Quadratic order conditions for bang-singular extremals

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2012, 2(3): 511-546. doi: 10.3934/naco.2012.2.511

Quadratic order conditions for bang-singular extremals

M. Soledad Aronna ^1, , J. Frédéric Bonnans ^2, , Andrei V. Dmitruk ^3, and Pablo A. Lotito ^4,

1.	CONICET CIFASIS, Argentina, INRIA Saclay - CMAP Ecole Polytechnique, Route de Saclay, 91128 Palaiseau, France
2.	INRIA Saclay - CMAP Ecole Polytechnique, Route de Saclay, 91128 Palaiseau, France
3.	Russian Academy of Sciences - CEMI and Moscow State University, 47 Nakhimovsky Prospect, 117418 Moscow, Russian Federation
4.	CONICET PLADEMA - Univ. Nacional de Centro de la Prov. de Buenos Aires, Campus Universitario Paraje Arroyo Seco, B7000 Tandil, Argentina

Received July 2011 Revised June 2012 Published August 2012

Abstract
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This paper deals with optimal control problems for systems affine in the control variable. We consider nonnegativity constraints on the control, and finitely many equality and inequality constraints on the final state. First, we obtain second order necessary optimality conditions. Secondly, we derive a second order sufficient condition for the scalar control case.

Keywords: second order condition, singular arc, control constraint, bang-singular solution., Optimal control.

Mathematics Subject Classification: Primary: 49K1.

Citation: M. Soledad Aronna, J. Frédéric Bonnans, Andrei V. Dmitruk, Pablo A. Lotito. Quadratic order conditions for bang-singular extremals. Numerical Algebra, Control and Optimization, 2012, 2 (3) : 511-546. doi: 10.3934/naco.2012.2.511

References:

