• Previous Article
    Error bounds for Euler approximation of linear-quadratic control problems with bang-bang solutions
  • NACO Home
  • This Issue
  • Next Article
    A direct method for the solution of an optimal control problem arising from image registration
2012, 2(3): 511-546. doi: 10.3934/naco.2012.2.511

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

Received  July 2011 Revised  June 2012 Published  August 2012

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.
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, 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, 87 (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. doi: 10.1137/S036301290138866X. Google Scholar

[4]

V. M. Alekseev, V. M. Tikhomirov and S. V. Fomin, "Optimal Control,", Nauka, (1979). 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, (7763). 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. 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, (2004). Google Scholar

[8]

A. E. Bryson, Jr. and Y. C. Ho, "Applied Optimal Control,", Hemisphere Publishing Corp. Washington, (1975). Google Scholar

[9]

Y. Chitour, F. Jean and E. Trélat, Genericity results for singular curves,, J. Differential Geom., 73 (2006), 45. Google Scholar

[10]

Y. Chitour, F. Jean and E. Trélat, Singular trajectories of control-affine systems,, SIAM J. Control Optim., 47 (2008), 1078. 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. 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. Google Scholar

[14]

A. V. Dmitruk, Jacobi-type conditions for the problem of Bolza with inequalities,, Math. Notes, 35 (1984), 427. 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. doi: 10.1070/IM1987v028n02ABEH000882. Google Scholar

[16]

A. V. Dmitruk, Jacobi type conditions for singular extremals,, Control & Cybernetics, 37 (2008), 285. 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. 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. 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. doi: 10.1016/0041-5553(65)90148-5. Google Scholar

[20]

N. Dunford and J. Schwartz, "Linear Operators, Vol I,", Interscience, (1958). Google Scholar

[21]

U. Felgenhauer, On stability of bang-bang type controls,, SIAM J. Control Optim., 41 (2003), 1843. 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. Google Scholar

[23]

U. Felgenhauer, Optimality properties of controls with bang-bang components in problems with semilinear state equation,, Control Cybernet., 34 (2005), 763. Google Scholar

[24]

R. Gabasov and F. M. Kirillova, High-order necessary conditions for optimality,, J. SIAM Control, 10 (1972), 127. 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. 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. doi: 10.1137/0304052. Google Scholar

[27]

B. S. Goh, The second variation for the singular Bolza problem,, J. SIAM Control, 4 (1966), 309. 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. Google Scholar

[29]

A. Hoffman, On approximate solutions of systems of linear inequalities,, Journal of Research of the National Bureau of Standards, 49 (1952), 263. 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. 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. 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. doi: 10.2514/3.2562. Google Scholar

[33]

H. J. Kelley, R. E. Kopp and H. G. Moyer, Singular extremals,, in, (1967), 63. 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. 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. doi: 10.1137/0315019. Google Scholar

[36]

S. Kurcyusz and J. Zowe, Regularity and stability for the mathematical programming problem in Banach spaces,, in, (1979), 49. 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, (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. 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. Google Scholar

[40]

H. Maurer, Numerical solution of singular control problems using multiple shooting techniques,, J. of Optimization Theory and Applications, 18 (1976), 235. 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. 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. 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. 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. 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. 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. 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, 366 (2007), 281. 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. Google Scholar

[51]

R. T. Rockafellar, "Convex Analysis,", Princeton University Press, (1970). Google Scholar

[52]

W. Rudin, "Real and Complex Analysis,", Mc Graw-Hill, (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. Google Scholar

[54]

H. Schättler, A local feedback synthesis of time-optimal stabilizing controls in dimension three,, Math. Control Signals Systems, 4 (1991), 293. 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. 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. 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. 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. 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. 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), 91. 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, 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, 87 (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. doi: 10.1137/S036301290138866X. Google Scholar

[4]

V. M. Alekseev, V. M. Tikhomirov and S. V. Fomin, "Optimal Control,", Nauka, (1979). 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, (7763). 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. 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, (2004). Google Scholar

[8]

A. E. Bryson, Jr. and Y. C. Ho, "Applied Optimal Control,", Hemisphere Publishing Corp. Washington, (1975). Google Scholar

[9]

Y. Chitour, F. Jean and E. Trélat, Genericity results for singular curves,, J. Differential Geom., 73 (2006), 45. Google Scholar

[10]

Y. Chitour, F. Jean and E. Trélat, Singular trajectories of control-affine systems,, SIAM J. Control Optim., 47 (2008), 1078. 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. 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. Google Scholar

[14]

A. V. Dmitruk, Jacobi-type conditions for the problem of Bolza with inequalities,, Math. Notes, 35 (1984), 427. 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. doi: 10.1070/IM1987v028n02ABEH000882. Google Scholar

[16]

A. V. Dmitruk, Jacobi type conditions for singular extremals,, Control & Cybernetics, 37 (2008), 285. 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. 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. 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. doi: 10.1016/0041-5553(65)90148-5. Google Scholar

[20]

N. Dunford and J. Schwartz, "Linear Operators, Vol I,", Interscience, (1958). Google Scholar

[21]

