June  2022, 27(6): 3325-3343. doi: 10.3934/dcdsb.2021187

Optimal spiral-like solutions near a singular extremal in a two-input control problem

1. 

HSE University, Moscow State Institute of Electronics and Mathematics, Moscow, Russia

2. 

National University of Oil and Gas ``Gubkin University", Moscow, Russia

* Corresponding author: Larisa Manita

Received  January 2020 Revised  May 2021 Published  June 2022 Early access  July 2021

We study an optimal control problem affine in two-dimensional bounded control, in which there is a singular point of the second order. In the neighborhood of the singular point we find optimal spiral-like solutions that attain the singular point in finite time, wherein the corresponding optimal controls perform an infinite number of rotations along the circle $ S^{1} $. The problem is related to the control of an inverted spherical pendulum in the neighborhood of the upper unstable equilibrium.

Citation: Larisa Manita, Mariya Ronzhina. Optimal spiral-like solutions near a singular extremal in a two-input control problem. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3325-3343. doi: 10.3934/dcdsb.2021187
References:
[1]

A. A. Agrachev and Y. L. Sachkov, Control Theory from the Geometric Viewpoint, vol. 87 of Encyclopaedia of Mathematical Sciences, Springer-Verlag Berlin Heidelberg, 2004. doi: 10.1007/978-3-662-06404-7.

[2]

D. Angeli, Almost global stabilization of the inverted pendulum via continuous state feedback, Automatica, 37 (2001), 1103-1108.  doi: 10.1016/S0005-1098(01)00064-4.

[3]

N. D. AnhH. MatsuhisaL. D. Viet and M. Yasuda, Vibration control of an inverted pendulum type structure by passive mass-spring-pendulum dynamic vibration absorber, Journal of Sound and Vibration, 307 (2007), 187-201.  doi: 10.1016/j.jsv.2007.06.060.

[4]

K. J. ÅströmJ. Aracil and F. Gordillo, A family of smooth controllers for swinging up a pendulum, Automatica J. IFAC, 44 (2008), 1841-1848.  doi: 10.1016/j.automatica.2007.10.040.

[5]

K. J. Åström and K. Furuta, Swinging up a pendulum by energy control, Automatica J. IFAC, 36 (2000), 287-295.  doi: 10.1016/S0005-1098(99)00140-5.

[6]

A. M. BlochN. E. Leonard and J. E. Marsden, Matching and stabilization by the method of controlled lagrangians, Proceedings of the 37th IEEE Conference on Decision and Control, 2 (1998), 1446-1451.  doi: 10.1109/CDC.1998.758490.

[7]

A. M. BlochN. E. Leonard and J. E. Marsden, Controlled Lagrangians and the stabilization of mechanical systems I: The first matching theorem, IEEE Trans. Automat. Control, 45 (2000), 2253-2270.  doi: 10.1109/9.895562.

[8]

F. BoarottoY. Chitour and M. Sigalotti, Fuller singularities for generic control-affine systems with an even number of controls, SIAM J. Control Optim., 58 (2020), 1207-1228.  doi: 10.1137/19M1285305.

[9]

F. Boarotto and M. Sigalotti, Time-optimal trajectories of generic control-affine systems have at worst iterated Fuller singularities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 36 (2019), 327-346.  doi: 10.1016/j.anihpc.2018.05.005.

[10]

B. Bonnard and M. Chyba, Singular Trajectories and their Role in Control Theory, 40 of Mathématiques et Applications, Springer-Verlag Berlin Heidelberg, 2003.

[11]

V. F. Borisov, Kelley condition and structure of Lagrange manifold in a neighborhood of a first-order singular extremal, J. Math. Sci. (N.Y.), 151 (2008), 3431-3472.  doi: 10.1007/s10958-008-9046-y.

[12]

M. CaponigroR. GhezziB. Piccoli and E. Trélat, Regularization of chattering phenomena via bounded variation controls, IEEE Trans. Automat. Control, 63 (2018), 2046-2060.  doi: 10.1109/TAC.2018.2810540.

