American Institute of Mathematical Sciences

Advanced
December  2013, 3(4): 447-466. doi: 10.3934/mcrf.2013.3.447

Sparse stabilization and optimal control of the Cucker-Smale model

Marco Caponigro 1, , Massimo Fornasier 2, , Benedetto Piccoli 3, and  Emmanuel Trélat 4,

1. 

Conservatoire National des Arts et Métiers, Département Ingénierie Mathématique (IMATH), Équipe M2N, 292 rue Saint-Martin, 75003, Paris,, France
2. 

Technische Universität München, Facultät Mathematik, Boltzmannstrasse 3, D-85748, Garching bei München, Germany
3. 

Rutgers University, Department of Mathematics, Business & Science Building Room 325, Camden, NJ 08102, United States
4. 

Université Pierre et Marie Curie (Univ. Paris 6) and Institut Universitaire de France and Team GECO Inria Saclay, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France

Received  November 2012 Revised  June 2013 Published  September 2013

This article is mainly based on the work [7], and it is dedicated to the 60th anniversary of B. Bonnard, held in Dijon in June 2012.
    We focus on a controlled Cucker--Smale model in finite dimension. Such dynamics model self-organization and consensus emergence in a group of agents. We explore how it is possible to control this model in order to enforce or facilitate pattern formation or convergence to consensus. In particular, we are interested in designing control strategies that are componentwise sparse in the sense that they require a small amount of external intervention, and also time sparse in the sense that such strategies are not chattering in time. These sparsity features are desirable in view of practical issues.
    We first show how very simple sparse feedback strategies can be designed with the use of a variational principle, in order to steer the system to consensus. These feedbacks are moreover optimal in terms of decay rate of some functional, illustrating the general principle according to which ``sparse is better''. We then combine these results with local controllability properties to get global controllability results. Finally, we explore the sparsity properties of the optimal control minimizing a combination of the distance from consensus and of a norm of the control.
Keywords: Cucker--Smale model, $l_1$-norm minimization, local controllability, sparse optimal control., consensus emergence, sparse stabilization.
Mathematics Subject Classification: Primary: 49J15, 93D15; Secondary: 65K10, 93B0.
Citation: Marco Caponigro, Massimo Fornasier, Benedetto Piccoli, Emmanuel Trélat. Sparse stabilization and optimal control of the Cucker-Smale model. Mathematical Control & Related Fields, 2013, 3 (4) : 447-466. doi: 10.3934/mcrf.2013.3.447
References:
[1]

PNAS, 105 (2008), 1232-1237. doi: 10.1073/pnas.0711437105.  Google Scholar

[2]

arXiv:1202.4554, 2012. Google Scholar

[3]

J. Funct. Anal., 262 (2012), 2142-2230. doi: 10.1016/j.jfa.2011.12.012.  Google Scholar

[4]

AIMS Series on Applied Mathematics, 2, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2007.  Google Scholar

[5]

Reprint of the 2001 original, Princeton Studies in Complexity, Princeton University Press, Princeton, NJ, 2003.  Google Scholar

[6]

Comm. Pure Appl. Math., 59 (2006), 1207-1223. doi: 10.1002/cpa.20124.  Google Scholar

[7]

M. Caponigro, M. Fornasier, B. Piccoli and E. Trélat, Sparse stabilization and control of the Cucker-Smale model,, , ().   Google Scholar

[8]

in "Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences" (eds. G. Naldi, L. Pareschi, G. Toscani and N. Bellomo), Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, (2010), 297-336. doi: 10.1007/978-0-8176-4946-3_12.  Google Scholar

[9]

SIAM J. Control Optim., 50 (2012), 1735-1752. doi: 10.1137/110843216.  Google Scholar

[10]

Applications of Mathematics (New York), 17, Springer-Verlag, New York, 1983.  Google Scholar

[11]

in "IEEE International Conference on Robotics and Automation," Roma, (2007), 2292-2299. doi: 10.1109/ROBOT.2007.363661.  Google Scholar

[12]

IEEE Trans. Automat. Control, 42 (1997), 1394-1407. doi: 10.1109/9.633828.  Google Scholar

[13]

