April  2022, 9(2): 165-189. doi: 10.3934/jdg.2022002

Control problems with vanishing Lie Bracket arising from complete odd circulant evolutionary games

1. 

Applied Research Laboratory, Penn State University, University Park, PA 16802, USA

2. 

Naval Postgraduate School, Monterey, CA 93940, USA

* Corresponding author: James Fan

Received  February 2019 Revised  November 2020 Published  April 2022 Early access  February 2022

Fund Project: CG is supported by NSF grant CMMI-1463482 and CMMI-1932991

We study an optimal control problem arising from a generalization of rock-paper-scissors in which the number of strategies may be selected from any positive odd number greater than 1 and in which the payoff to the winner is controlled by a control variable $ \gamma $. Using the replicator dynamics as the equations of motion, we show that a quasi-linearization of the problem admits a special optimal control form in which explicit dynamics for the controller can be identified. We show that all optimal controls must satisfy a specific second order differential equation parameterized by the number of strategies in the game. We show that as the number of strategies increases, a limiting case admits a closed form for the open-loop optimal control. In performing our analysis we show necessary conditions on an optimal control problem that allow this analytic approach to function.

Citation: Christopher Griffin, James Fan. Control problems with vanishing Lie Bracket arising from complete odd circulant evolutionary games. Journal of Dynamics and Games, 2022, 9 (2) : 165-189. doi: 10.3934/jdg.2022002
References:
[1]

J. Baillieul, Geometric methods for nonlinear optimal control problems, Journal of Optimization Theory and Applications, 25 (1978), 519-548.  doi: 10.1007/BF00933518.

[2]

J. V. Breakwell and Y.-C. Ho, On the conjugate point condition for the control problem, International Journal of Engineering Science, 2 (1965), 565-579.  doi: 10.1016/0020-7225(65)90037-6.

[3]

H. ChengN. YaoZ.-G. HuangJ. ParkY. Do and Y.-C. Lai, Mesoscopic interactions and species coexistence in evolutionary game dynamics of cyclic competitions, Scientific Reports, 4 (2014), 1-7.  doi: 10.1038/srep07486.

[4]

P. J. Davis, Circulant Matrices, 2nd edition, American Mathematical Society, 2012.

[5]

R. deForest and A. Belmonte, Spatial pattern dynamics due to the fitness gradient flux in evolutionary games, Phys. Rev. E, 87 (2013), 062138.  doi: 10.1103/PhysRevE.87.062138.

[6]

O. Diekmann and S. A. van Gils, On the cyclic replicator equation and the dynamics of semelparous populations, SIAM Journal on Applied Dynamical Systems, 8 (2009), 1160-1189.  doi: 10.1137/080722734.

[7]

G. B. Ermentrout, C. Griffin and A. Belmonte, Transition matrix model for evolutionary game dynamics, Phys. Rev. E, 93 (2016), 032138, 10 pp. doi: 10.1103/physreve.93.032138.

[8]

J. Fan and C. Griffin, Optimal digital product maintenance with a continuous revenue stream, Operations Research Letters, 45 (2017), 282-288.  doi: 10.1016/j.orl.2017.04.006.

[9]

T. L. Friesz, Dynamic Optimization and Differential Games, International Series in Operations Research & Management Science, 135, Springer, 2010. doi: 10.1007/978-0-387-72778-3.

[10]

O. GilgI. Hanski and B. Sittler, Cyclic dynamics in a simple vertebrate predator-prey community, Science, 302 (2003), 866-868.  doi: 10.1126/science.1087509.

[11]

D.-G. Granić and J. Kern, Circulant games, Theory and Decision, 80 (2016), 43-69.  doi: 10.1007/s11238-014-9478-4.

[12]

C. Griffin and A. Belmonte, Cyclic public goods games: Compensated coexistence among mutual cheaters stabilized by optimized penalty taxation, Phys. Rev. E, 95 (2017), 052309.  doi: 10.1103/PhysRevE.95.052309.

[13]

C. GriffinL. Jiang and R. Wu, Analysis of quasi-dynamic ordinary differential equations and the quasi-dynamic replicator, Physica A: Statistical Mechanics and its Applications, 555 (2020), 124422.  doi: 10.1016/j.physa.2020.124422.

