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Quadratic order conditions for bang-singular extremals
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 |
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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