Second order necessary and sufficient optimality conditions for singular solutions of partially-affine control problems

Previous Article
Recursive variational problems in nonreflexive Banach spaces with an infinite horizon: An existence result
DCDS-S Home
This Issue
Next Article
A study of structure-exploiting SQP algorithms for an optimal control problem with coupled hyperbolic and ordinary differential equation constraints

December 2018, 11(6): 1233-1258. doi: 10.3934/dcdss.2018070

Second order necessary and sufficient optimality conditions for singular solutions of partially-affine control problems

M. Soledad Aronna ^,

Escola de Matemática Aplicada, Fundação Getulio Vargas, Praia de Botafogo 190, 22250-900 Rio de Janeiro - RJ, Brazil

* Corresponding author: M. Soledad Aronna

Received March 2017 Revised June 2017 Published June 2018

Fund Project: This work was supported by the European Union under the 7th Framework Programme FP7-PEOPLE-2010-ITN Grant agreement number 264735-SADCO

In this article we study optimal control problems for systems that are affine with respect to some of the control variables and nonlinear in relation to the others. We consider finitely many equality and inequality constraints on the initial and final values of the state. We investigate singular optimal solutions for this class of problems, for which we obtain second order necessary and sufficient conditions for weak optimality in integral form. We also derive Goh pointwise necessary optimality conditions. We show an example to illustrate the results.

Keywords: Optimal control, singular control, second order optimality condition, Goh condition, Legendre-Clebsch, shooting algorithm.

Mathematics Subject Classification: Primary: 49K15.

Citation: M. Soledad Aronna. Second order necessary and sufficient optimality conditions for singular solutions of partially-affine control problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1233-1258. doi: 10.3934/dcdss.2018070

References:

