
-
Previous Article
Balanced model order reduction for linear random dynamical systems driven by Lévy noise
- JCD Home
- This Issue
- Next Article
Optimal transport over nonlinear systems via infinitesimal generators on graphs
1. | School for Engineering of Matter, Transport and Energy, Arizona State University, Tempe, AZ 85281, USA |
2. | Mitsubishi Electric Research Labs, 201 Broadway, Cambridge, MA 02139, USA |
We present a set-oriented graph-based computational framework for continuous-time optimal transport over nonlinear dynamical systems. We recover provably optimal control laws for steering a given initial distribution in phase space to a final distribution in prescribed finite time for the case of non-autonomous nonlinear control-affine systems, while minimizing a quadratic control cost. The resulting control law can be used to obtain approximate feedback laws for individual agents in a swarm control application. Using infinitesimal generators, the optimal control problem is reduced to a modified Monge-Kantorovich optimal transport problem, resulting in a convex Benamou-Brenier type fluid dynamics formulation on a graph. The well-posedness of this problem is shown to be a consequence of the graph being strongly-connected, which in turn is shown to result from controllability of the underlying dynamical system. Using our computational framework, we study optimal transport of distributions where the underlying dynamical systems are chaotic, and non-holonomic. The solutions to the optimal transport problem elucidate the role played by invariant manifolds, lobe-dynamics and almost-invariant sets in efficient transport of distributions in finite time. Our work connects set-oriented operator-theoretic methods in dynamical systems with optimal mass transportation theory, and opens up new directions in design of efficient feedback control strategies for nonlinear multi-agent and swarm systems operating in nonlinear ambient flow fields.
References:
[1] |
Intel Inc, Intel Xeon Processor X5690 https://ark.intel.com/products/52576/Intel-Xeon-Processor-X5690-12M-Cache-3_46-GHz-6_40-GTs-Intel-QPI, [Online; accessed 19-March-2018]. Google Scholar |
[2] |
A. Agrachev and P. Lee,
Optimal transportation under nonholonomic constraints, Transactions of the American Mathematical Society, 361 (2009), 6019-6047.
doi: 10.1090/S0002-9947-09-04813-2. |
[3] |
A. A. Agrachev and Y. Sachkov, Control Theory from the Geometric Viewpoint, Springer-Verlag, Berlin, 2004.
doi: 10.1007/978-3-662-06404-7. |
[4] |
M. Aicardi, G. Casalino, A. Bicchi and A. Balestrino,
Closed loop steering of unicycle like vehicles via Lyapunov techniques, IEEE Robotics & Automation Magazine, 2 (1995), 27-35.
doi: 10.1109/100.388294. |
[5] |
L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows: In Metric Spaces and in the Space of Probability Measures, Springer Science & Business Media, 2008. |
[6] |
J.-D. Benamou and Y. Brenier,
A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numerische Mathematik, 84 (2000), 375-393.
doi: 10.1007/s002110050002. |
[7] |
S. Berman, Á. Halász, M. Ani Hsieh and V. Kumar,
Optimized stochastic policies for task allocation in swarms of robots, IEEE Transactions on Robotics, 25 (2009), 927-937.
doi: 10.1109/TRO.2009.2024997. |
[8] |
P. Bernard and B. Buffoni,
Optimal mass transportation and mather theory, J. Eur. Math. Soc. (JEMS), 9 (2007), 85-121, math/0412299.
doi: 10.4171/JEMS/74. |
[9] |
A. Bloch, J. Baillieul, P. Crouch, J. E. Marsden, P. S. Krishnaprasad, R. M. Murray and D. Zenkov, Nonholonomic Mechanics and Control, vol. 24, Springer, 2003.
doi: 10.1007/b97376. |
[10] |
E. M. Bollt and N. Santitissadeekorn, Applied and Computational Measurable Dynamics, vol. 18, SIAM, 2013.
doi: 10.1137/1.9781611972641. |
[11] |
P. Boyland, H. Aref and M. Stremler,
Topological fluid mechanics of stirring, Journal of Fluid Mechanics, 403 (2000), 277-304.
doi: 10.1017/S0022112099007107. |
[12] |
M. Budišić, R. Mohr and I. Mezić, Applied koopmanism, Chaos: An Interdisciplinary Journal of Nonlinear Science, 22 (2012), 047510, 33pp.
doi: 10.1063/1.4772195. |
[13] |
A. Chapman and M. Mesbahi,
Advection on graphs, 2011 50th IEEE Conference on Decision and Control and European Control Conference, IEEE, (2011), 1461-1466.