[14]

Q. HeM. Mobilia and U. C. Täuber, Spatial rock-paper-scissors models with inhomogeneous reaction rates, Phys. Rev. E, 82 (2010), 051909.  doi: 10.1103/PhysRevE.82.051909.

[15]

J. Hofbauer and K. H. Schlag, Sophisticated imitation in cyclic games, Journal of Evolutionary Economics, 10 (2000), 523-543.  doi: 10.1007/s001910000049.

[16]

J. HofbauerP. SchusterK. Sigmund and R. Wolff, Dynamical systems under constant organization ii: Homogeneous growth functions of degree fanxiexian_myfhp = 2fanxiexian_myfh, SIAM Journal on Applied Mathematics, 38 (1980), 282-304.  doi: 10.1137/0138025.

[17]

J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, 1998. doi: 10.1017/CBO9781139173179.

[18]

J. Hofbauer and K. Sigmund, Evolutionary game dynamics, Bulletin of the American Mathematical Society, 40 (2003), 479-519.  doi: 10.1090/S0273-0979-03-00988-1.

[19]

W. HuG. ZhangH. Tian and Z. Wang, Chaotic dynamics in asymmetric rock-paper-scissors games, IEEE Access, 7 (2019), 175614-175621.  doi: 10.1109/ACCESS.2019.2956816.

[20]

J. B. C. Jackson and L. Buss, Alleopathy and spatial competition among coral reef invertebrates, Proceedings of the National Academy of Sciences, 72 (1975), 5160-5163.  doi: 10.1073/pnas.72.12.5160.

[21]

D. H. Jacobson, Sufficient conditions for non-negativity of the second variation in singular and non-singular control problems, SIAM J. Control, 8 (1970), 403-423.  doi: 10.1137/0308029.

[22]

G. KárolyiZ. Neufeld and I. Scheuring, Rock-scissors-paper game in a chaotic flow: The effect of dispersion on the cyclic competition of microorganisms, Journal of Theoretical Biology, 236 (2005), 12-20.  doi: 10.1016/j.jtbi.2005.02.012.

[23]

B. KerrM. A. RileyM. W. Feldman and B. J. M. Bohannan, Local dispersal promotes biodiversity in a real-life game of rock-paper-scissors, Nature, 418 (2002), 171-174.  doi: 10.1038/nature00823.

[24] D. E. Kirk, Optimal Control Theory: An Introduction, Dover Press, 2004. 
[25]

B. C. Kirkup and M. A. Riley, Antibiotic-mediated antagonism leads to a bacterial game of rock-paper-scissor in vivo, Nature, 428 (2004), 412–414, URL http://ezaccess.libraries.psu.edu/login?url=https://search.proquest.com/docview/204559118?accountid=13158, Copyright - Copyright Macmillan Journals Ltd. Mar 25, 2004; Document feature - charts; references; Last updated - 2014-04-21; CODEN - NATUAS. doi: 10.1038/nature02429.

[26]

O. L. Mangasarian, Sufficient conditions for the optimal control of nonlinear systems, SIAM J. Control, 4 (1966), 139-152.  doi: 10.1137/0304013.

[27]

R. M. May and W. J. Leonard, Nonlinear aspects of competition between three species, SIAM Journal on Applied Mathematics, 29 (1975), 243-253.  doi: 10.1137/0129022.

[28]

M. Mobilia, Oscillatory dynamics in rock–paper–scissors games with mutations, Journal of Theoretical Biology, 264 (2010), 1-10.  doi: 10.1016/j.jtbi.2010.01.008.

[29]

P. Morris, Introduction to Game Theory, Springer, 1994. doi: 10.1007/978-1-4612-4316-8.

[30]

J. R. NahumB. N. Harding and B. Kerr, Evolution of restraint in a structured rock–paper–scissors community, Proceedings of the National Academy of Sciences, 108 (2011), 10831-10838.  doi: 10.1073/pnas.1100296108.

[31]

M. Peltomäki and M. Alava, Three- and four-state rock-paper-scissors games with diffusion, Phys. Rev. E, 78 (2008), 031906.  doi: 10.1103/PhysRevE.78.031906.

[32]

C. M. Postlethwaite and A. M. Rucklidge, A trio of heteroclinic bifurcations arising from a model of spatially-extended rock–paper–scissors, Nonlinearity, 32 (2019), 1375-1407.  doi: 10.1088/1361-6544/aaf530.

