Towards the theory of strong minimum in calculus of variations and optimal control: a view from variational analysis
The paper offers a self-contained account of the theory of first and second order necessary conditions for optimal control problems (with state constraints) based on new principles coming from variational analysis. The key element of the theory is reduction of the problem to unconstrained minimizati...
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| Veröffentlicht in: | Calculus of variations and partial differential equations Jg. 59; H. 2 |
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| Sprache: | Englisch |
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01.04.2020
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| Abstract | The paper offers a self-contained account of the theory of first and second order necessary conditions for optimal control problems (with state constraints) based on new principles coming from variational analysis. The key element of the theory is reduction of the problem to unconstrained minimization of a Bolza-type functional with necessarily non-differentiable integrand and off-integral term. This allows to substantially shorten and simplify the proofs and to get new results not detected earlier by traditional variational techniques. This includes a totally new and easily verifiable second order necessary condition for a strong minimum in the classical problem of calculus of variations. The condition is a consequence of a new and more general second order necessary condition for optimal control problems with state constraints. Simple examples show that the new conditions may work when all known necessary conditions fail. |
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| AbstractList | The paper offers a self-contained account of the theory of first and second order necessary conditions for optimal control problems (with state constraints) based on new principles coming from variational analysis. The key element of the theory is reduction of the problem to unconstrained minimization of a Bolza-type functional with necessarily non-differentiable integrand and off-integral term. This allows to substantially shorten and simplify the proofs and to get new results not detected earlier by traditional variational techniques. This includes a totally new and easily verifiable second order necessary condition for a strong minimum in the classical problem of calculus of variations. The condition is a consequence of a new and more general second order necessary condition for optimal control problems with state constraints. Simple examples show that the new conditions may work when all known necessary conditions fail. |
| ArticleNumber | 83 |
| Author | Ioffe, A. D. |
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| Cites_doi | 10.1007/s10957-019-01485-z 10.1137/130917417 10.1090/crmp/002/04 10.1137/17M1160604 10.1137/0317019 10.1016/j.jde.2017.02.013 10.1090/mmono/180 10.1007/978-3-319-64277-2 10.1007/BFb0087685 10.1007/s10013-020-00397-0 10.1070/SM8721 10.1137/0314067 10.1007/s10958-012-0824-1 10.1287/moor.9.2.159 10.1090/S0002-9947-97-01795-9 10.1016/j.na.2006.08.046 10.1007/s10957-020-01647-4 |
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| References | FrankowskaHOsmolovskiiNPStrong local minimizers in optimal control. Problems with state constraints: second order necessary conditionsSIAM J. Control Optim.2018562353237638177551393.49016 IoffeADOn generalized Bolza problem and its application to dynamic optimizationJ. Optim. Theory Appl.201918228530939613601420.49027 Ioffe, A.D.: Elementary proof of the Pontryagin maximum principle. Vietnam J. Math. https://doi.org/10.1007/s10013-020-00397-0 Buttazzo, G.: Semicontinuity, relaxation and integral representation in the calculus of variations. Pitman Resarch Notes in Mathematics, vol. 207. Pitman (1989) IoffeADNecessary and sufficient conditions for a local minimum 1–3SIAM J. Control Optim.1979172452885250250417.49027 VinterRBOptimal Control2000BaselBirkhauser0952.49001 MordukhovichBSVariational Analysis and Generalized Differentiation2006BerlinSpringer ClarkeFHThe maximum principle under minimal hypothesesSIAM J. Contol Optim.197614107810914154530344.49009 IoffeADVariational Analysis of Regular Mappings2017BerlinSpringer1381.49001 OsmolovskiiNPNecessary second-order conditions for a strong local minimum in a problem with endpoint and control constraintsJ. Optim. Theory Appl.20201851164081239 