Quadratic order conditions  for bang-singular
extremals

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

doi:10.3934/naco.2012.2.511

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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 & 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). Google Scholar
[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. Google Scholar
[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. Google Scholar
[4]	V. M. Alekseev, V. M. Tikhomirov and S. V. Fomin, "Optimal Control," Nauka, Moscow, 1979, in Russian. Google Scholar
[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. Google Scholar
[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. Google Scholar
[7]	U. Boscain and B. Piccoli, "Optimal Syntheses for Control Systems on 2-D Manifolds," Springer-Verlag, Berlin, 2004. Google Scholar
[8]	A. E. Bryson, Jr. and Y. C. Ho, "Applied Optimal Control," Hemisphere Publishing Corp. Washington, D. C., 1975. Google Scholar
[9]	Y. Chitour, F. Jean and E. Trélat, Genericity results for singular curves, J. Differential Geom., 73 (2006), 45-73. Google Scholar
[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. Google Scholar
[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. Google Scholar
[12]	A. V. Dmitruk, Quadratic conditions for a weak minimum for singular regimes in optimal control problems, Soviet Math. Doklady, 18 (1977). Google Scholar
[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. Google Scholar
[14]	A. V. Dmitruk, Jacobi-type conditions for the problem of Bolza with inequalities, Math. Notes, 35 (1984), 427-435. doi: 10.1007/BF01139945. Google Scholar
[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. Google Scholar
[16]	A. V. Dmitruk, Jacobi type conditions for singular extremals, Control & Cybernetics, 37 (2008), 285-306. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[20]	N. Dunford and J. Schwartz, "Linear Operators, Vol I," Interscience, New York, 1958. Google Scholar
[21]	U. Felgenhauer, On stability of bang-bang type controls, SIAM J. Control Optim., 41 (2003), 1843-1867 (electronic). doi: 10.1137/S0363012901399271. Google Scholar
[22]	U. Felgenhauer, Optimality and sensitivity for semilinear bang-bang type optimal control problems, Int. J. Appl. Math. Comput. Sci., 14 (2004), 447-454. Google Scholar
[23]	U. Felgenhauer, Optimality properties of controls with bang-bang components in problems with semilinear state equation, Control Cybernet., 34 (2005), 763-785. Google Scholar
[24]	R. Gabasov and F. M. Kirillova, High-order necessary conditions for optimality, J. SIAM Control, 10 (1972), 127-168. doi: 10.1137/0310012. Google Scholar
[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. Google Scholar
[26]	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
[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]	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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[32]	H. J. Kelley, A second variation test for singular extremals, AIAA Journal, 2 (1964), 1380-1382. doi: 10.2514/3.2562. Google Scholar
[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. Google Scholar
[34]	R. E. Kopp and H. G. Moyer, Necessary conditions for singular extremals, AIAA Journal, 3 (1965), 1439-1444. doi: 10.2514/3.3165. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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). Google Scholar
[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. Google Scholar
[41]	H. Maurer and N. P. Osmolovskii, Second order optimality conditions for bang-bang control problems, Control and Cybernetics, 32 (2003), 555-584. Google Scholar
[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. Google Scholar
[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. Google Scholar
[44]	A. A. Milyutin and N. P. Osmolovskii, "Calculus of Variations and Optimal Control," American Mathematical Society, 1998. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[48]	L. Poggiolini and G. Stefani, On second order sufficient conditions for a bang-singular arc, Proceedings of science - SISSA, (2005). Google Scholar
[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. Google Scholar
[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. Google Scholar
[51]	R. T. Rockafellar, "Convex Analysis," Princeton University Press, Princeton, New Jersey, 1970. Google Scholar
[52]	W. Rudin, "Real and Complex Analysis," Mc Graw-Hill, New York, 1987. Google Scholar