[33]

C. Postlethwaite and A. Rucklidge, Spirals and heteroclinic cycles in a spatially extended rock-paper-scissors model of cyclic dominance, EPL (Europhysics Letters), 117 (2017), 48006.  doi: 10.1209/0295-5075/117/48006.

[34]

V. Raju and P. S. Krishnaprasad, Lie algebra structure of fitness and replicator control, arXiv preprint, arXiv: 2005.09792.

[35]

T. ReichenbachM. Mobilia and E. Frey, Mobility promotes and jeopardizes biodiversity in rock–paper–scissors games, Nature, 448 (2007), 1046-1049.  doi: 10.1038/nature06095.

[36]

T. ReichenbachM. Mobilia and E. Frey, Self-organization of mobile populations in cyclic competition, Journal of Theoretical Biology, 254 (2008), 368-383.  doi: 10.1016/j.jtbi.2008.05.014.

[37]

Y. SatoE. Akiyama and J. P. Crutchfield, Stability and diversity in collective adaptation, Physica D: Nonlinear Phenomena, 210 (2005), 21-57.  doi: 10.1016/j.physd.2005.06.031.

[38]

Y. SatoE. Akiyama and J. D. Farmer, Chaos in learning a simple two-person game, Proceedings of the National Academy of Sciences, 99 (2002), 4748-4751.  doi: 10.1073/pnas.032086299.

[39]

Y. Sato and J. P. Crutchfield, Coupled replicator equations for the dynamics of learning in multiagent systems, Physical Review E, 67 (2003), 015206.  doi: 10.1103/PhysRevE.67.015206.

[40]

H. Schättler and U. Ledzewicz, Geometric Optimal Control: Theory, Methods and Examples, Springer, New York, 2012. doi: 10.1007/978-1-4614-3834-2.

[41]

P. SchusterK. Sigmund and R. Wolff, Dynamical systems under constant organization i. topological analysis of a family of non-linear differential equations–-a model for catalytic hypercycles, Bulletin of Mathematical Biology, 40 (1978), 743-769. 

[42]

P. SchusterK. Sigmund and R. Wolff, Dynamical systems under constant organization. iii. cooperative and competitive behavior of hypercycles, Journal of Differential Equations, 32 (1979), 357-368.  doi: 10.1016/0022-0396(79)90039-1.

[43]

P. SchusterK. Sigmund and R. Wolff, Mass action kinetics of self-replication in flow reactors, Journal of Mathematical Analysis and Applications, 78 (1980), 88-112.  doi: 10.1016/0022-247X(80)90213-9.

[44]

H. ShiW.-X. WangR. Yang and Y.-C. Lai, Basins of attraction for species extinction and coexistence in spatial rock-paper-scissors games, Phys. Rev. E, 81 (2010), 030901.  doi: 10.1103/PhysRevE.81.030901.

[45]

B. Sinervo and C. M. Lively, The rock-paper-scissors game and the evolution of alternative male strategies, Nature, 380 (1996), 240-243.  doi: 10.1038/380240a0.

[46]

B. SzczesnyM. Mobilia and A. M. Rucklidge, When does cyclic dominance lead to stable spiral waves?, EPL (Europhysics Letters), 102 (2013), 28012.  doi: 10.1209/0295-5075/102/28012.

[47]

B. SzczesnyM. Mobilia and A. M. Rucklidge, Characterization of spiraling patterns in spatial rock-paper-scissors games, Phys. Rev. E, 90 (2014), 032704.  doi: 10.1103/PhysRevE.90.032704.

[48]

A. Szolnoki, M. Mobilia, L.-L. Jiang, B. Szczesny, A. M. Rucklidge and M. Perc, Cyclic dominance in evolutionary games: A review, Journal of the Royal Society Interface, 11 (2014). doi: 10.1098/rsif.2014.0735.

[49] J. W. Weibull, Evolutionary Game Theory, MIT Press, Cambridge, MA, 1995. 
[50]

E. C. Zeeman, Population dynamics from game theory, in Global Theory of Dynamical Systems, 819 in Springer Lecture Notes in Mathematics, Springer, 1980,471–497.

show all references

References:
[1]

J. Baillieul, Geometric methods for nonlinear optimal control problems, Journal of Optimization Theory and Applications, 25 (1978), 519-548.  doi: 10.1007/BF00933518.