PontryaginLSBoltyanskiiVGGamkrelidzeRVMishchenkoEFThe Mathemetical Theory of Optimal Processes, Fizmatgiz 19611964OxfordPergamon Press(in Russian) MilyutinAAOsmolovskiiNPCalculus of Variations and Optimal Control1998ProvidenceAMS IoffeADEuler-Lagrange and Hamiltonian formalisms in dynamic optimizationTrans. Am. Math. Soc.19973492871290013897790876.49024 GamkrelidzeRVOn some extremal problems in the theory of differential equations with applications to the theory of optimal controlSIAM J. Control196531061281929370296.49009 IoffeADNecessary conditions in nonsmooth optimizationMath. Oper. Res.198491591897422540548.90088 PalesZZeidanVFirst and second order optimality conditions in optimal control with pure state constraintsNonlinear Anal. TMA200767250625261130.49018 Dubovitzkii, A.Y., Milyutin, A.A.: Problems for extremum under constraints. Zh. Vychisl. Matematiki i Mat. Fiziki 5(3), 395–453 (1965) (in Russian; English translation, USSR Comput. Math. Math. Physics, 5 (1965)) Castaing, C., Valadier, M.: Convex analysis and measurable multifunctions. Lecture Notes in Mathematics, vol. 580. Springer (1977) ClarkeFHOptimization and Nonsmooth Analysis1983HobokenWiley0582.49001 Loewen, P.D.: Optimal control via nonsmooth analysis. CRM Proceedings and Lecture Notes, vol. 2. AMS (1993) Avakov, E.P., Magaril-Il’yaev, G.G.: Controllability and necessary optimality conditions of second order in optimal control. Matem. Sbornik 208, 3–37 (2017) (in Russian; English translation, Sb. Matem. 208 (2017), 585–619) OsmolovskiiNPNecessary quadratic conditions of extremum for discontinuous control in optimal control problems with mixed constraintsJ. Math. Sci.201218343557731739641263.49001 VinterRBThe Hamiltonian inclusion for nonconvex velocity setsSIAM J. Control Optim.2014521237125031915871304.49045 FrankowskaHHoehenerDPointwise second order necessary optimality conditions and second order sensitivity relations in optimal controlJ. Differ. Equ.20172625735577236245371373.49023 1736_CR2 1736_CR3 AD Ioffe (1736_CR11) 1984; 9 BS Mordukhovich (1736_CR18) 2006 RB Vinter (1736_CR23) 2000 AD Ioffe (1736_CR13) 2017 RB Vinter (1736_CR24) 2014; 52 AD Ioffe (1736_CR14) 2019; 182 cr-split#-1736_CR6.1 cr-split#-1736_CR6.2 AD Ioffe (1736_CR12) 1997; 349 1736_CR16 1736_CR15 AA Milyutin (1736_CR17) 1998 cr-split#-1736_CR1.2 cr-split#-1736_CR1.1 H Frankowska (1736_CR8) 2018; 56 LS Pontryagin (1736_CR22) 1964 NP Osmolovskii (1736_CR19) 2012; 183 FH Clarke (1736_CR4) 1976; 14 FH Clarke (1736_CR5) 1983 AD Ioffe (1736_CR10) 1979; 17 RV Gamkrelidze (1736_CR9) 1965; 3 H Frankowska (1736_CR7) 2017; 262 Z Pales (1736_CR21) 2007; 67 NP Osmolovskii (1736_CR20) 2020; 185 |
| References_xml | – reference: FrankowskaHOsmolovskiiNPStrong local minimizers in optimal control. Problems with state constraints: second order necessary conditionsSIAM J. Control Optim.2018562353237638177551393.49016 – reference: Loewen, P.D.: Optimal control via nonsmooth analysis. CRM Proceedings and Lecture Notes, vol. 2. AMS (1993) – reference: IoffeADNecessary and sufficient conditions for a local minimum 1–3SIAM J. Control Optim.1979172452885250250417.49027 – reference: GamkrelidzeRVOn some extremal problems in the theory of differential equations with applications to the theory of optimal controlSIAM J. Control196531061281929370296.49009 – reference: OsmolovskiiNPNecessary second-order conditions for a strong local minimum in a problem with endpoint and control constraintsJ. Optim. Theory Appl.20201851164081239 – reference: Castaing, C., Valadier, M.: Convex analysis and measurable multifunctions. Lecture Notes in Mathematics, vol. 580. Springer (1977) – reference: VinterRBOptimal Control2000BaselBirkhauser0952.49001 – reference: VinterRBThe Hamiltonian inclusion for nonconvex velocity setsSIAM J. Control Optim.2014521237125031915871304.49045 – reference: Avakov, E.P., Magaril-Il’yaev, G.G.: Controllability and necessary optimality conditions of second order in optimal control. Matem. Sbornik 208, 3–37 (2017) (in Russian; English translation, Sb. Matem. 