[53]	A. V. Sarychev, First- and second-order sufficient optimality conditions for bang-bang controls, SIAM J. Control Optim., 3 (1997), 565-588. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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). Google Scholar
[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. Google Scholar
[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. Google Scholar
[4]	V. M. Alekseev, V. M. Tikhomirov and S. V. Fomin, "Optimal Control," Nauka, Moscow, 1979, in Russian. Google Scholar
[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. Google Scholar
[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. Google Scholar
[7]	U. Boscain and B. Piccoli, "Optimal Syntheses for Control Systems on 2-D Manifolds," Springer-Verlag, Berlin, 2004. Google Scholar
[8]	A. E. Bryson, Jr. and Y. C. Ho, "Applied Optimal Control," Hemisphere Publishing Corp. Washington, D. C., 1975. Google Scholar
[9]	Y. Chitour, F. Jean and E. Trélat, Genericity results for singular curves, J. Differential Geom., 73 (2006), 45-73. Google Scholar
[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. Google Scholar
[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. Google Scholar
[12]	A. V. Dmitruk, Quadratic conditions for a weak minimum for singular regimes in optimal control problems, Soviet Math. Doklady, 18 (1977). Google Scholar
[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. Google Scholar
[14]	A. V. Dmitruk, Jacobi-type conditions for the problem of Bolza with inequalities, Math. Notes, 35 (1984), 427-435. doi: 10.1007/BF01139945. Google Scholar
[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. Google Scholar
[16]	A. V. Dmitruk, Jacobi type conditions for singular extremals, Control & Cybernetics, 37 (2008), 285-306. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[20]	N. Dunford and J. Schwartz, "Linear Operators, Vol I," Interscience, New York, 1958. Google Scholar
[21]	U. Felgenhauer, On stability of bang-bang type controls, SIAM J. Control Optim., 41 (2003), 1843-1867 (electronic). doi: 10.1137/S0363012901399271. Google Scholar
[22]	U. Felgenhauer, Optimality and sensitivity for semilinear bang-bang type optimal control problems, Int. J. Appl. Math. Comput. Sci., 14 (2004), 447-454. Google Scholar
[23]	U. Felgenhauer, Optimality properties of controls with bang-bang components in problems with semilinear state equation, Control Cybernet., 34 (2005), 763-785. Google Scholar
[24]	R. Gabasov and F. M. Kirillova, High-order necessary conditions for optimality, J. SIAM Control, 10 (1972), 127-168. doi: 10.1137/0310012. Google Scholar
[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. Google Scholar
[26]	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
[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]	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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[32]	H. J. Kelley, A second variation test for singular extremals, AIAA Journal, 2 (1964), 1380-1382. doi: 10.2514/3.2562. Google Scholar
[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. Google Scholar
[34]	R. E. Kopp and H. G. Moyer, Necessary conditions for singular extremals, AIAA Journal, 3 (1965), 1439-1444. doi: 10.2514/3.3165. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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). Google Scholar
[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. Google Scholar
[41]	H. Maurer and N. P. Osmolovskii, Second order optimality conditions for bang-bang control problems, Control and Cybernetics, 32 (2003), 555-584. Google Scholar
[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. Google Scholar
[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. Google Scholar
[44]	A. A. Milyutin and N. P. Osmolovskii, "Calculus of Variations and Optimal Control," American Mathematical Society, 1998. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[48]	L. Poggiolini and G. Stefani, On second order sufficient conditions for a bang-singular arc, Proceedings of science - SISSA, (2005). Google Scholar
[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. Google Scholar
[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. Google Scholar
[51]	R. T. Rockafellar, "Convex Analysis," Princeton University Press, Princeton, New Jersey, 1970. Google Scholar
[52]	W. Rudin, "Real and Complex Analysis," Mc Graw-Hill, New York, 1987. Google Scholar
[53]	A. V. Sarychev, First- and second-order sufficient optimality conditions for bang-bang controls, SIAM J. Control Optim., 3 (1997), 565-588. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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