[1]	R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. Google Scholar
[2]	A. A. Agrachev and Y. L. Sachkov, Control Theory from the Geometric Viewpoint, Encyclopaedia of Mathematical Sciences, 87, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-662-06404-7. Google Scholar
[3]	V. M. Alekseev, V. M. Tikhomirov and S. V. Fomin, Optimal Control, Nauka, Moscow, 1979. [in Russian]. Google Scholar
[4]	M. S. Aronna, Convergence of the shooting algorithm for singular optimal control problems, in Proceedings of the IEEE European Control Conference (ECC), July 2013, 215–220.Google Scholar
[5]	M. S. Aronna, Singular Solutions in Optimal Control: Second Order Conditions and a Shooting Algorithm, Technical Report, Inria RR-7764, 2013. arXiv: 1210.7425, Inria RR-7764.Google Scholar
[6]	M. S. Aronna, J. F. Bonnans, A. V. Dmitruk and P. A. Lotito, Quadratic order conditions for bang-singular extremals, Numer. Algebra Control Optim., 2 (2012), 511-546. doi: 10.3934/naco.2012.2.511. Google Scholar
[7]	D. M. Azimov, Active sections of rocket trajectories. A survey of research, Avtomat. i Telemekh., 11 (2005), 14-34. doi: 10.1007/s10513-005-0207-x. Google Scholar
[8]	D. J. Bell and D. H. Jacobson, Singular Optimal Control Problems, Academic Press, 1975. Google Scholar
[9]	D. S. Bernstein and V. Zeidan, The singular linear-quadratic regulator problem and the Goh-Riccati equation, Proceedings of the IEEE Conference on Decision and Control, 1 (1990), 334-339. doi: 10.1109/CDC.1990.203608. Google Scholar
[10]	G. A. Bliss, Lectures on the Calculus of Variations, University of Chicago Press, Chicago, Illinois, 1946. Google Scholar
[11]	F. Bonnans, J. Laurent-Varin, P. Martinon and E. Trélat, Numerical study of optimal trajectories with singular arcs for an Ariane 5 launcher, J. Guidance Control Dynam., 32 (2009), 51-55. Google Scholar
[12]	J. F. Bonnans, Optimisation Continue, Dunod, 2006.Google Scholar
[13]	H. J. Bortolossi, M. V. Pereira and C. Tomei, Optimal hydrothermal scheduling with variable production coefficient, Math. Methods Oper. Res., 55 (2002), 11-36. doi: 10.1007/s001860200174. Google Scholar
[14]	H. Brézis, Analyse Fonctionnelle, Masson, Paris, 1983. Google Scholar
[15]	A. E. Bryson, Jr. and Y. C. Ho, Applied Optimal Control, Hemisphere Publishing Corp. Washington, D. C., 1975. Optimization, estimation, and control, Revised printing. Google Scholar
[16]	D. I. Cho, P. L. Abad and M. Parlar, Optimal production and maintenance decisions when a system experience age-dependent deterioration, Optimal Control Appl. Methods, 14 (1993), 153-167. doi: 10.1002/oca.4660140302. Google Scholar
[17]	A. V. Dmitruk, Quadratic conditions for a weak minimum fo control problems, Soviet Math. Doklady, 18 (1977). Google Scholar
[18]	A. V. Dmitruk, Jacobi-type conditions for the problem of Bolza with inequalities, Math. Notes, 35 (1984), 427-435. Google Scholar
[19]	A. V. Dmitruk, Quadratic order conditions for a Pontryagin minimum in an optimal control problem linear in the control, Math. USSR Izvestiya, 28 (1987), 275-303. Google Scholar
[20]	A. V. Dmitruk, Jacobi type conditions for singular extremals, Control & Cybernetics, 37 (2008), 285-306. Google Scholar
[21]	A. V. Dmitruk, Quadratic order optimality conditions for extremals completely singular in part of controls, in Operations Research Proceedings, Selected Papers of the Annual International Conference of the German Operations Research Society, (2011), 341–346. doi: 10.1007/978-3-642-20009-0_54. Google Scholar
[22]	A. V. Dmitruk and K. K. Shishov, Analysis of a quadratic functional with a partly singular Legendre condition, Moscow University Comput. Math. and Cybernetics, 34 (2010), 56-65. doi: 10.3103/S0278641910020020. Google Scholar
[23]	H. Frankowska and D. Tonon, Pointwise second-order necessary optimality conditions for the Mayer problem with control constraints, SIAM J. Control Optim., 51 (2013), 3814-3843. doi: 10.1137/130906799. Google Scholar
[24]	R. H. Goddard, A Method of Reaching Extreme Altitudes, Smithsonian Miscellaneous Collections, 71(2), Smithsonian Institution, City of Washington, 1919.Google Scholar