[2]

J. V. Breakwell and Y.-C. Ho, On the conjugate point condition for the control problem, International Journal of Engineering Science, 2 (1965), 565-579.  doi: 10.1016/0020-7225(65)90037-6.

[3]

H. ChengN. YaoZ.-G. HuangJ. ParkY. Do and Y.-C. Lai, Mesoscopic interactions and species coexistence in evolutionary game dynamics of cyclic competitions, Scientific Reports, 4 (2014), 1-7.  doi: 10.1038/srep07486.

[4]

P. J. Davis, Circulant Matrices, 2nd edition, American Mathematical Society, 2012.

[5]

R. deForest and A. Belmonte, Spatial pattern dynamics due to the fitness gradient flux in evolutionary games, Phys. Rev. E, 87 (2013), 062138.  doi: 10.1103/PhysRevE.87.062138.

[6]

O. Diekmann and S. A. van Gils, On the cyclic replicator equation and the dynamics of semelparous populations, SIAM Journal on Applied Dynamical Systems, 8 (2009), 1160-1189.  doi: 10.1137/080722734.

[7]

G. B. Ermentrout, C. Griffin and A. Belmonte, Transition matrix model for evolutionary game dynamics, Phys. Rev. E, 93 (2016), 032138, 10 pp. doi: 10.1103/physreve.93.032138.

[8]

J. Fan and C. Griffin, Optimal digital product maintenance with a continuous revenue stream, Operations Research Letters, 45 (2017), 282-288.  doi: 10.1016/j.orl.2017.04.006.

[9]

T. L. Friesz, Dynamic Optimization and Differential Games, International Series in Operations Research & Management Science, 135, Springer, 2010. doi: 10.1007/978-0-387-72778-3.

[10]

O. GilgI. Hanski and B. Sittler, Cyclic dynamics in a simple vertebrate predator-prey community, Science, 302 (2003), 866-868.  doi: 10.1126/science.1087509.

[11]

D.-G. Granić and J. Kern, Circulant games, Theory and Decision, 80 (2016), 43-69.  doi: 10.1007/s11238-014-9478-4.

[12]

C. Griffin and A. Belmonte, Cyclic public goods games: Compensated coexistence among mutual cheaters stabilized by optimized penalty taxation, Phys. Rev. E, 95 (2017), 052309.  doi: 10.1103/PhysRevE.95.052309.

[13]

C. GriffinL. Jiang and R. Wu, Analysis of quasi-dynamic ordinary differential equations and the quasi-dynamic replicator, Physica A: Statistical Mechanics and its Applications, 555 (2020), 124422.  doi: 10.1016/j.physa.2020.124422.

[14]

Q. HeM. Mobilia and U. C. Täuber, Spatial rock-paper-scissors models with inhomogeneous reaction rates, Phys. Rev. E, 82 (2010), 051909.  doi: 10.1103/PhysRevE.82.051909.

[15]

J. Hofbauer and K. H. Schlag, Sophisticated imitation in cyclic games, Journal of Evolutionary Economics, 10 (2000), 523-543.  doi: 10.1007/s001910000049.

[16]

J. HofbauerP. SchusterK. Sigmund and R. Wolff, Dynamical systems under constant organization ii: Homogeneous growth functions of degree fanxiexian_myfhp = 2fanxiexian_myfh, SIAM Journal on Applied Mathematics, 38 (1980), 282-304.  doi: 10.1137/0138025.

[17]

J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, 1998. doi: 10.1017/CBO9781139173179.

[18]

J. Hofbauer and K. Sigmund, Evolutionary game dynamics, Bulletin of the American Mathematical Society, 40 (2003), 479-519.  doi: 10.1090/S0273-0979-03-00988-1.

[19]

W. HuG. ZhangH. Tian and Z. Wang, Chaotic dynamics in asymmetric rock-paper-scissors games, IEEE Access, 7 (2019), 175614-175621.  doi: 10.1109/ACCESS.2019.2956816.

[20]

J. B. C. Jackson and L. Buss, Alleopathy and spatial competition among coral reef invertebrates, Proceedings of the National Academy of Sciences, 72 (1975), 5160-5163.  doi: 10.1073/pnas.72.12.5160.