[13]

N. A. ChaturvediN. H. McClamroch and D. S. Bernstein, Asymptotic smooth stabilization of the inverted 3-d pendulum, IEEE Trans. Automat. Control, 54 (2009), 1204-1215.  doi: 10.1109/TAC.2009.2019792.

[14]

F. L. Chernousko and S. A. Reshmin, Time-optimal swing-up feedback control of a pendulum, Nonlinear Dynam., 47 (2007), 65-73.  doi: 10.1007/s11071-006-9059-3.

[15]

S. V. Chukanov and A. A. Milyutin, Qualitative study of singularities for extremals of quadratic optimal control problem, Russian J. Math. Phys., 2 (1994), 31-48. 

[16]

M. ChybaN. E. Leonard and E. D. Sontag, Singular trajectories in multi-input time-optimal problems: Application to controlled mechanical systems, J. Dynam. Control Systems, 9 (2003), 103-129.  doi: 10.1023/A:1022159318457.

[17]

A. Elhasairi and A. Pechev, Humanoid robot balance control using the spherical inverted pendulum mode, Frontiers in Robotics and AI, 2 (2015), 1-13.  doi: 10.3389/frobt.2015.00021.

[18]

M. Farkas, Periodic Motions, New York: Springer-Verlag, 1994. doi: 10.1007/978-1-4757-4211-4.

[19]

R. Fujimoto and N. Sakamoto, The stable manifold approach for optimal swing up and stabilization of an inverted pendulum with input saturation, IFAC Proceedings Volumes, 44 (2011), 8046-8051.  doi: 10.3182/20110828-6-IT-1002.01504.

[20]

B. S. Goh, Optimal singular rocket and aircraft trajectories, in 2008 Chinese Control and Decision Conference, (2008), 1531–1536. doi: 10.1109/CCDC.2008.4597574.

[21]

O. O. Gutiérrez F.C. Aguilar Ibáñez and H. Sossa A., Stabilization of the inverted spherical pendulum via Lyapunov approach, Asian J. Control, 11 (2009), 587-594.  doi: 10.1002/asjc.140.

[22]

P. Hartman, Ordinary Differential Equations, J. Wiley & Sons, New York, London, Sydney, 1964.

[23]

P. L. Kapitza, Dynamic stability of the pendulum with vibrating suspension point, Soviet Physics –- JETP, 21 (1951), 588-597. 

[24]

I. A. K. Kupka, The ubiquity of Fuller's phenomenon, Nonlinear Controllability and Optimal Control, 133 (1990), 313-350. 

[25]

U. Ledzewicz and H. Schättler, Singular controls and chattering arcs in optimal control problems arising in biomedicine, Control Cybernet., 38 (2009), 1501-1523. 

[26]

U. Ledzewicz and H. Schättler, Multi-input optimal control problems for combined tumor anti-angiogenic and radiotherapy treatments, J. Optim. Theory Appl., 153 (2012), 195-224.  doi: 10.1007/s10957-011-9954-8.

[27]

T. Lee, M. Leok and N. H. McClamroch, Dynamics and control of a chain pendulum on a cart, in 2012 IEEE 51st IEEE Conference on Decision and Control (CDC), (2012), 2502–2508. doi: 10.1109/CDC.2012.6427059.

[28]

G. LiuD. Nešić and I. Mareels, Non-local stabilization of a spherical inverted pendulum, Internat. J. Control, 81 (2008), 1035-1053.  doi: 10.1080/00207170701397541.

[29]

L. V. Lokutsievskiy and V. A. Myrikova, Optimal synthesis in a model problem with two-dimensional control lying in an arbitrary convex set, Math. Notes, 105 (2019), 36-55.  doi: 10.1134/S000143461901005X.

[30]

L. A. Manita, Optimal operating modes with chattering switching in manipulator control problems, J. Appl. Math. Mech., 64 (2000), 17-24.  doi: 10.1016/S0021-8928(00)00021-6.