[25]	B. S. Goh, Necessary conditions for singular extremals involving multiple control variables, J. SIAM Control, 4 (1966), 716-731. doi: 10.1137/0304052. Google Scholar
[26]	B. S. Goh, Necessary Conditions for the Singular Extremals in the Calculus of Variations, PhD thesis, University of Canterbury, 1966.Google Scholar
[27]	B. S. Goh, The second variation for the singular Bolza problem, J. SIAM Control, 4 (1966), 309-325. doi: 10.1137/0304026. Google Scholar
[28]	B. S. Goh, Optimal singular control for multi-input linear systems, J. Math. Anal. Appl., 20 (1967), 534-539. doi: 10.1016/0022-247X(67)90079-0. Google Scholar
[29]	B. S. Goh, Optimal singular rocket and aircraft trajectories, in Control and Decision Conference, CCDC 2008, (2008), 1531–1536. doi: 10.1109/CCDC.2008.4597574. Google Scholar
[30]	M. R. Hestenes, Applications of the theory of quadratic forms in Hilbert space to the calculus of variations, Pacific J. Math., 1 (1951), 525-581. doi: 10.2140/pjm.1951.1.525. Google Scholar
[31]	D. G. Hull, Optimal guidance for quasi-planar lunar ascent, J. Optim. Theory Appl., 151 (2011), 353-372. doi: 10.1007/s10957-011-9884-5. Google Scholar
[32]	S. Kurcyusz and J. Zowe, Regularity and stability for the mathematical programming problem in Banach spaces, Applied Mathematics and Optimization, 5 (1979), 49-62. doi: 10.1007/BF01442543. Google Scholar
[33]	D. F. Lawden, Optimal Trajectories for Space Navigation, Butterworths, London, 1963. Google Scholar
[34]	E. S. Levitin, A. A. Milyutin and N. P. Osmolovskii, Theory of higher-order conditions in smooth constrained extremal problems, Theoretical and Applied Optimal Control Problems, (1985), 4-40. Google Scholar
[35]	O. Mangasarian and S. Fromovitz, The Fritz-John necessary optimality conditions in the presence of equality and inequality constraints, J. Math. Anal. Appl., 17 (1967), 37-47. doi: 10.1016/0022-247X(67)90163-1. Google Scholar
[36]	H. Maurer, J.-H. Kim and G. Vossen, On a state-constrained control problem in optimal production and maintenance, in Optimal Control and Dynamic Games (eds. C. Deissenberg, R. Hartl, H. M. Amman and B. Rustem), Advances in Computational Management Science, 7, Springer, 2005,289–308. doi: 10.1007/0-387-25805-1_17. Google Scholar
[37]	H. Maurer and N. P. Osmolovskii, Second order sufficient optimality conditions for a control problem with continuous and bang-bang control components: Riccati approach, in System Modeling and Optimization, IFIP Adv. Inf. Commun. Technol., 312, Springer, Berlin, 2009,411–429. doi: 10.1007/978-3-642-04802-9_24. Google Scholar
[38]	A. A. Milyutin, On quadratic conditions for an extremum in smooth problems with a finite-dimensional range, Methods of the Theory of Extremal Problems in Economics, (1981), 138-177. Google Scholar
[39]	A. A. Milyutin and N. P. Osmolovskii, Calculus of Variations and Optimal Control, American Mathematical Society, 1998. Google Scholar
[40]	H. J. Oberle, On the numerical computation of minimum-fuel, Earth-Mars transfer, J. Optim. Theory Appl., 22 (1977), 447-453. doi: 10.1007/BF00932866. Google Scholar
[41]	H. J. Oberle, Numerical computation of singular control functions in trajectory optimization problems, J. Guidance Control Dynam., 13 (1990), 153-159. doi: 10.2514/3.20529. Google Scholar
[42]	R. E. O'Malley Jr., Partially Singular Control Problems as Singular Singular-Perturbation Problems, Technical Report, Arizona Univ. Tucson, Department of Mathematics, 1977.Google Scholar
[43]	L. Poggiolini and G. Stefani, Minimum time optimality of a partially singular arc: Second order conditions, in Lagrangian and Hamiltonian Methods for Nonlinear Control 2006, Lecture Notes in Control and Inform. Sci., 366, Springer, Berlin, 2007,281–291. doi: 10.1007/978-3-540-73890-9_22. Google Scholar
[44]	D. J. W. Ruxton and D. J. Bell, Junction times in singular optimal control, Applied Mathematics and Computation, 70 (1995), 143-154. doi: 10.1016/0096-3003(94)00115-K. Google Scholar
[45]	A. Shapiro. On duality theory of conic linear problems, in Semi-Infinite Programming (Alicante, 1999), Nonconvex Optim. Appl., 57, Kluwer Acad. Publ., Dordrecht, 2001,135–165. doi: 10.1007/978-1-4757-3403-4_7. Google Scholar