ESAIM Control Optim. Calc. Var., 17 (2011), 243-266. doi: 10.1051/cocv/2010003.  Google Scholar

[14]

Comput. Optim. Appl., 53 (2012), 155-171. doi: 10.1007/s10589-011-9444-9.  Google Scholar

[15]

SIAM J. Control Optim., 43 (2004), 549-569 (electronic). doi: 10.1137/S036301290342471X.  Google Scholar

[16]

Commun. Contemp. Math., 8 (2006), 535-567. doi: 10.1142/S0219199706002209.  Google Scholar

[17]

Proc. R. Soc. Lond. B, 270 (2002), 139-146. doi: 10.1098/rspb.2002.2210.  Google Scholar

[18]

Nature, 433 (2005), 513-516. doi: 10.1038/nature03236.  Google Scholar

[19]

J. Basic Engineering, 87 (1965), 39-57. doi: 10.1115/1.3650527.  Google Scholar

[20]

in "Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences," (eds. G. Naldi, L. Pareschi, G. Toscani and N. Bellomo), Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 2010. doi: 10.1007/978-0-8176-4946-3_13.  Google Scholar

[21]

Multiscale Model. Simul., 9 (2011), 155-182. doi: 10.1137/100797515.  Google Scholar

[22]

IEEE Trans. Automat. Control, 52 (2007), 852-862. doi: 10.1109/TAC.2007.895842.  Google Scholar

[23]

Jpn. J. Math., 2 (2007), 197-227. doi: 10.1007/s11537-007-0647-x.  Google Scholar

[24]

Found. Comput. Math., 4 (2004), 315-343. doi: 10.1007/s10208-003-0101-2.  Google Scholar

[25]

IEEE Trans. Inform. Theory, 52 (2006), 1289-1306. doi: 10.1109/TIT.2006.871582.  Google Scholar

[26]

Phys. Rev. E, 78 (2008), 056103, 12 pp. doi: 10.1103/PhysRevE.78.056103.  Google Scholar

[27]

IEEE Trans. Inform. Theory, 56 (2010), 505-519. doi: 10.1109/TIT.2009.2034789.  Google Scholar

[28]

SIAM J. Numer. Anal., 46 (2008), 577-613. doi: 10.1137/0606668909.  Google Scholar

[29]

chapter Compressive Sensing, Springer-Verlag, 2010. Google Scholar

[30]

IEEE Trans. Automat. Control, 55 (2010), 1679-1683. doi: 10.1109/TAC.2010.2046113.  Google Scholar

[31]

SIAM J. Control and Optimization, 50 (2012), 943-963. doi: 10.1137/100815037.  Google Scholar

[32]

Jahresber. Dtsch. Math.-Ver., 105 (2003), 103-165.  Google Scholar

[33]

Jahresber. Dtsch. Math.-Ver., 106 (2004), 51-69.  Google Scholar

[34]

[IEEE Trans. Automat. Control 48 (2003), 988-1001; MR 1986266] IEEE Trans. Automat. Control, 48 (2003), 1675.  Google Scholar

[35]

Complexity, 7 (2002), 41-54. doi: 10.1002/cplx.10030.  Google Scholar

[36]

J. Theor. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[37]

Trans. Res.: Part B: Methodological, 45 (2011), 1572-1589. doi: 10.1016/j.trb.2011.07.011.  Google Scholar

[38]

Jpn. J. Math. (3), 2 (2007), 229-260. doi: 10.1007/s11537-007-0657-8.  Google Scholar

[39]

Computer Graphics Forum, 31 (2012), 489-498. doi: 10.1111/j.1467-8659.2012.03028.x.  Google Scholar

[40]

in "Proc. 40th IEEE Conf. Decision Contr.," (2001), 2968-2973. Google Scholar

[41]

Third edition, Elsevier/Academic Press, Amsterdam, 2009.  Google Scholar

[42]

PLoS Computational Biology, 8 (2012), e1002442. Google Scholar

[43]

J. Theor. Biol., 171 (1994), 123-136. doi: 10.1006/jtbi.1994.1218.  Google Scholar

[44]

Science, 294 (1999), 99-101. doi: 10.1126/science.284.5411.99.  Google Scholar

[45]