[31]

L. Manita and M. Ronzhina, Optimal control of a spherical inverted pendulum, Lobachevskii J. Math., 38 (2017), 954-957.  doi: 10.1134/S1995080217050262.

[32]

L. A. Manita and M. I. Ronzhina, Optimal synthesis in the control problem of an $n$-link inverted pendulum with a moving base, J. Math. Sci. (N.Y.), 221 (2017), 137-153.  doi: 10.1007/s10958-017-3222-x.

[33]

Yu. G. Martynenko and A. M. Formal'skii, Controlled pendulum on a movable base, Mechanics of Solids, 43 (2013), 6-18.  doi: 10.3103/S0025654413010020.

[34]

F. NicolosiP. D. Vecchia and D. Ciliberti, An investigation on vertical tailplane contribution to aircraft sideforce, Aerospace Science and Technology, 28 (2013), 401-416.  doi: 10.1016/j.ast.2012.12.006.

[35]

R. Olfati-Saber, Fixed point controllers and stabilization of the cart-pole system and the rotating pendulum, Proceedings of the 38th IEEE Conference on Decision and Control, 2 (1999), 1174-1181.  doi: 10.1109/CDC.1999.830086.

[36]

C. Park, Necessary conditions for the optimality of singular arcs of spacecraft trajectories subject to multiple gravitational bodies, Advances in Space Research, 51 (2013), 2125-2135.  doi: 10.1016/j.asr.2013.01.005.

[37]

L. Postelnik, G. Liu, K. Stol and A. Swain, Approximate output regulation for a spherical inverted pendulum, in Proceedings of the 2011 American Control Conference, (2011), 539–544. doi: 10.1109/ACC.2011.5991533.

[38]

R.-E. Precup, S. Preitl, J. Fodor, I.-B. Ursache, P. A. Clep and S. Kilyeni, Experimental validation of iterative feedback tuning solutions for inverted pendulum crane mode control, in 2008 Conference on Human System Interactions, (2008), 536–541. doi: 10.1109/HSI.2008.4581496.

[39]

D. B. Reister and S. M. Lenhart, Time-optimal paths for high-speed maneuvering, The International Journal of Robotics Research, 14 (1995), 184-194.  doi: 10.1177/027836499501400208.

[40]

H. M. Robbins, Optimality of intermediate-thrust arcs of rocket trajectories, AIAA J., 3 (1965), 1094-1098.  doi: 10.2514/3.3060.

[41]

M. I. Ronzhina, L. A. Manita and L. V. Lokutsievskii, Solutions of a Hamiltonian system with two-dimensional control in the neighborhood of a singular extremal of the second order, Russian Mathematical Surveys, (in the press) (2021), 2 pp.

[42]

H. Schättler and U. Ledzewicz, Geometric Optimal Control, vol. 38 of ser. Interdisciplinary Applied Mathematics, Springer, New York, 2012. doi: 10.1007/978-1-4614-3834-2.

[43]

H. Seywald and R. R. Kumar, Singular control in minimum time spacecraft reorientation, Journal of Guidance, Control, and Dynamics, 16 (1993), 686-697.  doi: 10.2514/6.1991-2645.

[44]

H. Shen and P. Tsiotras, Time-optimal control of axi-symmetric rigid spacecraft using two controls, Journal of Guidance, Control, and Dynamics, 22 (1999), 682-694.  doi: 10.2514/2.4436.

[45]

A. S. ShiriaevH. Ludvigsen and O. Egeland, Swinging up the spherical pendulum via stabilization of its first integrals, Automatica J. IFAC, 40 (2004), 73-85.  doi: 10.1016/j.automatica.2003.07.009.

[46]

A. Stephenson, On induced stability, Philosophical Magazine, 15 (1908), 233-236.  doi: 10.1080/14786440809463763.

[47]

Y. XuM. Iwase and K. Furuta, Time optimal swing-up control of single pendulum, J. Dyn. Sys., Meas., Control, 123 (2001), 518-527.  doi: 10.1115/1.1383027.