U. Felgenhauer, On stability of bang-bang type controls,, SIAM J. Control Optim., 41 (2003), 1843. 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. Google Scholar

[23]

U. Felgenhauer, Optimality properties of controls with bang-bang components in problems with semilinear state equation,, Control Cybernet., 34 (2005), 763. Google Scholar

[24]

R. Gabasov and F. M. Kirillova, High-order necessary conditions for optimality,, J. SIAM Control, 10 (1972), 127. 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. 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. doi: 10.1137/0304052. Google Scholar

[27]

B. S. Goh, The second variation for the singular Bolza problem,, J. SIAM Control, 4 (1966), 309. 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. Google Scholar

[29]

A. Hoffman, On approximate solutions of systems of linear inequalities,, Journal of Research of the National Bureau of Standards, 49 (1952), 263. 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. 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. 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. doi: 10.2514/3.2562. Google Scholar

[33]

H. J. Kelley, R. E. Kopp and H. G. Moyer, Singular extremals,, in, (1967), 63. 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. 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. doi: 10.1137/0315019. Google Scholar

[36]

S. Kurcyusz and J. Zowe, Regularity and stability for the mathematical programming problem in Banach spaces,, in, (1979), 49. 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, (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. 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. Google Scholar

[40]

H. Maurer, Numerical solution of singular control problems using multiple shooting techniques,, J. of Optimization Theory and Applications, 18 (1976), 235. 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. 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. 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. 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. 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. 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. 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, 366 (2007), 281. 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. Google Scholar

[51]

R. T. Rockafellar, "Convex Analysis,", Princeton University Press, (1970). Google Scholar

[52]

W. Rudin, "Real and Complex Analysis,", Mc Graw-Hill, (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. Google Scholar

[54]

H. Schättler, A local feedback synthesis of time-optimal stabilizing controls in dimension three,, Math. Control Signals Systems, 4 (1991), 293. 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. 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. 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. 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. 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. 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), 91. Google Scholar

[1]

Térence Bayen, Marc Mazade, Francis Mairet. Analysis of an optimal control problem connected to bioprocesses involving a saturated singular arc. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 39-58. doi: 10.3934/dcdsb.2015.20.39

[2]

Shihchung Chiang. Numerical optimal unbounded control with a singular integro-differential equation as a constraint. Conference Publications, 2013, 2013 (special) : 129-137. doi: 10.3934/proc.2013.2013.129

[3]

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

[4]

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

[5]

Galina V. Grishina. On positive solution to a second order elliptic equation with a singular nonlinearity. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1335-1343. doi: 10.3934/cpaa.2010.9.1335

[6]

M. Delgado-Téllez, Alberto Ibort. On the geometry and topology of singular optimal control problems and their solutions. Conference Publications, 2003, 2003 (Special) : 223-233. doi: 10.3934/proc.2003.2003.223

[7]

Shanjian Tang. A second-order maximum principle for singular optimal stochastic controls. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1581-1599. doi: 10.3934/dcdsb.2010.14.1581

[8]

Pierluigi Colli, Gianni Gilardi, Gabriela Marinoschi, Elisabetta Rocca. Optimal control for a phase field system with a possibly singular potential. Mathematical Control & Related Fields, 2016, 6 (1) : 95-112. doi: 10.3934/mcrf.2016.6.95

[9]

Pierluigi Colli, Gianni Gilardi, Gabriela Marinoschi, Elisabetta Rocca. Optimal control for a conserved phase field system with a possibly singular potential. Evolution Equations & Control Theory, 2018, 7 (1) : 95-116. doi: 10.3934/eect.2018006

[10]

Z. Foroozandeh, Maria do rosário de Pinho, M. Shamsi. On numerical methods for singular optimal control problems: An application to an AUV problem. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2219-2235. doi: 10.3934/dcdsb.2019092

[11]

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

[12]

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

[13]

Jérome Lohéac, Jean-François Scheid. Time optimal control for a nonholonomic system with state constraint. Mathematical Control & Related Fields, 2013, 3 (2) : 185-208. doi: 10.3934/mcrf.2013.3.185

[14]

Alain Bensoussan, John Liu, Jiguang Yuan. Singular control and impulse control: A common approach. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 27-57. doi: 10.3934/dcdsb.2010.13.27

[15]

Alexander Arguchintsev, Vasilisa Poplevko. An optimal control problem by parabolic equation with boundary smooth control and an integral constraint. Numerical Algebra, Control & Optimization, 2018, 8 (2) : 193-202. doi: 10.3934/naco.2018011

[16]

Helmut Maurer, Tanya Tarnopolskaya, Neale Fulton. Computation of bang-bang and singular controls in collision avoidance. Journal of Industrial & Management Optimization, 2014, 10 (2) : 443-460. doi: 10.3934/jimo.2014.10.443

[17]

Sofia O. Lopes, Fernando A. C. C. Fontes, Maria do Rosário de Pinho. On constraint qualifications for nondegenerate necessary conditions of optimality applied to optimal control problems. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 559-575. doi: 10.3934/dcds.2011.29.559

[18]

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

[19]

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

[20]

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

 Impact Factor: 

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (0)

[Back to Top]