Biol. Bull., 202 (2002), 296-305. doi: 10.2307/1543482.  Google Scholar

[46]

Bull. Math. Biophys., 15 (1953), 311-338. doi: 10.1007/BF02476407.  Google Scholar

[47]

AIAA Journal of Guidance, Control, and Dynamics, 32 (2009), 527-537. doi: 10.2514/1.36269.  Google Scholar

[48]

Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2007.  Google Scholar

[49]

SIAM J. Control Optim., 51 (2013), 2788-2808. doi: 10.1137/120889137.  Google Scholar

[50]

Interscience Publishers John Wiley & Sons, Inc., New York-London, 1962.  Google Scholar

[51]

hal-00813647, version 1, 2013. Google Scholar

[52]

to appear in Ann. Inst. H. Poincaré Anal. Non Linéaire, (2013). Google Scholar

[53]

J. Fourier Anal. Appl., 19 (2013), 514-544. doi: 10.1007/s00041-013-9267-4.  Google Scholar

[54]

SIAM J. Control and Optimization, 48 (2009), 162-186. doi: 10.1137/060674909.  Google Scholar

[55]

Ecol. Model., 92 (1996), 65-77. doi: 10.1016/0304-3800(95)00202-2.  Google Scholar

[56]

IEEE Transactions on Automatic Control, 52 (2007), 811-824. doi: 10.1109/TAC.2007.898077.  Google Scholar

[57]

Math. Models Methods Appl. Sci., 18 (2008), 1249-1267. doi: 10.1142/S0218202508003029.  Google Scholar

[58]

Comput. Optim. Appl., 44 (2009), 159-181. doi: 10.1007/s10589-007-9150-9.  Google Scholar

[59]

Physica D, 100 (1997), 343-354. Google Scholar

[60]

Penguin, 2010. Google Scholar

[61]

Phys. Rev. Lett., 75 (1995), 4326-4329. doi: 10.1103/PhysRevLett.75.4326.  Google Scholar

[62]

Vuibert, Paris, 2005.  Google Scholar

[63]

Phys. Rev. Lett., 75 (1995), 1226-1229. doi: 10.1103/PhysRevLett.75.1226.  Google Scholar

[64]

Optimal Control Applications and Methods, 27 (2006), 301-321. doi: 10.1002/oca.781.  Google Scholar

[65]

ESAIM, Control Optim. Calc. Var., 17 (2011), 858-886. doi: 10.1051/cocv/2010027.  Google Scholar

[66]

Proceedings of the National Academy of Sciences, 106 (2009), 5464-5469. doi: 10.1073/pnas.0811195106.  Google Scholar

[67]

Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1994. doi: 10.1007/978-1-4612-2702-1.  Google Scholar

show all references

References:
[1]

PNAS, 105 (2008), 1232-1237. doi: 10.1073/pnas.0711437105.  Google Scholar

[2]

arXiv:1202.4554, 2012. Google Scholar

[3]

J. Funct. Anal., 262 (2012), 2142-2230. doi: 10.1016/j.jfa.2011.12.012.  Google Scholar

[4]

AIMS Series on Applied Mathematics, 2, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2007.  Google Scholar

[5]

Reprint of the 2001 original, Princeton Studies in Complexity, Princeton University Press, Princeton, NJ, 2003.  Google Scholar

[6]

Comm. Pure Appl. Math., 59 (2006), 1207-1223. doi: 10.1002/cpa.20124.  Google Scholar

[7]

M. Caponigro, M. Fornasier, B. Piccoli and E. Trélat, Sparse stabilization and control of the Cucker-Smale model,, , ().   Google Scholar

[8]

in "Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences" (eds. G. Naldi, L. Pareschi, G. Toscani and N. Bellomo), Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, (2010), 297-336. doi: 10.1007/978-0-8176-4946-3_12.  Google Scholar

[9]

SIAM J. Control Optim., 50 (2012), 1735-1752. doi: 10.1137/110843216.  Google Scholar

[10]

Applications of Mathematics (New York), 17, Springer-Verlag, New York, 1983.  Google Scholar