[48]

I. YegorovA. Bratus and Y. Todorov, Synthesis of optimal control in a mathematical model of economic growth under R & D investments, Applied Mathematical Sciences, 9 (2015), 4523-4564.  doi: 10.12988/ams.2015.55404.

[49]

I. YegorovF. Mairet and J.-L. Gouzé, Optimal feedback strategies for bacterial growth with degradation, recycling and effect of temperature, Optimal Control Applications and Methods, 39 (2018), 1084-1109.  doi: 10.1002/oca.2398.

[50]

M. I. Zelikin, One-parameter families of solutions to a class of PDE optimal control problems, Contemp. Math., 209 (1997), 339-349.  doi: 10.1090/conm/209/02774.

[51]

M. I. Zelikin and V. F. Borisov, Theory of Chattering Control with Applications to Astronautics, Robotics, Economics, and Engineering, Birkhäuser, Boston, 1994. doi: 10.1007/978-1-4612-2702-1.

[52]

M. I. Zelikin and V. F. Borisov, Optimal chattering feedback control, Journal of Mathematical Sciences, 114 (2003), 1227-1344.  doi: 10.1023/A:1022082011808.

[53]

M. I. Zelikin and V. F. Borisov, Singular optimal regimes in problems of mathematical economics, J. Math. Sci. (N.Y.), 130 (2005), 4409-4570.  doi: 10.1007/s10958-005-0350-5.

[54]

M. I. ZelikinL. V. Lokutsievskii and R. Hildebrand, Typicality of chaotic fractal behavior of integral vortices in Hamiltonian systems with discontinuous right hand side, J. Math. Sci. (N.Y.), 221 (2017), 1-136.  doi: 10.1007/s10958-017-3221-y.

[55]

M. I. Zelikin and L. A. Manita, Optimal control for a Timoshenko beam, Comptes Rendus Mécanique, 334 (2006), 292-297.  doi: 10.1016/j.crme.2006.03.011.

[56]

M. I. Zelikin and L. F. Zelikina, The deviation of a functional from its optimal value under chattering decreases exponentially as the number of switchings grows, Differential Equations, 35 (1999), 1489-1493. 

[57]

J. ZhuE. Trélat and M. Cerf, Minimum time control of the rocket attitude reorientation associated with orbit dynamics, SIAM J. Control Optim., 54 (2016), 391-422.  doi: 10.1137/15M1028716.

[58]

J. ZhuE. Trélat and M. Cerf, Planar tilting maneuver of a spacecraft: Singular arcs in the minimum time problem and chattering, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1347-1388.  doi: 10.3934/dcdsb.2016.21.1347.

show all references

References:
[1]

A. A. Agrachev and Y. L. Sachkov, Control Theory from the Geometric Viewpoint, vol. 87 of Encyclopaedia of Mathematical Sciences, Springer-Verlag Berlin Heidelberg, 2004. doi: 10.1007/978-3-662-06404-7.

[2]

D. Angeli, Almost global stabilization of the inverted pendulum via continuous state feedback, Automatica, 37 (2001), 1103-1108.  doi: 10.1016/S0005-1098(01)00064-4.

[3]

N. D. AnhH. MatsuhisaL. D. Viet and M. Yasuda, Vibration control of an inverted pendulum type structure by passive mass-spring-pendulum dynamic vibration absorber, Journal of Sound and Vibration, 307 (2007), 187-201.  doi: 10.1016/j.jsv.2007.06.060.

[4]

K. J. ÅströmJ. Aracil and F. Gordillo, A family of smooth controllers for swinging up a pendulum, Automatica J. IFAC, 44 (2008), 1841-1848.  doi: 10.1016/j.automatica.2007.10.040.

[5]

K. J. Åström and K. Furuta, Swinging up a pendulum by energy control, Automatica J. IFAC, 36 (2000), 287-295.  doi: 10.1016/S0005-1098(99)00140-5.

[6]

A. M. BlochN. E. Leonard and J. E. Marsden, Matching and stabilization by the method of controlled lagrangians, Proceedings of the 37th IEEE Conference on Decision and Control, 2 (1998), 1446-1451.  doi: 10.1109/CDC.1998.758490.