show all references

References:

[1]	R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. Google Scholar
[2]	A. A. Agrachev and Y. L. Sachkov, Control Theory from the Geometric Viewpoint, Encyclopaedia of Mathematical Sciences, 87, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-662-06404-7. Google Scholar
[3]	V. M. Alekseev, V. M. Tikhomirov and S. V. Fomin, Optimal Control, Nauka, Moscow, 1979. [in Russian]. Google Scholar
[4]	M. S. Aronna, Convergence of the shooting algorithm for singular optimal control problems, in Proceedings of the IEEE European Control Conference (ECC), July 2013, 215–220.Google Scholar
[5]	M. S. Aronna, Singular Solutions in Optimal Control: Second Order Conditions and a Shooting Algorithm, Technical Report, Inria RR-7764, 2013. arXiv: 1210.7425, Inria RR-7764.Google Scholar
[6]	M. S. Aronna, J. F. Bonnans, A. V. Dmitruk and P. A. Lotito, Quadratic order conditions for bang-singular extremals, Numer. Algebra Control Optim., 2 (2012), 511-546. doi: 10.3934/naco.2012.2.511. Google Scholar
[7]	D. M. Azimov, Active sections of rocket trajectories. A survey of research, Avtomat. i Telemekh., 11 (2005), 14-34. doi: 10.1007/s10513-005-0207-x. Google Scholar
[8]	D. J. Bell and D. H. Jacobson, Singular Optimal Control Problems, Academic Press, 1975. Google Scholar
[9]	D. S. Bernstein and V. Zeidan, The singular linear-quadratic regulator problem and the Goh-Riccati equation, Proceedings of the IEEE Conference on Decision and Control, 1 (1990), 334-339. doi: 10.1109/CDC.1990.203608. Google Scholar
[10]	G. A. Bliss, Lectures on the Calculus of Variations, University of Chicago Press, Chicago, Illinois, 1946. Google Scholar
[11]	F. Bonnans, J. Laurent-Varin, P. Martinon and E. Trélat, Numerical study of optimal trajectories with singular arcs for an Ariane 5 launcher, J. Guidance Control Dynam., 32 (2009), 51-55. Google Scholar
[12]	J. F. Bonnans, Optimisation Continue, Dunod, 2006.Google Scholar
[13]	H. J. Bortolossi, M. V. Pereira and C. Tomei, Optimal hydrothermal scheduling with variable production coefficient, Math. Methods Oper. Res., 55 (2002), 11-36. doi: 10.1007/s001860200174. Google Scholar
[14]	H. Brézis, Analyse Fonctionnelle, Masson, Paris, 1983. Google Scholar
[15]	A. E. Bryson, Jr. and Y. C. Ho, Applied Optimal Control, Hemisphere Publishing Corp. Washington, D. C., 1975. Optimization, estimation, and control, Revised printing. Google Scholar
[16]	D. I. Cho, P. L. Abad and M. Parlar, Optimal production and maintenance decisions when a system experience age-dependent deterioration, Optimal Control Appl. Methods, 14 (1993), 153-167. doi: 10.1002/oca.4660140302. Google Scholar
[17]	A. V. Dmitruk, Quadratic conditions for a weak minimum fo control problems, Soviet Math. Doklady, 18 (1977). Google Scholar
[18]	A. V. Dmitruk, Jacobi-type conditions for the problem of Bolza with inequalities, Math. Notes, 35 (1984), 427-435. Google Scholar
[19]	A. V. Dmitruk, Quadratic order conditions for a Pontryagin minimum in an optimal control problem linear in the control, Math. USSR Izvestiya, 28 (1987), 275-303. Google Scholar
[20]	A. V. Dmitruk, Jacobi type conditions for singular extremals, Control & Cybernetics, 37 (2008), 285-306. Google Scholar
[21]	A. V. Dmitruk, Quadratic order optimality conditions for extremals completely singular in part of controls, in Operations Research Proceedings, Selected Papers of the Annual International Conference of the German Operations Research Society, (2011), 341–346. doi: 10.1007/978-3-642-20009-0_54. Google Scholar
[22]	A. V. Dmitruk and K. K. Shishov, Analysis of a quadratic functional with a partly singular Legendre condition, Moscow University Comput. Math. and Cybernetics, 34 (2010), 56-65. doi: 10.3103/S0278641910020020. Google Scholar
[23]	H. Frankowska and D. Tonon, Pointwise second-order necessary optimality conditions for the Mayer problem with control constraints, SIAM J. Control Optim., 51 (2013), 3814-3843. doi: 10.1137/130906799. Google Scholar
[24]	R. H. Goddard, A Method of Reaching Extreme Altitudes, Smithsonian Miscellaneous Collections, 71(2), Smithsonian Institution, City of Washington, 1919.Google Scholar