[11]

in "IEEE International Conference on Robotics and Automation," Roma, (2007), 2292-2299. doi: 10.1109/ROBOT.2007.363661.  Google Scholar

[12]

IEEE Trans. Automat. Control, 42 (1997), 1394-1407. doi: 10.1109/9.633828.  Google Scholar

[13]

ESAIM Control Optim. Calc. Var., 17 (2011), 243-266. doi: 10.1051/cocv/2010003.  Google Scholar

[14]

Comput. Optim. Appl., 53 (2012), 155-171. doi: 10.1007/s10589-011-9444-9.  Google Scholar

[15]

SIAM J. Control Optim., 43 (2004), 549-569 (electronic). doi: 10.1137/S036301290342471X.  Google Scholar

[16]

Commun. Contemp. Math., 8 (2006), 535-567. doi: 10.1142/S0219199706002209.  Google Scholar

[17]

Proc. R. Soc. Lond. B, 270 (2002), 139-146. doi: 10.1098/rspb.2002.2210.  Google Scholar

[18]

Nature, 433 (2005), 513-516. doi: 10.1038/nature03236.  Google Scholar

[19]

J. Basic Engineering, 87 (1965), 39-57. doi: 10.1115/1.3650527.  Google Scholar

[20]

in "Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences," (eds. G. Naldi, L. Pareschi, G. Toscani and N. Bellomo), Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 2010. doi: 10.1007/978-0-8176-4946-3_13.  Google Scholar

[21]

Multiscale Model. Simul., 9 (2011), 155-182. doi: 10.1137/100797515.  Google Scholar

[22]

IEEE Trans. Automat. Control, 52 (2007), 852-862. doi: 10.1109/TAC.2007.895842.  Google Scholar

[23]

Jpn. J. Math., 2 (2007), 197-227. doi: 10.1007/s11537-007-0647-x.  Google Scholar

[24]

Found. Comput. Math., 4 (2004), 315-343. doi: 10.1007/s10208-003-0101-2.  Google Scholar

[25]

IEEE Trans. Inform. Theory, 52 (2006), 1289-1306. doi: 10.1109/TIT.2006.871582.  Google Scholar

[26]

Phys. Rev. E, 78 (2008), 056103, 12 pp. doi: 10.1103/PhysRevE.78.056103.  Google Scholar

[27]

IEEE Trans. Inform. Theory, 56 (2010), 505-519. doi: 10.1109/TIT.2009.2034789.  Google Scholar

[28]

SIAM J. Numer. Anal., 46 (2008), 577-613. doi: 10.1137/0606668909.  Google Scholar

[29]

chapter Compressive Sensing, Springer-Verlag, 2010. Google Scholar

[30]

IEEE Trans. Automat. Control, 55 (2010), 1679-1683. doi: 10.1109/TAC.2010.2046113.  Google Scholar

[31]

SIAM J. Control and Optimization, 50 (2012), 943-963. doi: 10.1137/100815037.  Google Scholar

[32]

Jahresber. Dtsch. Math.-Ver., 105 (2003), 103-165.  Google Scholar

[33]

Jahresber. Dtsch. Math.-Ver., 106 (2004), 51-69.  Google Scholar

[34]

[IEEE Trans. Automat. Control 48 (2003), 988-1001; MR 1986266] IEEE Trans. Automat. Control, 48 (2003), 1675.  Google Scholar

[35]

Complexity, 7 (2002), 41-54. doi: 10.1002/cplx.10030.  Google Scholar

[36]

J. Theor. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[37]

Trans. Res.: Part B: Methodological, 45 (2011), 1572-1589. doi: 10.1016/j.trb.2011.07.011.  Google Scholar

[38]

Jpn. J. Math. (3), 2 (2007), 229-260. doi: 10.1007/s11537-007-0657-8.  Google Scholar

[39]

Computer Graphics Forum, 31 (2012), 489-498. doi: 10.1111/j.1467-8659.2012.03028.x.  Google Scholar

[40]

in "Proc. 40th IEEE Conf. Decision Contr.," (2001), 2968-2973. Google Scholar

[41]

Third edition, Elsevier/Academic Press, Amsterdam, 2009.  Google Scholar

[42]