[21]

D. H. Jacobson, Sufficient conditions for non-negativity of the second variation in singular and non-singular control problems, SIAM J. Control, 8 (1970), 403-423.  doi: 10.1137/0308029.

[22]

G. KárolyiZ. Neufeld and I. Scheuring, Rock-scissors-paper game in a chaotic flow: The effect of dispersion on the cyclic competition of microorganisms, Journal of Theoretical Biology, 236 (2005), 12-20.  doi: 10.1016/j.jtbi.2005.02.012.

[23]

B. KerrM. A. RileyM. W. Feldman and B. J. M. Bohannan, Local dispersal promotes biodiversity in a real-life game of rock-paper-scissors, Nature, 418 (2002), 171-174.  doi: 10.1038/nature00823.

[24] D. E. Kirk, Optimal Control Theory: An Introduction, Dover Press, 2004. 
[25]

B. C. Kirkup and M. A. Riley, Antibiotic-mediated antagonism leads to a bacterial game of rock-paper-scissor in vivo, Nature, 428 (2004), 412–414, URL http://ezaccess.libraries.psu.edu/login?url=https://search.proquest.com/docview/204559118?accountid=13158, Copyright - Copyright Macmillan Journals Ltd. Mar 25, 2004; Document feature - charts; references; Last updated - 2014-04-21; CODEN - NATUAS. doi: 10.1038/nature02429.

[26]

O. L. Mangasarian, Sufficient conditions for the optimal control of nonlinear systems, SIAM J. Control, 4 (1966), 139-152.  doi: 10.1137/0304013.

[27]

R. M. May and W. J. Leonard, Nonlinear aspects of competition between three species, SIAM Journal on Applied Mathematics, 29 (1975), 243-253.  doi: 10.1137/0129022.

[28]

M. Mobilia, Oscillatory dynamics in rock–paper–scissors games with mutations, Journal of Theoretical Biology, 264 (2010), 1-10.  doi: 10.1016/j.jtbi.2010.01.008.

[29]

P. Morris, Introduction to Game Theory, Springer, 1994. doi: 10.1007/978-1-4612-4316-8.

[30]

J. R. NahumB. N. Harding and B. Kerr, Evolution of restraint in a structured rock–paper–scissors community, Proceedings of the National Academy of Sciences, 108 (2011), 10831-10838.  doi: 10.1073/pnas.1100296108.

[31]

M. Peltomäki and M. Alava, Three- and four-state rock-paper-scissors games with diffusion, Phys. Rev. E, 78 (2008), 031906.  doi: 10.1103/PhysRevE.78.031906.

[32]

C. M. Postlethwaite and A. M. Rucklidge, A trio of heteroclinic bifurcations arising from a model of spatially-extended rock–paper–scissors, Nonlinearity, 32 (2019), 1375-1407.  doi: 10.1088/1361-6544/aaf530.

[33]

C. Postlethwaite and A. Rucklidge, Spirals and heteroclinic cycles in a spatially extended rock-paper-scissors model of cyclic dominance, EPL (Europhysics Letters), 117 (2017), 48006.  doi: 10.1209/0295-5075/117/48006.

[34]

V. Raju and P. S. Krishnaprasad, Lie algebra structure of fitness and replicator control, arXiv preprint, arXiv: 2005.09792.

[35]

T. ReichenbachM. Mobilia and E. Frey, Mobility promotes and jeopardizes biodiversity in rock–paper–scissors games, Nature, 448 (2007), 1046-1049.  doi: 10.1038/nature06095.

[36]

T. ReichenbachM. Mobilia and E. Frey, Self-organization of mobile populations in cyclic competition, Journal of Theoretical Biology, 254 (2008), 368-383.  doi: 10.1016/j.jtbi.2008.05.014.

[37]

Y. SatoE. Akiyama and J. P. Crutchfield, Stability and diversity in collective adaptation, Physica D: Nonlinear Phenomena, 210 (2005), 21-57.  doi: 10.1016/j.physd.2005.06.031.

[38]

Y. SatoE. Akiyama and J. D. Farmer, Chaos in learning a simple two-person game, Proceedings of the National Academy of Sciences, 99 (2002), 4748-4751.  doi: 10.1073/pnas.032086299.

[39]

Y. Sato and J. P. Crutchfield, Coupled replicator equations for the dynamics of learning in multiagent systems, Physical Review E, 67 (2003), 015206.  doi: 10.1103/PhysRevE.67.015206.