[25]	B. S. Goh, Necessary conditions for singular extremals involving multiple control variables, J. SIAM Control, 4 (1966), 716-731. doi: 10.1137/0304052. Google Scholar
[26]	B. S. Goh, Necessary Conditions for the Singular Extremals in the Calculus of Variations, PhD thesis, University of Canterbury, 1966.Google Scholar
[27]	B. S. Goh, The second variation for the singular Bolza problem, J. SIAM Control, 4 (1966), 309-325. doi: 10.1137/0304026. Google Scholar
[28]	B. S. Goh, Optimal singular control for multi-input linear systems, J. Math. Anal. Appl., 20 (1967), 534-539. doi: 10.1016/0022-247X(67)90079-0. Google Scholar
[29]	B. S. Goh, Optimal singular rocket and aircraft trajectories, in Control and Decision Conference, CCDC 2008, (2008), 1531–1536. doi: 10.1109/CCDC.2008.4597574. Google Scholar
[30]	M. R. Hestenes, Applications of the theory of quadratic forms in Hilbert space to the calculus of variations, Pacific J. Math., 1 (1951), 525-581. doi: 10.2140/pjm.1951.1.525. Google Scholar
[31]	D. G. Hull, Optimal guidance for quasi-planar lunar ascent, J. Optim. Theory Appl., 151 (2011), 353-372. doi: 10.1007/s10957-011-9884-5. Google Scholar
[32]	S. Kurcyusz and J. Zowe, Regularity and stability for the mathematical programming problem in Banach spaces, Applied Mathematics and Optimization, 5 (1979), 49-62. doi: 10.1007/BF01442543. Google Scholar
[33]	D. F. Lawden, Optimal Trajectories for Space Navigation, Butterworths, London, 1963. Google Scholar
[34]	E. S. Levitin, A. A. Milyutin and N. P. Osmolovskii, Theory of higher-order conditions in smooth constrained extremal problems, Theoretical and Applied Optimal Control Problems, (1985), 4-40. Google Scholar
[35]	O. Mangasarian and S. Fromovitz, The Fritz-John necessary optimality conditions in the presence of equality and inequality constraints, J. Math. Anal. Appl., 17 (1967), 37-47. doi: 10.1016/0022-247X(67)90163-1. Google Scholar
[36]	H. Maurer, J.-H. Kim and G. Vossen, On a state-constrained control problem in optimal production and maintenance, in Optimal Control and Dynamic Games (eds. C. Deissenberg, R. Hartl, H. M. Amman and B. Rustem), Advances in Computational Management Science, 7, Springer, 2005,289–308. doi: 10.1007/0-387-25805-1_17. Google Scholar
[37]	H. Maurer and N. P. Osmolovskii, Second order sufficient optimality conditions for a control problem with continuous and bang-bang control components: Riccati approach, in System Modeling and Optimization, IFIP Adv. Inf. Commun. Technol., 312, Springer, Berlin, 2009,411–429. doi: 10.1007/978-3-642-04802-9_24. Google Scholar
[38]	A. A. Milyutin, On quadratic conditions for an extremum in smooth problems with a finite-dimensional range, Methods of the Theory of Extremal Problems in Economics, (1981), 138-177. Google Scholar
[39]	A. A. Milyutin and N. P. Osmolovskii, Calculus of Variations and Optimal Control, American Mathematical Society, 1998. Google Scholar
[40]	H. J. Oberle, On the numerical computation of minimum-fuel, Earth-Mars transfer, J. Optim. Theory Appl., 22 (1977), 447-453. doi: 10.1007/BF00932866. Google Scholar
[41]	H. J. Oberle, Numerical computation of singular control functions in trajectory optimization problems, J. Guidance Control Dynam., 13 (1990), 153-159. doi: 10.2514/3.20529. Google Scholar
[42]	R. E. O'Malley Jr., Partially Singular Control Problems as Singular Singular-Perturbation Problems, Technical Report, Arizona Univ. Tucson, Department of Mathematics, 1977.Google Scholar
[43]	L. Poggiolini and G. Stefani, Minimum time optimality of a partially singular arc: Second order conditions, in Lagrangian and Hamiltonian Methods for Nonlinear Control 2006, Lecture Notes in Control and Inform. Sci., 366, Springer, Berlin, 2007,281–291. doi: 10.1007/978-3-540-73890-9_22. Google Scholar
[44]	D. J. W. Ruxton and D. J. Bell, Junction times in singular optimal control, Applied Mathematics and Computation, 70 (1995), 143-154. doi: 10.1016/0096-3003(94)00115-K. Google Scholar
[45]	A. Shapiro. On duality theory of conic linear problems, in Semi-Infinite Programming (Alicante, 1999), Nonconvex Optim. Appl., 57, Kluwer Acad. Publ., Dordrecht, 2001,135–165. doi: 10.1007/978-1-4757-3403-4_7. Google Scholar