PLoS Computational Biology, 8 (2012), e1002442. Google Scholar

[43]

J. Theor. Biol., 171 (1994), 123-136. doi: 10.1006/jtbi.1994.1218.  Google Scholar

[44]

Science, 294 (1999), 99-101. doi: 10.1126/science.284.5411.99.  Google Scholar

[45]

Biol. Bull., 202 (2002), 296-305. doi: 10.2307/1543482.  Google Scholar

[46]

Bull. Math. Biophys., 15 (1953), 311-338. doi: 10.1007/BF02476407.  Google Scholar

[47]

AIAA Journal of Guidance, Control, and Dynamics, 32 (2009), 527-537. doi: 10.2514/1.36269.  Google Scholar

[48]

Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2007.  Google Scholar

[49]

SIAM J. Control Optim., 51 (2013), 2788-2808. doi: 10.1137/120889137.  Google Scholar

[50]

Interscience Publishers John Wiley & Sons, Inc., New York-London, 1962.  Google Scholar

[51]

hal-00813647, version 1, 2013. Google Scholar

[52]

to appear in Ann. Inst. H. Poincaré Anal. Non Linéaire, (2013). Google Scholar

[53]

J. Fourier Anal. Appl., 19 (2013), 514-544. doi: 10.1007/s00041-013-9267-4.  Google Scholar

[54]

SIAM J. Control and Optimization, 48 (2009), 162-186. doi: 10.1137/060674909.  Google Scholar

[55]

Ecol. Model., 92 (1996), 65-77. doi: 10.1016/0304-3800(95)00202-2.  Google Scholar

[56]

IEEE Transactions on Automatic Control, 52 (2007), 811-824. doi: 10.1109/TAC.2007.898077.  Google Scholar

[57]

Math. Models Methods Appl. Sci., 18 (2008), 1249-1267. doi: 10.1142/S0218202508003029.  Google Scholar

[58]

Comput. Optim. Appl., 44 (2009), 159-181. doi: 10.1007/s10589-007-9150-9.  Google Scholar

[59]

Physica D, 100 (1997), 343-354. Google Scholar

[60]

Penguin, 2010. Google Scholar

[61]

Phys. Rev. Lett., 75 (1995), 4326-4329. doi: 10.1103/PhysRevLett.75.4326.  Google Scholar

[62]

Vuibert, Paris, 2005.  Google Scholar

[63]

Phys. Rev. Lett., 75 (1995), 1226-1229. doi: 10.1103/PhysRevLett.75.1226.  Google Scholar

[64]

Optimal Control Applications and Methods, 27 (2006), 301-321. doi: 10.1002/oca.781.  Google Scholar

[65]

ESAIM, Control Optim. Calc. Var., 17 (2011), 858-886. doi: 10.1051/cocv/2010027.  Google Scholar

[66]

Proceedings of the National Academy of Sciences, 106 (2009), 5464-5469. doi: 10.1073/pnas.0811195106.  Google Scholar

[67]

Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1994. doi: 10.1007/978-1-4612-2702-1.  Google Scholar

[1]

Pia Heins, Michael Moeller, Martin Burger. Locally sparse reconstruction using the $l^{1,\infty}$-norm. Inverse Problems & Imaging, 2015, 9 (4) : 1093-1137. doi: 10.3934/ipi.2015.9.1093
[2]

Seung-Yeal Ha, Jinwook Jung, Peter Kuchling. Emergence of anomalous flocking in the fractional Cucker-Smale model. Discrete & Continuous Dynamical Systems, 2019, 39 (9) : 5465-5489. doi: 10.3934/dcds.2019223
[3]

Ewa Girejko, Luís Machado, Agnieszka B. Malinowska, Natália Martins. On consensus in the Cucker–Smale type model on isolated time scales. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 77-89. doi: 10.3934/dcdss.2018005
[4]

Lei Wu, Zhe Sun. A new spectral method for $l_1$-regularized minimization. Inverse Problems & Imaging, 2015, 9 (1) : 257-272. doi: 10.3934/ipi.2015.9.257
[5]