208 (2017), 585–619) – reference: PalesZZeidanVFirst and second order optimality conditions in optimal control with pure state constraintsNonlinear Anal. TMA200767250625261130.49018 – reference: ClarkeFHOptimization and Nonsmooth Analysis1983HobokenWiley0582.49001 – reference: Dubovitzkii, A.Y., Milyutin, A.A.: Problems for extremum under constraints. Zh. Vychisl. Matematiki i Mat. Fiziki 5(3), 395–453 (1965) (in Russian; English translation, USSR Comput. Math. Math. Physics, 5 (1965)) – reference: IoffeADVariational Analysis of Regular Mappings2017BerlinSpringer1381.49001 – reference: PontryaginLSBoltyanskiiVGGamkrelidzeRVMishchenkoEFThe Mathemetical Theory of Optimal Processes, Fizmatgiz 19611964OxfordPergamon Press(in Russian) – reference: IoffeADEuler-Lagrange and Hamiltonian formalisms in dynamic optimizationTrans. Am. Math. Soc.19973492871290013897790876.49024 – reference: Buttazzo, G.: Semicontinuity, relaxation and integral representation in the calculus of variations. Pitman Resarch Notes in Mathematics, vol. 207. Pitman (1989) – reference: IoffeADNecessary conditions in nonsmooth optimizationMath. Oper. Res.198491591897422540548.90088 – reference: MilyutinAAOsmolovskiiNPCalculus of Variations and Optimal Control1998ProvidenceAMS – reference: OsmolovskiiNPNecessary quadratic conditions of extremum for discontinuous control in optimal control problems with mixed constraintsJ. Math. Sci.201218343557731739641263.49001 – reference: FrankowskaHHoehenerDPointwise second order necessary optimality conditions and second order sensitivity relations in optimal controlJ. Differ. Equ.20172625735577236245371373.49023 – reference: MordukhovichBSVariational Analysis and Generalized Differentiation2006BerlinSpringer – reference: IoffeADOn generalized Bolza problem and its application to dynamic optimizationJ. Optim. Theory Appl.201918228530939613601420.49027 – reference: ClarkeFHThe maximum principle under minimal hypothesesSIAM J. Contol Optim.197614107810914154530344.49009 – reference: Ioffe, A.D.: Elementary proof of the Pontryagin maximum principle. Vietnam J. Math. https://doi.org/10.1007/s10013-020-00397-0 – volume-title: Optimal Control year: 2000 ident: 1736_CR23 – volume: 182 start-page: 285 year: 2019 ident: 1736_CR14 publication-title: J. Optim. Theory Appl. doi: 10.1007/s10957-019-01485-z – volume: 52 start-page: 1237 year: 2014 ident: 1736_CR24 publication-title: SIAM J. Control Optim. doi: 10.1137/130917417 – ident: 1736_CR16 doi: 10.1090/crmp/002/04 – volume: 56 start-page: 2353 year: 2018 ident: 1736_CR8 publication-title: SIAM J. Control Optim. doi: 10.1137/17M1160604 – volume: 17 start-page: 245 year: 1979 ident: 1736_CR10 publication-title: SIAM J. Control Optim. doi: 10.1137/0317019 – volume: 262 start-page: 5735 year: 2017 ident: 1736_CR7 publication-title: J. Differ. Equ. doi: 10.1016/j.jde.2017.02.013 – volume-title: Calculus of Variations and Optimal Control year: 1998 ident: 1736_CR17 doi: 10.1090/mmono/180 – ident: #cr-split#-1736_CR6.1 – volume-title: Variational Analysis of Regular Mappings year: 2017 ident: 1736_CR13 doi: 10.1007/978-3-319-64277-2 – volume-title: Optimization and Nonsmooth Analysis year: 1983 ident: 1736_CR5 – volume-title: The Mathemetical Theory of Optimal Processes, Fizmatgiz 1961 year: 1964 ident: 1736_CR22 – ident: 1736_CR3 doi: 10.1007/BFb0087685 – volume: 3 start-page: 106 year: 1965 ident: 1736_CR9 publication-title: SIAM J. Control – ident: 1736_CR15 doi: 10.1007/s10013-020-00397-0 – ident: #cr-split#-1736_CR1.2 doi: 10.1070/SM8721 – volume: 14 start-page: 1078 year: 1976 ident: 1736_CR4 publication-title: SIAM J. Contol Optim. doi: 10.1137/0314067 – volume: 183 start-page: 435 year: 2012 ident: 1736_CR19 publication-title: J. Math. Sci. doi: 10.1007/s10958-012-0824-1 – volume-title: Variational Analysis and Generalized Differentiation year: 2006 ident: 1736_CR18 – volume: 9 start-page: 159 year: 1984 ident: 1736_CR11 publication-title: Math. Oper. Res. doi: 10.1287/moor.9.2.159 – volume: 349 start-page: 2871 year: 1997 ident: 1736_CR12 publication-title: Trans. Am. Math. Soc. doi: 10.1090/S0002-9947-97-01795-9 – volume: 67 start-page: 2506 year: 2007 ident: 1736_CR21 publication-title: Nonlinear Anal. TMA doi: 10.1016/j.na.2006.08.046 – ident: 1736_CR2 – ident: #cr-split#-1736_CR1.1 – volume: 185 start-page: 1 year: 2020 ident: 1736_CR20 publication-title: J. Optim. Theory Appl. doi: 10.1007/s10957-020-01647-4 – ident: #cr-split#-1736_CR6.2 |
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| SubjectTerms | Analysis Calculus of variations Calculus of Variations and Optimal Control; Optimization Control Mathematical analysis Mathematical and Computational Physics Mathematics Mathematics and Statistics Optimal control Optimization Systems Theory Theoretical |
| Title | Towards the theory of strong minimum in calculus of variations and optimal control: a view from variational analysis |
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