[1]	Ana P. Lemos-Paião, Cristiana J. Silva, Delfim F. M. Torres. A sufficient optimality condition for delayed state-linear optimal control problems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2293-2313. doi: 10.3934/dcdsb.2019096
[2]	Lucas Bonifacius, Ira Neitzel. Second order optimality conditions for optimal control of quasilinear parabolic equations. Mathematical Control & Related Fields, 2018, 8 (1) : 1-34. doi: 10.3934/mcrf.2018001
[3]	François Gay-Balmaz, Tudor S. Ratiu. Clebsch optimal control formulation in mechanics. Journal of Geometric Mechanics, 2011, 3 (1) : 41-79. doi: 10.3934/jgm.2011.3.41
[4]	Lihua Li, Yan Gao, Hongjie Wang. Second order sufficient optimality conditions for hybrid control problems with state jump. Journal of Industrial & Management Optimization, 2015, 11 (1) : 329-343. doi: 10.3934/jimo.2015.11.329
[5]	Omid S. Fard, Javad Soolaki, Delfim F. M. Torres. A necessary condition of Pontryagin type for fuzzy fractional optimal control problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 59-76. doi: 10.3934/dcdss.2018004
[6]	Anthony M. Bloch, Peter E. Crouch, Nikolaj Nordkvist. Continuous and discrete embedded optimal control problems and their application to the analysis of Clebsch optimal control problems and mechanical systems. Journal of Geometric Mechanics, 2013, 5 (1) : 1-38. doi: 10.3934/jgm.2013.5.1
[7]	J.-P. Raymond, F. Tröltzsch. Second order sufficient optimality conditions for nonlinear parabolic control problems with state constraints. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 431-450. doi: 10.3934/dcds.2000.6.431
[8]	Leonardo Colombo, David Martín de Diego. Second-order variational problems on Lie groupoids and optimal control applications. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6023-6064. doi: 10.3934/dcds.2016064
[9]	Ciro D'Apice, Olha P. Kupenko, Rosanna Manzo. On boundary optimal control problem for an arterial system: First-order optimality conditions. Networks & Heterogeneous Media, 2018, 13 (4) : 585-607. doi: 10.3934/nhm.2018027
[10]	RazIye Mert, A. Zafer. A necessary and sufficient condition for oscillation of second order sublinear delay dynamic equations. Conference Publications, 2011, 2011 (Special) : 1061-1067. doi: 10.3934/proc.2011.2011.1061
[11]	Hancheng Guo, Jie Xiong. A second-order stochastic maximum principle for generalized mean-field singular control problem. Mathematical Control & Related Fields, 2018, 8 (2) : 451-473. doi: 10.3934/mcrf.2018018
[12]	Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. II. Convergence of the method of finite differences. Inverse Problems & Imaging, 2016, 10 (4) : 869-898. doi: 10.3934/ipi.2016025
[13]	Elimhan N. Mahmudov. Optimal control of second order delay-discrete and delay-differential inclusions with state constraints. Evolution Equations & Control Theory, 2018, 7 (3) : 501-529. doi: 10.3934/eect.2018024
[14]	Rui Li, Yingjing Shi. Finite-time optimal consensus control for second-order multi-agent systems. Journal of Industrial & Management Optimization, 2014, 10 (3) : 929-943. doi: 10.3934/jimo.2014.10.929
[15]	Hongwei Lou, Jiongmin Yong. Second-order necessary conditions for optimal control of semilinear elliptic equations with leading term containing controls. Mathematical Control & Related Fields, 2018, 8 (1) : 57-88. doi: 10.3934/mcrf.2018003
[16]	Hongwei Lou. Second-order necessary/sufficient conditions for optimal control problems in the absence of linear structure. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1445-1464. doi: 10.3934/dcdsb.2010.14.1445
[17]	Leonardo Colombo. Second-order constrained variational problems on Lie algebroids: Applications to Optimal Control. Journal of Geometric Mechanics, 2017, 9 (1) : 1-45. doi: 10.3934/jgm.2017001
[18]	Muhammad I. Mustafa. On the control of the wave equation by memory-type boundary condition. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1179-1192. doi: 10.3934/dcds.2015.35.1179
[19]	Gaoxi Li, Zhongping Wan, Jia-wei Chen, Xiaoke Zhao. Necessary optimality condition for trilevel optimization problem. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-16. doi: 10.3934/jimo.2018140
[20]	B. Bonnard, J.-B. Caillau, E. Trélat. Second order optimality conditions with applications. Conference Publications, 2007, 2007 (Special) : 145-154. doi: 10.3934/proc.2007.2007.145

American Institute of Mathematical Sciences