Yingying Li, Stanley Osher, Richard Tsai. Heat source identification based on $l_1$ constrained minimization. Inverse Problems & Imaging, 2014, 8 (1) : 199-221. doi: 10.3934/ipi.2014.8.199
[6]

Gero Friesecke, Felix Henneke, Karl Kunisch. Frequency-sparse optimal quantum control. Mathematical Control & Related Fields, 2018, 8 (1) : 155-176. doi: 10.3934/mcrf.2018007
[7]

Agnieszka B. Malinowska, Tatiana Odzijewicz. Optimal control of the discrete-time fractional-order Cucker-Smale model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 347-357. doi: 10.3934/dcdsb.2018023
[8]

Seung-Yeal Ha, Shi Jin. Local sensitivity analysis for the Cucker-Smale model with random inputs. Kinetic & Related Models, 2018, 11 (4) : 859-889. doi: 10.3934/krm.2018034
[9]

Eduardo Casas, Fredi Tröltzsch. Sparse optimal control for the heat equation with mixed control-state constraints. Mathematical Control & Related Fields, 2020, 10 (3) : 471-491. doi: 10.3934/mcrf.2020007
[10]

Lidan Li, Hongwei Zhang, Liwei Zhang. Inverse quadratic programming problem with $ l_1 $ norm measure. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2425-2437. doi: 10.3934/jimo.2019061
[11]

Ying Lin, Rongrong Lin, Qi Ye. Sparse regularized learning in the reproducing kernel banach spaces with the $ \ell^1 $ norm. Mathematical Foundations of Computing, 2020, 3 (3) : 205-218. doi: 10.3934/mfc.2020020
[12]

Zhaohui Guo, Stanley Osher. Template matching via $l_1$ minimization and its application to hyperspectral data. Inverse Problems & Imaging, 2011, 5 (1) : 19-35. doi: 10.3934/ipi.2011.5.19
[13]

Jiying Liu, Jubo Zhu, Fengxia Yan, Zenghui Zhang. Compressive sampling and $l_1$ minimization for SAR imaging with low sampling rate. Inverse Problems & Imaging, 2013, 7 (4) : 1295-1305. doi: 10.3934/ipi.2013.7.1295
[14]

Vladimir Gaitsgory, Tanya Tarnopolskaya. Threshold value of the penalty parameter in the minimization of $L_1$-penalized conditional value-at-risk. Journal of Industrial & Management Optimization, 2013, 9 (1) : 191-204. doi: 10.3934/jimo.2013.9.191
[15]

Duo Wang, Zheng-Fen Jin, Youlin Shang. A penalty decomposition method for nuclear norm minimization with l1 norm fidelity term. Evolution Equations & Control Theory, 2019, 8 (4) : 695-708. doi: 10.3934/eect.2019034
[16]

Seung-Yeal Ha, Bora Moon. Quantitative local sensitivity estimates for the random kinetic Cucker-Smale model with chemotactic movement. Kinetic & Related Models, 2020, 13 (5) : 889-931. doi: 10.3934/krm.2020031
[17]

Moon-Jin Kang, Seung-Yeal Ha, Jeongho Kim, Woojoo Shim. Hydrodynamic limit of the kinetic thermomechanical Cucker-Smale model in a strong local alignment regime. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1233-1256. doi: 10.3934/cpaa.2020057
[18]

Jerzy Klamka, Helmut Maurer, Andrzej Swierniak. Local controllability and optimal control for\newline a model of combined anticancer therapy with control delays. Mathematical Biosciences & Engineering, 2017, 14 (1) : 195-216. doi: 10.3934/mbe.2017013
[19]

M. Sumon Hossain, M. Monir Uddin. Iterative methods for solving large sparse Lyapunov equations and application to model reduction of index 1 differential-algebraic-equations. Numerical Algebra, Control & Optimization, 2019, 9 (2) : 173-186. doi: 10.3934/naco.2019013
[20]

Lining Ru, Xiaoping Xue. Flocking of Cucker-Smale model with intrinsic dynamics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6817-6835. doi: 10.3934/dcdsb.2019168

2019 Impact Factor: 0.857

Tools

Metrics

  • PDF downloads (80)
  • HTML views (0)
  • Cited by (42)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences