doi: 10.1137/0315023 – volume: 12 start-page: 563 issue: 4 year: 2006 ident: 560_CR9 publication-title: J. Dyn. Control Syst. doi: 10.1007/s10883-006-0005-y – volume-title: Second order parabolic differential equations year: 1996 ident: 560_CR29 doi: 10.1142/3302 – volume: 37 start-page: 285 issue: 2 year: 2008 ident: 560_CR28 publication-title: Control & Cybernetics – volume: 15 start-page: 687 year: 1996 ident: 560_CR11 publication-title: J. for Analysis and its Applications (ZAA) – volume: 2 start-page: 1380 year: 1964 ident: 560_CR3 publication-title: AIAA J. doi: 10.2514/3.2562 |
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| Title | State-Constrained Control-Affine Parabolic Problems I: First and Second Order Necessary Optimality Conditions |
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