[7]

A. M. BlochN. E. Leonard and J. E. Marsden, Controlled Lagrangians and the stabilization of mechanical systems I: The first matching theorem, IEEE Trans. Automat. Control, 45 (2000), 2253-2270.  doi: 10.1109/9.895562.

[8]

F. BoarottoY. Chitour and M. Sigalotti, Fuller singularities for generic control-affine systems with an even number of controls, SIAM J. Control Optim., 58 (2020), 1207-1228.  doi: 10.1137/19M1285305.

[9]

F. Boarotto and M. Sigalotti, Time-optimal trajectories of generic control-affine systems have at worst iterated Fuller singularities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 36 (2019), 327-346.  doi: 10.1016/j.anihpc.2018.05.005.

[10]

B. Bonnard and M. Chyba, Singular Trajectories and their Role in Control Theory, 40 of Mathématiques et Applications, Springer-Verlag Berlin Heidelberg, 2003.

[11]

V. F. Borisov, Kelley condition and structure of Lagrange manifold in a neighborhood of a first-order singular extremal, J. Math. Sci. (N.Y.), 151 (2008), 3431-3472.  doi: 10.1007/s10958-008-9046-y.

[12]

M. CaponigroR. GhezziB. Piccoli and E. Trélat, Regularization of chattering phenomena via bounded variation controls, IEEE Trans. Automat. Control, 63 (2018), 2046-2060.  doi: 10.1109/TAC.2018.2810540.

[13]

N. A. ChaturvediN. H. McClamroch and D. S. Bernstein, Asymptotic smooth stabilization of the inverted 3-d pendulum, IEEE Trans. Automat. Control, 54 (2009), 1204-1215.  doi: 10.1109/TAC.2009.2019792.

[14]

F. L. Chernousko and S. A. Reshmin, Time-optimal swing-up feedback control of a pendulum, Nonlinear Dynam., 47 (2007), 65-73.  doi: 10.1007/s11071-006-9059-3.

[15]

S. V. Chukanov and A. A. Milyutin, Qualitative study of singularities for extremals of quadratic optimal control problem, Russian J. Math. Phys., 2 (1994), 31-48. 

[16]

M. ChybaN. E. Leonard and E. D. Sontag, Singular trajectories in multi-input time-optimal problems: Application to controlled mechanical systems, J. Dynam. Control Systems, 9 (2003), 103-129.  doi: 10.1023/A:1022159318457.

[17]

A. Elhasairi and A. Pechev, Humanoid robot balance control using the spherical inverted pendulum mode, Frontiers in Robotics and AI, 2 (2015), 1-13.  doi: 10.3389/frobt.2015.00021.

[18]

M. Farkas, Periodic Motions, New York: Springer-Verlag, 1994. doi: 10.1007/978-1-4757-4211-4.

[19]

R. Fujimoto and N. Sakamoto, The stable manifold approach for optimal swing up and stabilization of an inverted pendulum with input saturation, IFAC Proceedings Volumes, 44 (2011), 8046-8051.  doi: 10.3182/20110828-6-IT-1002.01504.

[20]

B. S. Goh, Optimal singular rocket and aircraft trajectories, in 2008 Chinese Control and Decision Conference, (2008), 1531–1536. doi: 10.1109/CCDC.2008.4597574.

[21]

O. O. Gutiérrez F.C. Aguilar Ibáñez and H. Sossa A., Stabilization of the inverted spherical pendulum via Lyapunov approach, Asian J. Control, 11 (2009), 587-594.  doi: 10.1002/asjc.140.

[22]

P. Hartman, Ordinary Differential Equations, J. Wiley & Sons, New York, London, Sydney, 1964.

[23]

P. L. Kapitza, Dynamic stability of the pendulum with vibrating suspension point, Soviet Physics –- JETP, 21 (1951), 588-597. 

[24]

I. A. K. Kupka, The ubiquity of Fuller's phenomenon, Nonlinear Controllability and Optimal Control, 133 (1990), 313-350. 

[25]

U. Ledzewicz and H. Schättler, Singular controls and chattering arcs in optimal control problems arising in biomedicine, Control Cybernet., 38 (2009), 1501-1523. 

[26]

U. Ledzewicz and H. Schättler, Multi-input optimal control problems for combined tumor anti-angiogenic and radiotherapy treatments, J. Optim. Theory Appl., 153 (2012), 195-224.  doi: 10.1007/s10957-011-9954-8.

[27]

T. Lee, M. Leok and N. H. McClamroch, Dynamics and control of a chain pendulum on a cart, in 2012 IEEE 51st IEEE Conference on Decision and Control (CDC), (2012), 2502–2508. doi: 10.1109/CDC.2012.6427059.

[28]

G. LiuD. Nešić and I. Mareels, Non-local stabilization of a spherical inverted pendulum, Internat. J. Control, 81 (2008), 1035-1053.  doi: 10.1080/00207170701397541.

[29]

L. V. Lokutsievskiy and V. A. Myrikova, Optimal synthesis in a model problem with two-dimensional control lying in an arbitrary convex set, Math. Notes, 105 (2019), 36-55.  doi: 10.1134/S000143461901005X.

[30]

L. A. Manita, Optimal operating modes with chattering switching in manipulator control problems, J. Appl. Math. Mech., 64 (2000), 17-24.  doi: 10.1016/S0021-8928(00)00021-6.

[31]

L. Manita and M. Ronzhina, Optimal control of a spherical inverted pendulum, Lobachevskii J. Math., 38 (2017), 954-957.  doi: 10.1134/S1995080217050262.

[32]

L. A. Manita and M. I. Ronzhina, Optimal synthesis in the control problem of an $n$-link inverted pendulum with a moving base, J. Math. Sci. (N.Y.), 221 (2017), 137-153.  doi: 10.1007/s10958-017-3222-x.

[33]

Yu. G. Martynenko and A. M. Formal'skii, Controlled pendulum on a movable base, Mechanics of Solids, 43 (2013), 6-18.  doi: 10.3103/S0025654413010020.

[34]

F. NicolosiP. D. Vecchia and D. Ciliberti, An investigation on vertical tailplane contribution to aircraft sideforce, Aerospace Science and Technology, 28 (2013), 401-416.  doi: 10.1016/j.ast.2012.12.006.

[35]

R. Olfati-Saber, Fixed point controllers and stabilization of the cart-pole system and the rotating pendulum, Proceedings of the 38th IEEE Conference on Decision and Control, 2 (1999), 1174-1181.  doi: 10.1109/CDC.1999.830086.

[36]

C. Park, Necessary conditions for the optimality of singular arcs of spacecraft trajectories subject to multiple gravitational bodies, Advances in Space Research, 51 (2013), 2125-2135.  doi: 10.1016/j.asr.2013.01.005.

[37]

L. Postelnik, G. Liu, K. Stol and A. Swain, Approximate output regulation for a spherical inverted pendulum, in Proceedings of the 2011 American Control Conference, (2011), 539–544. doi: 10.1109/ACC.2011.5991533.

[38]

R.-E. Precup, S. Preitl, J. Fodor, I.-B. Ursache, P. A. Clep and S. Kilyeni, Experimental validation of iterative feedback tuning solutions for inverted pendulum crane mode control, in 2008 Conference on Human System Interactions, (2008), 536–541. doi: 10.1109/HSI.2008.4581496.

[39]

D. B. Reister and S. M. Lenhart, Time-optimal paths for high-speed maneuvering, The International Journal of Robotics Research, 14 (1995), 184-194.  doi: 10.1177/027836499501400208.

[40]

H. M. Robbins, Optimality of intermediate-thrust arcs of rocket trajectories, AIAA J., 3 (1965), 1094-1098.  doi: 10.2514/3.3060.

[41]

M. I. Ronzhina, L. A. Manita and L. V. Lokutsievskii, Solutions of a Hamiltonian system with two-dimensional control in the neighborhood of a singular extremal of the second order, Russian Mathematical Surveys, (in the press) (2021), 2 pp.

[42]

H. Schättler and U. Ledzewicz, Geometric Optimal Control, vol. 38 of ser. Interdisciplinary Applied Mathematics, Springer, New York, 2012. doi: 10.1007/978-1-4614-3834-2.

[43]

H. Seywald and R. R. Kumar, Singular control in minimum time spacecraft reorientation, Journal of Guidance, Control, and Dynamics, 16 (1993), 686-697.  doi: 10.2514/6.1991-2645.

[44]

H. Shen and P. Tsiotras, Time-optimal control of axi-symmetric rigid spacecraft using two controls, Journal of Guidance, Control, and Dynamics, 22 (1999), 682-694.  doi: 10.2514/2.4436.

[45]

A. S. ShiriaevH. Ludvigsen and O. Egeland, Swinging up the spherical pendulum via stabilization of its first integrals, Automatica J. IFAC, 40 (2004), 73-85.  doi: 10.1016/j.automatica.2003.07.009.

[46]

A. Stephenson, On induced stability, Philosophical Magazine, 15 (1908), 233-236.  doi: 10.1080/14786440809463763.

[47]

Y. XuM. Iwase and K. Furuta, Time optimal swing-up control of single pendulum, J. Dyn. Sys., Meas., Control, 123 (2001), 518-527.  doi: 10.1115/1.1383027.

[48]

I. YegorovA. Bratus and Y. Todorov, Synthesis of optimal control in a mathematical model of economic growth under R & D investments, Applied Mathematical Sciences, 9 (2015), 4523-4564.  doi: 10.12988/ams.2015.55404.

[49]

I. YegorovF. Mairet and J.-L. Gouzé, Optimal feedback strategies for bacterial growth with degradation, recycling and effect of temperature, Optimal Control Applications and Methods, 39 (2018), 1084-1109.  doi: 10.1002/oca.2398.

[50]

M. I. Zelikin, One-parameter families of solutions to a class of PDE optimal control problems, Contemp. Math., 209 (1997), 339-349.  doi: 10.1090/conm/209/02774.

[51]

M. I. Zelikin and V. F. Borisov, Theory of Chattering Control with Applications to Astronautics, Robotics, Economics, and Engineering, Birkhäuser, Boston, 1994. doi: 10.1007/978-1-4612-2702-1.

[52]

M. I. Zelikin and V. F. Borisov, Optimal chattering feedback control, Journal of Mathematical Sciences, 114 (2003), 1227-1344.  doi: 10.1023/A:1022082011808.

[53]

M. I. Zelikin and V. F. Borisov, Singular optimal regimes in problems of mathematical economics, J. Math. Sci. (N.Y.), 130 (2005), 4409-4570.  doi: 10.1007/s10958-005-0350-5.

[54]

M. I. ZelikinL. V. Lokutsievskii and R. Hildebrand, Typicality of chaotic fractal behavior of integral vortices in Hamiltonian systems with discontinuous right hand side, J. Math. Sci. (N.Y.), 221 (2017), 1-136.  doi: 10.1007/s10958-017-3221-y.

[55]

M. I. Zelikin and L. A. Manita, Optimal control for a Timoshenko beam, Comptes Rendus Mécanique, 334 (2006), 292-297.  doi: 10.1016/j.crme.2006.03.011.

[56]

M. I. Zelikin and L. F. Zelikina, The deviation of a functional from its optimal value under chattering decreases exponentially as the number of switchings grows, Differential Equations, 35 (1999), 1489-1493. 

[57]

J. ZhuE. Trélat and M. Cerf, Minimum time control of the rocket attitude reorientation associated with orbit dynamics, SIAM J. Control Optim., 54 (2016), 391-422.  doi: 10.1137/15M1028716.

[58]

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Figure 1.  Optimal chattering solutions in P1
Figure 2.  Solutions of the blown-up Hamiltonian system that lie on $ Q $ and tend to $ \xi^{0} $
Figure 3.  The inverted spherical pendulum
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