[1]	A. A. Agrachev and R. V. Gamkrelidze, Second order optimality principle for a time-optimal problem, Math. USSR, Sbornik, 100, (1976).
[2]	A. A. Agrachev and Y. L. Sachkov, "Control Theory from the Geometric Viewpoint," volume 87 of Encyclopaedia of Mathematical Sciences, Springer-Verlag, Berlin, 2004.
[3]	A. A. Agrachev, G. Stefani and P. L. Zezza, Strong optimality for a bang-bang trajectory, SIAM J. Control and Optimization, 41 (2002), 991-1014. doi: 10.1137/S036301290138866X.
[4]	V. M. Alekseev, V. M. Tikhomirov and S. V. Fomin, "Optimal Control," Nauka, Moscow, 1979, in Russian.
[5]	M. S. Aronna, J. F. Bonnans and P. Martinon, A shooting algorithm for problems with singular arcs, INRIA Research Rapport Nr. 7763, October 2011, arXiv:1206.0839.
[6]	B. Bonnard, J. B. Caillau and E. Trélat, Geometric optimal control of elliptic Keplerian orbits, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 929-956 (electronic). doi: 10.3934/dcdsb.2005.5.929.
[7]	U. Boscain and B. Piccoli, "Optimal Syntheses for Control Systems on 2-D Manifolds," Springer-Verlag, Berlin, 2004.
[8]	A. E. Bryson, Jr. and Y. C. Ho, "Applied Optimal Control," Hemisphere Publishing Corp. Washington, D. C., 1975.
[9]	Y. Chitour, F. Jean and E. Trélat, Genericity results for singular curves, J. Differential Geom., 73 (2006), 45-73.
[10]	Y. Chitour, F. Jean and E. Trélat, Singular trajectories of control-affine systems, SIAM J. Control Optim., 47 (2008), 1078-1095. doi: 10.1137/060663003.
[11]	R. Cominetti and J.-Penot, Tangent sets of order one and two to the positive cones of some functional spaces, Applied Mathematics and Optimization, 36 (1997), 291-312. doi: 10.1007/s002459900064.
[12]	A. V. Dmitruk, Quadratic conditions for a weak minimum for singular regimes in optimal control problems, Soviet Math. Doklady, 18 (1977).
[13]	A. V. Dmitruk, Quadratic conditions for a Pontryagin minimum in an optimal control problems, linear in the control, with a constraint on the control, Dokl. Akad. Nauk SSSR, 28 (1983), 364-368.
[14]	A. V. Dmitruk, Jacobi-type conditions for the problem of Bolza with inequalities, Math. Notes, 35 (1984), 427-435. doi: 10.1007/BF01139945.
[15]	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. doi: 10.1070/IM1987v028n02ABEH000882.
[16]	A. V. Dmitruk, Jacobi type conditions for singular extremals, Control & Cybernetics, 37 (2008), 285-306.
[17]	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), 16-25. doi: 10.3103/S0278641910020020.
[18]	L. E. Dubins, On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents, Amer. J. Math., 79 (1957), 497-516. doi: 10.2307/2372560.
[19]	A. Ya. Dubovitskii and A. A. Milyutin, Extremum problems with constraints, USSR Comp. Math. and Math. Phys., 5 (1965), 1-80. doi: 10.1016/0041-5553(65)90148-5.
[20]	N. Dunford and J. Schwartz, "Linear Operators, Vol I," Interscience, New York, 1958.
[21]	U. Felgenhauer, On stability of bang-bang type controls, SIAM J. Control Optim., 41 (2003), 1843-1867 (electronic). doi: 10.1137/S0363012901399271.
[22]	U. Felgenhauer, Optimality and sensitivity for semilinear bang-bang type optimal control problems, Int. J. Appl. Math. Comput. Sci., 14 (2004), 447-454.
[23]	U. Felgenhauer, Optimality properties of controls with bang-bang components in problems with semilinear state equation, Control Cybernet., 34 (2005), 763-785.
[24]	R. Gabasov and F. M. Kirillova, High-order necessary conditions for optimality, J. SIAM Control, 10 (1972), 127-168. doi: 10.1137/0310012.
[25]	P. Gajardo, H. Ramírez C. and A. Rapaport, Minimal time sequential batch reactors with bounded and impulse controls for one or more species, SIAM J. Control Optim., 47 (2008), 2827-2856. doi: 10.1137/070695204.
[26]	B. S. Goh, Necessary conditions for singular extremals involving multiple control variables, J. SIAM Control, 4 (1966), 716-731. doi: 10.1137/0304052.
[27]	B. S. Goh, The second variation for the singular Bolza problem, J. SIAM Control, 4 (1966), 309-325. doi: 10.1137/0304026.
[28]	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.
[29]	A. Hoffman, On approximate solutions of systems of linear inequalities, Journal of Research of the National Bureau of Standards, Section B, Mathematical Sciences, 49 (1952), 263-265.
[30]	D. H. Jacobson and J. L. Speyer, Necessary and sufficient conditions for optimality for singular control problems: A limit approach, J. Math. Anal. Appl., 34 (1971), 239-266. doi: 10.1016/0022-247X(71)90111-9.
[31]	D. H. Jacobson, M. M. Lele and J. L. Speyer, New necessary conditions of optimality for control problems with state-variable inequality constraints, Journal of Mathematical Analysis and Applications, 35 (1971), 255-284. doi: 10.1016/0022-247X(71)90219-8.