[40]

H. Schättler and U. Ledzewicz, Geometric Optimal Control: Theory, Methods and Examples, Springer, New York, 2012. doi: 10.1007/978-1-4614-3834-2.

[41]

P. SchusterK. Sigmund and R. Wolff, Dynamical systems under constant organization i. topological analysis of a family of non-linear differential equations–-a model for catalytic hypercycles, Bulletin of Mathematical Biology, 40 (1978), 743-769. 

[42]

P. SchusterK. Sigmund and R. Wolff, Dynamical systems under constant organization. iii. cooperative and competitive behavior of hypercycles, Journal of Differential Equations, 32 (1979), 357-368.  doi: 10.1016/0022-0396(79)90039-1.

[43]

P. SchusterK. Sigmund and R. Wolff, Mass action kinetics of self-replication in flow reactors, Journal of Mathematical Analysis and Applications, 78 (1980), 88-112.  doi: 10.1016/0022-247X(80)90213-9.

[44]

H. ShiW.-X. WangR. Yang and Y.-C. Lai, Basins of attraction for species extinction and coexistence in spatial rock-paper-scissors games, Phys. Rev. E, 81 (2010), 030901.  doi: 10.1103/PhysRevE.81.030901.

[45]

B. Sinervo and C. M. Lively, The rock-paper-scissors game and the evolution of alternative male strategies, Nature, 380 (1996), 240-243.  doi: 10.1038/380240a0.

[46]

B. SzczesnyM. Mobilia and A. M. Rucklidge, When does cyclic dominance lead to stable spiral waves?, EPL (Europhysics Letters), 102 (2013), 28012.  doi: 10.1209/0295-5075/102/28012.

[47]

B. SzczesnyM. Mobilia and A. M. Rucklidge, Characterization of spiraling patterns in spatial rock-paper-scissors games, Phys. Rev. E, 90 (2014), 032704.  doi: 10.1103/PhysRevE.90.032704.

[48]

A. Szolnoki, M. Mobilia, L.-L. Jiang, B. Szczesny, A. M. Rucklidge and M. Perc, Cyclic dominance in evolutionary games: A review, Journal of the Royal Society Interface, 11 (2014). doi: 10.1098/rsif.2014.0735.

[49] J. W. Weibull, Evolutionary Game Theory, MIT Press, Cambridge, MA, 1995. 
[50]

E. C. Zeeman, Population dynamics from game theory, in Global Theory of Dynamical Systems, 819 in Springer Lecture Notes in Mathematics, Springer, 1980,471–497.

Figure 1.  The phase portrait of the first order system representing the limiting behavior of the open loop control $ \gamma $ and it's first derivative
Figure 2.  The control function, its approximations and the $ \left\lVert{{\bf{H}}^T\mathit{\boldsymbol{\lambda}}^T}\right\rVert^2 $, used in determining whether the solution to the necessary conditions are sufficient for an optimal control for the 3 strategy cyclic game (rock-paper-scissors)
Figure 3.  The control function, its approximations and the $ \left\lVert{{\bf{H}}^T\mathit{\boldsymbol{\lambda}}^T}\right\rVert^2 $, used in determining whether the solution to the necessary conditions are sufficient for an optimal control for the 5 strategy cyclic game (rock-paper-scissors-Spock-lizard)
Figure 4.  The control function, its approximations and the $ \left\lVert{{\bf{H}}^T\mathit{\boldsymbol{\lambda}}^T}\right\rVert^2 $, used in determining whether the solution to the necessary conditions are sufficient for an optimal control for the 7 strategy cyclic game
Figure 5.  The control function, its approximations and the $ \left\lVert{{\bf{H}}^T\mathit{\boldsymbol{\lambda}}^T}\right\rVert^2 $, used in determining whether the solution to the necessary conditions are sufficient for an optimal control for the 9 strategy cyclic game
Figure 6.  A solution that violates the sufficient condition for optimality $ r > {\bf{H}}^T\mathit{\boldsymbol{\lambda}} $ but can be shown to be optimal by appealing to the matrix Riccatti equation, which has bounded solutions for $ t \in [0,t_f] $. We compare the solution to the ordinary controller derived from the ordinary Euler-Lagrange equations and the controller that is directly computed using Lemma 4.2. Note they are identical as expected
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