[32]	H. J. Kelley, A second variation test for singular extremals, AIAA Journal, 2 (1964), 1380-1382. doi: 10.2514/3.2562.
[33]	H. J. Kelley, R. E. Kopp and H. G. Moyer, Singular extremals, in "Topics in Optimization", Academic Press, New York, (1967), 63-101. doi: 10.1016/S0076-5392(09)60039-4.
[34]	R. E. Kopp and H. G. Moyer, Necessary conditions for singular extremals, AIAA Journal, 3 (1965), 1439-1444. doi: 10.2514/3.3165.
[35]	A. J. Krener, The high order maximal principle and its application to singular extremals, SIAM J. on Control, 15 (1977), 256-293. doi: 10.1137/0315019.
[36]	S. Kurcyusz and J. Zowe, Regularity and stability for the mathematical programming problem in Banach spaces, in "Applied Mathematics and Optimization", Springer, New York, (1979), 49-62.
[37]	U. Ledzewicz and H. Schättler, Multi-input optimal control problems for combined tumor anti-angiogenic and radiotherapy treatments, Journal of Optimization Theory and Applications, published as "online first", to appear 2012.
[38]	E. S. Levitin, A. A. Milyutin and N. P. Osmolovskiĭ, Higher order conditions for local minima in problems with constraints, Uspekhi Mat. Nauk, 33 (1978), 85-148, 272.
[39]	A. A. Markov, Some examples of the solution of a special kind of problem on greatest and least quantities, Soobshch. Karkovsk. Mat. Obshch., 1 (1887), 250-276, (in Russian).
[40]	H. Maurer, Numerical solution of singular control problems using multiple shooting techniques, J. of Optimization Theory and Applications, 18 (1976), 235-257. doi: 10.1007/BF00935706.
[41]	H. Maurer and N. P. Osmolovskii, Second order optimality conditions for bang-bang control problems, Control and Cybernetics, 32 (2003), 555-584.
[42]	H. Maurer and N. P. Osmolovskii, Second order sufficient conditions for time-optimal bang-bang control, SIAM J. Control Optim., 42 (2003), 2239-2263. doi: 10.1137/S0363012902402578.
[43]	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.
[44]	A. A. Milyutin and N. P. Osmolovskii, "Calculus of Variations and Optimal Control," American Mathematical Society, 1998.
[45]	H. G. Moyer, Sufficient conditions for a strong minimum in singular control problems, SIAM J. Control, 11 (1973), 620-636. doi: 10.1137/0311048.
[46]	N. P. Osmolovskii, Quadratic extremality conditions for broken extremals in the general problem of the calculus of variations, J. Math. Sci. (N. Y.), 123 (2004), 3987-4122. doi: 10.1023/B:JOTH.0000036707.55314.d3.
[47]	L. Poggiolini and M. Spadini, Strong local optimality for a bang-bang trajectory in a Mayer problem, SIAM J. Control Optimization, 49 (2011), 140-161. doi: 10.1137/090771405.
[48]	L. Poggiolini and G. Stefani, On second order sufficient conditions for a bang-singular arc, Proceedings of science - SISSA, (2005).
[49]	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.), Springer, Berlin, 366 (2007), 281-291. doi: 10.1007/978-3-540-73890-9_22.
[50]	L. Poggiolini and G. Stefani, Sufficient optimality conditions for a bang-singular extremal in the minimum time problem, Control Cybernet., 37 (2008), 469-490.
[51]	R. T. Rockafellar, "Convex Analysis," Princeton University Press, Princeton, New Jersey, 1970.
[52]	W. Rudin, "Real and Complex Analysis," Mc Graw-Hill, New York, 1987.
[53]	A. V. Sarychev, First- and second-order sufficient optimality conditions for bang-bang controls, SIAM J. Control Optim., 3 (1997), 565-588.
[54]	H. Schättler, A local feedback synthesis of time-optimal stabilizing controls in dimension three, Math. Control Signals Systems, 4 (1991), 293-313. doi: 10.1007/BF02551282.
[55]	H. Schättler and M. Jankovic, A synthesis of time-optimal controls in the presence of saturated singular arcs, Forum Math., 5 (1993), 203-241. doi: 10.1515/form.1993.5.203.
[56]	P. Souères and J. P. Laumond, Shortest paths synthesis for a car-like robot, IEEE Trans. Automat. Control, 41 (1996), 672-688. doi: 10.1109/9.489204.
[57]	H. J. Sussmann, Regular synthesis for time-optimal control of single-input real analytic systems in the plane, SIAM J. Control Optim., 25 (1987), 1145-1162. doi: 10.1137/0325062.
[58]	H. J. Sussmann, The structure of time-optimal trajectories for single-input systems in the plane: the $C^\infty$ nonsingular case, SIAM J. Control Optim., 25 (1987), 433-465. doi: 10.1137/0325025.
[59]	H. J. Sussmann, The structure of time-optimal trajectories for single-input systems in the plane: the general real analytic case, SIAM J. Control Optim., 25 (1987), 868-904. doi: 10.1137/0325048.
[60]	H. J. Sussmann and G. Tang, Shortest paths for the reeds-shepp car: A worked out example of the use of geometric techniques in nonlinear optimal control, Rutgers Center for Systems and Control Technical Report 91-10, 1991.

show all references

References:

[1]	A. A. Agrachev and R. V. Gamkrelidze, Second order optimality principle for a time-optimal problem, Math. USSR, Sbornik, 100, (1976).
[2]	A. A. Agrachev and Y. L. Sachkov, "Control Theory from the Geometric Viewpoint," volume 87 of Encyclopaedia of Mathematical Sciences, Springer-Verlag, Berlin, 2004.
[3]	A. A. Agrachev, G. Stefani and P. L. Zezza, Strong optimality for a bang-bang trajectory, SIAM J. Control and Optimization, 41 (2002), 991-1014. doi: 10.1137/S036301290138866X.
[4]	V. M. Alekseev, V. M. Tikhomirov and S. V. Fomin, "Optimal Control," Nauka, Moscow, 1979, in Russian.
[5]	M. S. Aronna, J. F. Bonnans and P. Martinon, A shooting algorithm for problems with singular arcs, INRIA Research Rapport Nr. 7763, October 2011, arXiv:1206.0839.
[6]	B. Bonnard, J. B. Caillau and E. Trélat, Geometric optimal control of elliptic Keplerian orbits, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 929-956 (electronic). doi: 10.3934/dcdsb.2005.5.929.
[7]	U. Boscain and B. Piccoli, "Optimal Syntheses for Control Systems on 2-D Manifolds," Springer-Verlag, Berlin, 2004.
[8]	A. E. Bryson, Jr. and Y. C. Ho, "Applied Optimal Control," Hemisphere Publishing Corp. Washington, D. C., 1975.
[9]	Y. Chitour, F. Jean and E. Trélat, Genericity results for singular curves, J. Differential Geom., 73 (2006), 45-73.
[10]	Y. Chitour, F. Jean and E. Trélat, Singular trajectories of control-affine systems, SIAM J. Control Optim., 47 (2008), 1078-1095. doi: 10.1137/060663003.
[11]	R. Cominetti and J.-Penot, Tangent sets of order one and two to the positive cones of some functional spaces, Applied Mathematics and Optimization, 36 (1997), 291-312. doi: 10.1007/s002459900064.
[12]	A. V. Dmitruk, Quadratic conditions for a weak minimum for singular regimes in optimal control problems, Soviet Math. Doklady, 18 (1977).
[13]	A. V. Dmitruk, Quadratic conditions for a Pontryagin minimum in an optimal control problems, linear in the control, with a constraint on the control, Dokl. Akad. Nauk SSSR, 28 (1983), 364-368.
[14]	A. V. Dmitruk, Jacobi-type conditions for the problem of Bolza with inequalities, Math. Notes, 35 (1984), 427-435. doi: 10.1007/BF01139945.
[15]	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. doi: 10.1070/IM1987v028n02ABEH000882.
[16]	A. V. Dmitruk, Jacobi type conditions for singular extremals, Control & Cybernetics, 37 (2008), 285-306.
[17]	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), 16-25. doi: 10.3103/S0278641910020020.
[18]	L. E. Dubins, On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents, Amer. J. Math., 79 (1957), 497-516. doi: 10.2307/2372560.
[19]	A. Ya. Dubovitskii and A. A. Milyutin, Extremum problems with constraints, USSR Comp. Math. and Math. Phys., 5 (1965), 1-80. doi: 10.1016/0041-5553(65)90148-5.
[20]	N. Dunford and J. Schwartz, "Linear Operators, Vol I," Interscience, New York, 1958.
[21]	U. Felgenhauer, On stability of bang-bang type controls, SIAM J. Control Optim., 41 (2003), 1843-1867 (electronic). doi: 10.1137/S0363012901399271.
[22]	U. Felgenhauer, Optimality and sensitivity for semilinear bang-bang type optimal control problems, Int. J. Appl. Math. Comput. Sci., 14 (2004), 447-454.
[23]	U. Felgenhauer, Optimality properties of controls with bang-bang components in problems with semilinear state equation, Control Cybernet., 34 (2005), 763-785.
[24]	R. Gabasov and F. M. Kirillova, High-order necessary conditions for optimality, J. SIAM Control, 10 (1972), 127-168. doi: 10.1137/0310012.
[25]	P. Gajardo, H. Ramírez C. and A. Rapaport, Minimal time sequential batch reactors with bounded and impulse controls for one or more species, SIAM J. Control Optim., 47 (2008), 2827-2856. doi: 10.1137/070695204.
[26]	B. S. Goh, Necessary conditions for singular extremals involving multiple control variables, J. SIAM Control, 4 (1966), 716-731. doi: 10.1137/0304052.
[27]	B. S. Goh, The second variation for the singular Bolza problem, J. SIAM Control, 4 (1966), 309-325. doi: 10.1137/0304026.
[28]	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.
[29]	A. Hoffman, On approximate solutions of systems of linear inequalities, Journal of Research of the National Bureau of Standards, Section B, Mathematical Sciences, 49 (1952), 263-265.
[30]	D. H. Jacobson and J. L. Speyer, Necessary and sufficient conditions for optimality for singular control problems: A limit approach, J. Math. Anal. Appl., 34 (1971), 239-266. doi: 10.1016/0022-247X(71)90111-9.
[31]	D. H. Jacobson, M. M. Lele and J. L. Speyer, New necessary conditions of optimality for control problems with state-variable inequality constraints, Journal of Mathematical Analysis and Applications, 35 (1971), 255-284. doi: 10.1016/0022-247X(71)90219-8.
[32]	H. J. Kelley, A second variation test for singular extremals, AIAA Journal, 2 (1964), 1380-1382. doi: 10.2514/3.2562.
[33]	H. J. Kelley, R. E. Kopp and H. G. Moyer, Singular extremals, in "Topics in Optimization", Academic Press, New York, (1967), 63-101. doi: 10.1016/S0076-5392(09)60039-4.
[34]	R. E. Kopp and H. G. Moyer, Necessary conditions for singular extremals, AIAA Journal, 3 (1965), 1439-1444. doi: 10.2514/3.3165.
[35]	A. J. Krener, The high order maximal principle and its application to singular extremals, SIAM J. on Control, 15 (1977), 256-293. doi: 10.1137/0315019.
[36]	S. Kurcyusz and J. Zowe, Regularity and stability for the mathematical programming problem in Banach spaces, in "Applied Mathematics and Optimization", Springer, New York, (1979), 49-62.
[37]	U. Ledzewicz and H. Schättler, Multi-input optimal control problems for combined tumor anti-angiogenic and radiotherapy treatments, Journal of Optimization Theory and Applications, published as "online first", to appear 2012.
[38]	E. S. Levitin, A. A. Milyutin and N. P. Osmolovskiĭ, Higher order conditions for local minima in problems with constraints, Uspekhi Mat. Nauk, 33 (1978), 85-148, 272.
[39]	A. A. Markov, Some examples of the solution of a special kind of problem on greatest and least quantities, Soobshch. Karkovsk. Mat. Obshch., 1 (1887), 250-276, (in Russian).
[40]	H. Maurer, Numerical solution of singular control problems using multiple shooting techniques, J. of Optimization Theory and Applications, 18 (1976), 235-257. doi: 10.1007/BF00935706.
[41]	H. Maurer and N. P. Osmolovskii, Second order optimality conditions for bang-bang control problems, Control and Cybernetics, 32 (2003), 555-584.
[42]	H. Maurer and N. P. Osmolovskii, Second order sufficient conditions for time-optimal bang-bang control, SIAM J. Control Optim., 42 (2003), 2239-2263. doi: 10.1137/S0363012902402578.
[43]	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.
[44]	A. A. Milyutin and N. P. Osmolovskii, "Calculus of Variations and Optimal Control," American Mathematical Society, 1998.
[45]	H. G. Moyer, Sufficient conditions for a strong minimum in singular control problems, SIAM J. Control, 11 (1973), 620-636. doi: 10.1137/0311048.
[46]	N. P. Osmolovskii, Quadratic extremality conditions for broken extremals in the general problem of the calculus of variations, J. Math. Sci. (N. Y.), 123 (2004), 3987-4122. doi: 10.1023/B:JOTH.0000036707.55314.d3.
[47]	L. Poggiolini and M. Spadini, Strong local optimality for a bang-bang trajectory in a Mayer problem, SIAM J. Control Optimization, 49 (2011), 140-161. doi: 10.1137/090771405.
[48]	L. Poggiolini and G. Stefani, On second order sufficient conditions for a bang-singular arc, Proceedings of science - SISSA, (2005).
[49]	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.), Springer, Berlin, 366 (2007), 281-291. doi: 10.1007/978-3-540-73890-9_22.
[50]	L. Poggiolini and G. Stefani, Sufficient optimality conditions for a bang-singular extremal in the minimum time problem, Control Cybernet., 37 (2008), 469-490.
[51]	R. T. Rockafellar, "Convex Analysis," Princeton University Press, Princeton, New Jersey, 1970.
[52]	W. Rudin, "Real and Complex Analysis," Mc Graw-Hill, New York, 1987.
[53]	A. V. Sarychev, First- and second-order sufficient optimality conditions for bang-bang controls, SIAM J. Control Optim., 3 (1997), 565-588.
[54]	H. Schättler, A local feedback synthesis of time-optimal stabilizing controls in dimension three, Math. Control Signals Systems, 4 (1991), 293-313. doi: 10.1007/BF02551282.
[55]	H. Schättler and M. Jankovic, A synthesis of time-optimal controls in the presence of saturated singular arcs, Forum Math., 5 (1993), 203-241. doi: 10.1515/form.1993.5.203.
[56]	P. Souères and J. P. Laumond, Shortest paths synthesis for a car-like robot, IEEE Trans. Automat. Control, 41 (1996), 672-688. doi: 10.1109/9.489204.
[57]	H. J. Sussmann, Regular synthesis for time-optimal control of single-input real analytic systems in the plane, SIAM J. Control Optim., 25 (1987), 1145-1162. doi: 10.1137/0325062.
[58]	H. J. Sussmann, The structure of time-optimal trajectories for single-input systems in the plane: the $C^\infty$ nonsingular case, SIAM J. Control Optim., 25 (1987), 433-465. doi: 10.1137/0325025.
[59]	H. J. Sussmann, The structure of time-optimal trajectories for single-input systems in the plane: the general real analytic case, SIAM J. Control Optim., 25 (1987), 868-904. doi: 10.1137/0325048.
[60]	H. J. Sussmann and G. Tang, Shortest paths for the reeds-shepp car: A worked out example of the use of geometric techniques in nonlinear optimal control, Rutgers Center for Systems and Control Technical Report 91-10, 1991.

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M. Soledad Aronna J. Frédéric Bonnans Andrei V. Dmitruk Pablo A. Lotito
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