doi: 10.1109/CDC.2011.6161471. |
[14] |
U. K. Cheang, K. Lee, A. A. Julius and M. J. Kim, Multiple-robot drug delivery strategy through coordinated teams of microswimmers, Applied Physics Letters, 105 (2014), 083705.
doi: 10.1063/1.4893695. |
[15] |
Y. Chen, T. T. Georgiou and M. Pavon,
On the relation between optimal transport and Schrödinger bridges: A stochastic control viewpoint, Journal of Optimization Theory and Applications, 169 (2016), 671-691.
doi: 10.1007/s10957-015-0803-z. |
[16] |
____,
Optimal transport over a linear dynamical system, IEEE Transactions on Automatic Control, 62 (2017), 2137-2152.
doi: 10.1109/TAC.2016.2602103. |
[17] |
M. Dellnitz, G. Froyland and O. Junge, The algorithms behind GAIO - set oriented numerical methods for dynamical systems, Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems (B. Fiedler, ed. ), Springer, Berlin-Heidelberg-New York, 2001,145-174. |
[18] |
M. Dellnitz, O. Junge, W. S. Koon, F. Lekien, M. W. Lo, J. E. Marsden, K. Padberg, R. Preis, S. D. Ross and B. Thiere,
Transport in dynamical astronomy and multibody problems, Int. J. Bifurc. Chaos, 15 (2005), 699-727.
doi: 10.1142/S0218127405012545. |
[19] |
M. Dellnitz and O. Junge,
On the approximation of complicated dynamical behavior, SIAM Journal on Numerical Analysis, 36 (1998), 491-515.
doi: 10.1137/S0036142996313002. |
[20] |
J. Eckstein and W. Yao, Augmented Lagrangian and alternating direction methods for convex optimization: A tutorial and some illustrative computational results, RUTCOR Research Reports, 32 (2012). Google Scholar |
[21] |
K. Elamvazhuthi and S. Berman,
Optimal control of stochastic coverage strategies for robotic swarms, 2015 IEEE International Conference on Robotics and Automation (ICRA), IEEE, (2015), 1822-1829.
doi: 10.1109/ICRA.2015.7139435. |
[22] |
K. Elamvazhuthi, V. Deshmukh, M. Kawski and S. Berman, Mean-field controllability and decentralized stabilization of Markov chains, part ⅰ: Global controllability and rational feedbacks, arXiv preprint, arXiv: 1703.08243, (2017). Google Scholar |
[23] |
D. L. Elliott, Bilinear Control Systems: Matrices in Action, vol. 169, Springer Science & Business Media, 2009.
doi: 10.1023/b101451. |
[24] |
K.-J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations vol. 194, Springer-Verlag, New York, 2000. |
[25] |
H. O. Fattorini, The Cauchy Problem, vol. 13517, Encyclopedia of Mathematics and its Applications, 18. Addison-Wesley Publishing Co., Reading, Mass., 1983. |
[26] |
A. Figalli and L. Rifford,
Mass transportation on sub-riemannian manifolds, Geometric And Functional Analysis, 20 (2010), 124-159.
doi: 10.1007/s00039-010-0053-z. |
[27] |
M. D. Finn and J.-L. Thiffeault,
Topological optimization of rod-stirring devices, SIAM Review, 53 (2011), 723-743.
doi: 10.1137/100791828. |
[28] |
G. Froyland and M. Dellnitz,
Detecting and locating near-optimal almost-invariant sets and cycles, SIAM Journal on Scientific Computing, 24 (2003), 1839-1863.
doi: 10.1137/S106482750238911X. |
[29] |
G. Froyland and K. Padberg,
Almost-invariant sets and invariant manifolds - connecting probabilistic and geometric descriptions of coherent structures in flows, Physica D, 238 (2009), 1507-1523.
doi: 10.1016/j.physd.2009.03.002. |
[30] |
G. Froyland, N. Santitissadeekorn and A. Monahan, Transport in time-dependent dynamical systems: Finite-time coherent sets, Chaos, 20 (2010), 043116, 10pp.
doi: 10.1063/1.3502450. |
[31] |
G. Froyland,
Dynamic isoperimetry and the geometry of Lagrangian coherent structures, Nonlinearity, 28 (2015), 3587-3622.
doi: 10.1088/0951-7715/28/10/3587. |
[32] |
G. Froyland, C. González-Tokman and T. M. Watson,
Optimal mixing enhancement by local perturbation, SIAM Review, 58 (2016), 494-513.
doi: 10.1137/15M1023221. |
[33] |
G. Froyland, O. Junge and P. Koltai,
Estimating long-term behavior of flows without trajectory integration: The infinitesimal generator approach, SIAM Journal on Numerical Analysis, 51 (2013), 223-247.
doi: 10.1137/110819986. |
[34] |
G. Froyland and N. Santitissadeekorn, Optimal mixing enhancement, SIAM J. Appl. Math., 77 (2017), 1444-1470, arXiv: 1610.01651.
doi: 10.1137/16M1091496. |
[35] |
A. Ghosh and P. Fischer,
Controlled propulsion of artificial magnetic nanostructured propellers, Nano letters, 9 (2009), 2243-2245.
doi: 10.1021/nl900186w. |
[36] |
N. Gigli and J. Maas,
Gromov-hausdorff convergence of discrete transportation metrics, SIAM Journal on Mathematical Analysis, 45 (2013), 879-899.
doi: 10.1137/120886315. |
[37] |
R. Gilmore,
Topological analysis of chaotic dynamical systems, Reviews of Modern Physics, 70 (1998), 1455-1529.
doi: 10.1103/RevModPhys.70.1455. |
[38] |
M. Grant, S. Boyd and Y. Ye, CVX: Matlab software for disciplined convex programming, Global Optimization, Nonconvex Optim. Appl., Springer, New York, 84 (2006), 155-210.
doi: 10.1007/0-387-30528-9_7. |
[39] |
P. Grover and K. Elamvazhuthi,
Optimal perturbations for nonlinear systems using graph-based optimal transport, Communications in Nonlinear Science and Numerical Simulation, 59 (2018), 197-215.
doi: 10.1016/j.cnsns.2017.09.020. |
[40] |
P. Grover, S. D. Ross, M. A. Stremler and P. Kumar, Topological chaos, braiding and bifurcation of almost-cyclic sets, Chaos: An Interdisciplinary Journal of Nonlinear Science, 22 (2012), 043135, 16pp.
doi: 10.1063/1.4768666. |
[41] |
G. Haller,
Lagrangian coherent structures, Annual Review of Fluid Mechanics, 47 (2015), 137-162.
|
[42] |
D. Damir Harabor, A. Grastien, et al., Online graph pruning for pathfinding on grid maps, AAAI, 2011. Google Scholar |
[43] |
A. Hindawi, J.-B. Pomet and L. Rifford,
Mass transportation with LQ cost functions, Acta Applicandae Mathematicae, 113 (2011), 215-229.
doi: 10.1007/s10440-010-9595-1. |
[44] |
S. Jerg, O. Junge and M. Post,
Global optimal feedbacks for stochastic quantized nonlinear event systems, AIMS, 1 (2014), 163-176.
doi: 10.3934/jcd.2014.1.163. |
[45] |
O. Junge and H. M. Osinga,
A set oriented approach to global optimal control, ESAIM: Control, Optimisation and Calculus of Variations, 10 (2004), 259-270.
doi: 10.1051/cocv:2004006. |
[46] |
E. W. Justh and P. S. Krishnaprasad, Optimality, reduction and collective motion, Proc. R. Soc. A, The Royal Society, 471 (2015), 20140606, 22pp.
doi: 10.1098/rspa.2014.0606. |
[47] |
E. Kirillova and K. Spindler,
Optimal control of a vehicle during parking, 2004 IFAC Symposium on Nonlinear Control Systems (NOLCOS), 37 (2004), 925-930.
doi: 10.1016/S1474-6670(17)31344-7. |
[48] |
W. S. Koon, M. Lo, J. E. Marsden and S. D. Ross, Dynamical Systems, The Three-Body Problem and Space Mission Design, Marsden Books, 2008. Google Scholar |
[49] |
P. S. Krishnaprasad, Optimal Control and Poisson Reduction, Tech. report, DTIC Document, 1993. Google Scholar |
[50] |
A. Lasota and M. Mackey, Chaos, Fractals and Noise, Springer-Verlag, New York, 1994.
doi: 10.1007/978-1-4612-4286-4. |
[51] |
J. B. Lasserre, D. Henrion, C. Prieur and E. Trélat,
Nonlinear optimal control via occupation measures and LMI-relaxations, SIAM Journal on Control and Optimization, 47 (2008), 1643-1666.
doi: 10.1137/070685051. |
[52] |
F. Lekien and C. Coulliette,
Chaotic stirring in quasi-turbulent flows, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 365 (2007), 3061-3084.
doi: 10.1098/rsta.2007.0020. |
[53] |
P. F. J. Lermusiaux, T. Lolla, P. J. Haley Jr, K. Yigit, M. P. Ueckermann, T. Sondergaard and W. G. Leslie,
Science of Autonomy: Time-optimal Path Planning and Adaptive Sampling for Swarms of Ocean Vehicles, Springer Handbook of Ocean Engineering: Autonomous Ocean Vehicles, Subsystems and Control, (2015), 481-498.
doi: 10.1007/978-3-319-16649-0_21. |
[54] |
J. Maas,
Gradient flows of the entropy for finite markov chains, Journal of Functional Analysis, 261 (2011), 2250-2292.
doi: 10.1016/j.jfa.2011.06.009. |
[55] |
J. D. Meiss,
Symplectic maps, variational principles, and transport, Rev. Mod. Phys., 64 (1992), 795-848.
doi: 10.1103/RevModPhys.64.795. |
[56] |
I. Mezic,
Analysis of fluid flows via spectral properties of the Koopman operator, Annual Review of Fluid Mechanics, 45 (2013), 357-378.
doi: 10.1146/annurev-fluid-011212-140652. |
[57] |
A. Mielke,
Geodesic convexity of the relative entropy in reversible markov chains, Calculus of Variations and Partial Differential Equations, 48 (2013), 1-31.
doi: 10.1007/s00526-012-0538-8. |
[58] |
R. M. Murray and S. S. Sastry,
Nonholonomic motion planning: Steering using sinusoids, IEEE Transactions on Automatic Control, 38 (1993), 700-716.
doi: 10.1109/9.277235. |
[59] |
B. O'Donoghue, E. Chu, N. Parikh and S. Boyd, Operator splitting for conic optimization via homogeneous self-dual embedding, Journal of Optimization Theory and Applications, 159 (2016), 1042-1068, arXiv: 1312.3039.
doi: 10.1007/s10957-016-0892-3. |
[60] |
J. M. Ottino and S. Wiggins,
Introduction: Mixing in microfluidics, Philosophical Transactions: Mathematical, Physical and Engineering Sciences, 362 (2004), 923-935.
doi: 10.1098/rsta.2003.1355. |
[61] |
N. Papadakis, G. Peyré and E. Oudet,
Optimal transport with proximal splitting, SIAM Journal on Imaging Sciences, 7 (2014), 212-238.
doi: 10.1137/130920058. |
[62] |
K. E. Peyer, L. Zhang and B. J. Nelson,
Bio-inspired magnetic swimming microrobots for biomedical applications, Nanoscale, 5 (2013), 1259-1272.
doi: 10.1039/C2NR32554C. |
[63] |
H. Poincaré, Les Méthodes Nouvelles de la Mécanique Céleste: Méthodes de mm. Newcomb, Glydén, Lindstedt et Bohlin, (French) Dover Publications, Inc., New York, N. Y., 1957. |
[64] |
A. C. Poje and G. Haller,
Geometry of cross-stream mixing in a double-gyre ocean model, Journal of Physical Oceanography, 29 (1999), 1649-1665.
doi: 10.1175/1520-0485(1999)029<1649:GOCSMI>2.0.CO;2. |
[65] |
A. Raghunathan and U. Vaidya,
Optimal stabilization using Lyapunov measures, IEEE Transactions on Automatic Control, 59 (2014), 1316-1321.
doi: 10.1109/TAC.2013.2289707. |
[66] |
L. Rifford, Sub-riemannian Geometry and Optimal Transport, Springer, 2014.
doi: 10.1007/978-3-319-04804-8. |
[67] |
S. D Ross, S. Jerg and O. Junge,
Optimal capture trajectories using multiple gravity assists, Communications in Nonlinear Science and Numerical Simulations, 14 (2009), 4168-4175.
doi: 10.1016/j.cnsns.2008.12.009. |
[68] |
J. Sabuco, M. A. F. Sanjuán and J. A. Yorke, Dynamics of partial control, Chaos, 22 (2012), 047507, 9pp.
doi: 10.1063/1.4754874. |
[69] |
C. Senatore and S. D. Ross,
Fuel-efficient navigation in complex flows, 2008 American Control Conference, IEEE, (2008), 1244-1248.
doi: 10.1109/ACC.2008.4586663. |
[70] |
J. Solomon, R. Rustamov, L. Guibas and A. Butscher, Continuous-flow graph transportation distances, arXiv preprint, arXiv: 1603.06927, (2016). Google Scholar |
[71] |
M. A. Stremler, S. D. Ross, P. Grover and P. Kumar, Topological chaos and periodic braiding of almost-cyclic sets, Physical Review Letters, 106 (2011), 114101. Google Scholar |
[72] |
H. J Sussmann,
A general theorem on local controllability, SIAM Journal on Control and Optimization, 25 (1987), 158-194.
doi: 10.1137/0325011. |
[73] |
P. Tallapragada and S. D. Ross,
A set oriented definition of finite-time Lyapunov exponents and coherent sets, Communications in Nonlinear Science and Numerical Simulation, 18 (2013), 1106-1126.
doi: 10.1016/j.cnsns.2012.09.017. |
[74] |
J.-L. Thiffeault and M. D. Finn,
Topology, braids and mixing in fluids, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 364 (2006), 3251-3266.
doi: 10.1098/rsta.2006.1899. |
[75] |
S. M. Ulam, Problems in Modern Mathematics, Science Editions John Wiley & Sons, Inc., New York, 1964. |
[76] |
U. Vaidya and P. G. Mehta,
Lyapunov measure for almost everywhere stability, IEEE Transactions on Automatic Control, 53 (2008), 307-323.
doi: 10.1109/TAC.2007.914955. |
[77] |
U. Vaidya, P. G. Mehta and U. V. Shanbhag,
Nonlinear stabilization via control Lyapunov measure, IEEE Transactions on Automatic Control, 55 (2010), 1314-1328.
doi: 10.1109/TAC.2010.2042226. |
[78] |
U. Vaidya and I. Mezić,
Controllability for a class of area-preserving twist maps, Physica D: Nonlinear Phenomena, 189 (2004), 234-246.
doi: 10.1016/j.physd.2003.10.008. |
[79] |
D. L. Vainchtein, A. I. Neishtadt and I. Mezic, On passage through resonances in volume-preserving systems, Chaos: An Interdisciplinary Journal of Nonlinear Science, 16 (2006), 043123, 11pp.
doi: 10.1063/1.2404585. |
[80] |
C. Villani, Topics in Optimal Transportation no. 58, American Mathematical Society, 2003.
doi: 10.1007/b12016. |
[81] |
S. Wiggins,
The dynamical systems approach to Lagrangian transport in oceanic flows, Annu. Rev. Fluid Mech., 37 (2005), 295-328.
doi: 10.1146/annurev.fluid.37.061903.175815. |
[82] |
____, Chaotic Transport in Dynamical Systems, vol. 2, Springer Science & Business Media, 2013. Google Scholar |
[83] |
S. Wiggins and J. M. Ottino,
Foundations of chaotic mixing, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 362 (2004), 937-970.
doi: 10.1098/rsta.2003.1356. |
[84] |
R. Wood, R. Nagpal and G.-Y. Wei,
Flight of the robobees, Scientific American, 308 (2013), 60-65.
doi: 10.1038/scientificamerican0313-60. |
show all references
References:
[1] |
Intel Inc, Intel Xeon Processor X5690 https://ark.intel.com/products/52576/Intel-Xeon-Processor-X5690-12M-Cache-3_46-GHz-6_40-GTs-Intel-QPI, [Online; accessed 19-March-2018]. Google Scholar |
[2] |
A. Agrachev and P. Lee,
Optimal transportation under nonholonomic constraints, Transactions of the American Mathematical Society, 361 (2009), 6019-6047.
doi: 10.1090/S0002-9947-09-04813-2. |
[3] |
A. A. Agrachev and Y. Sachkov, Control Theory from the Geometric Viewpoint, Springer-Verlag, Berlin, 2004.
doi: 10.1007/978-3-662-06404-7. |
[4] |
M. Aicardi, G. Casalino, A. Bicchi and A. Balestrino,
Closed loop steering of unicycle like vehicles via Lyapunov techniques, IEEE Robotics & Automation Magazine, 2 (1995), 27-35.
doi: 10.1109/100.388294. |
[5] |
L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows: In Metric Spaces and in the Space of Probability Measures, Springer Science & Business Media, 2008. |
[6] |
J.-D. Benamou and Y. Brenier,
A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numerische Mathematik, 84 (2000), 375-393.
doi: 10.1007/s002110050002. |
[7] |
S. Berman, Á. Halász, M. Ani Hsieh and V. Kumar,
Optimized stochastic policies for task allocation in swarms of robots, IEEE Transactions on Robotics, 25 (2009), 927-937.
doi: 10.1109/TRO.2009.2024997. |
[8] |
P. Bernard and B. Buffoni,
Optimal mass transportation and mather theory, J. Eur. Math. Soc. (JEMS), 9 (2007), 85-121, math/0412299.
doi: 10.4171/JEMS/74. |
[9] |
A. Bloch, J. Baillieul, P. Crouch, J. E. Marsden, P. S. Krishnaprasad, R. M. Murray and D. Zenkov, Nonholonomic Mechanics and Control, vol. 24, Springer, 2003.
doi: 10.1007/b97376. |
[10] |
E. M. Bollt and N. Santitissadeekorn, Applied and Computational Measurable Dynamics, vol. 18, SIAM, 2013.
doi: 10.1137/1.9781611972641. |
[11] |
P. Boyland, H. Aref and M. Stremler,
Topological fluid mechanics of stirring, Journal of Fluid Mechanics, 403 (2000), 277-304.
doi: 10.1017/S0022112099007107. |
[12] |
M. Budišić, R. Mohr and I. Mezić, Applied koopmanism, Chaos: An Interdisciplinary Journal of Nonlinear Science, 22 (2012), 047510, 33pp.
doi: 10.1063/1.4772195. |
[13] |
A. Chapman and M. Mesbahi,
Advection on graphs, 2011 50th IEEE Conference on Decision and Control and European Control Conference, IEEE, (2011), 1461-1466.
doi: 10.1109/CDC.2011.6161471. |
[14] |
U. K. Cheang, K. Lee, A. A. Julius and M. J. Kim, Multiple-robot drug delivery strategy through coordinated teams of microswimmers, Applied Physics Letters, 105 (2014), 083705.
doi: 10.1063/1.4893695. |
[15] |
Y. Chen, T. T. Georgiou and M. Pavon,
On the relation between optimal transport and Schrödinger bridges: A stochastic control viewpoint, Journal of Optimization Theory and Applications, 169 (2016), 671-691.
doi: 10.1007/s10957-015-0803-z. |
[16] |
____,
Optimal transport over a linear dynamical system, IEEE Transactions on Automatic Control, 62 (2017), 2137-2152.
doi: 10.1109/TAC.2016.2602103. |
[17] |
M. Dellnitz, G. Froyland and O. Junge, The algorithms behind GAIO - set oriented numerical methods for dynamical systems, Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems (B. Fiedler, ed. ), Springer, Berlin-Heidelberg-New York, 2001,145-174. |
[18] |
M. Dellnitz, O. Junge, W. S. Koon, F. Lekien, M. W. Lo, J. E. Marsden, K. Padberg, R. Preis, S. D. Ross and B. Thiere,
Transport in dynamical astronomy and multibody problems, Int. J. Bifurc. Chaos, 15 (2005), 699-727.
doi: 10.1142/S0218127405012545. |
[19] |
M. Dellnitz and O. Junge,
On the approximation of complicated dynamical behavior, SIAM Journal on Numerical Analysis, 36 (1998), 491-515.
doi: 10.1137/S0036142996313002. |
[20] |
J. Eckstein and W. Yao, Augmented Lagrangian and alternating direction methods for convex optimization: A tutorial and some illustrative computational results, RUTCOR Research Reports, 32 (2012). Google Scholar |
[21] |
K. Elamvazhuthi and S. Berman,
Optimal control of stochastic coverage strategies for robotic swarms, 2015 IEEE International Conference on Robotics and Automation (ICRA), IEEE, (2015), 1822-1829.
doi: 10.1109/ICRA.2015.7139435. |
[22] |
K. Elamvazhuthi, V. Deshmukh, M. Kawski and S. Berman, Mean-field controllability and decentralized stabilization of Markov chains, part ⅰ: Global controllability and rational feedbacks, arXiv preprint, arXiv: 1703.08243, (2017). Google Scholar |
[23] |
D. L. Elliott, Bilinear Control Systems: Matrices in Action, vol. 169, Springer Science & Business Media, 2009.
doi: 10.1023/b101451. |
[24] |
K.-J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations vol. 194, Springer-Verlag, New York, 2000. |
[25] |
H. O. Fattorini, The Cauchy Problem, vol. 13517, Encyclopedia of Mathematics and its Applications, 18. Addison-Wesley Publishing Co., Reading, Mass., 1983. |
[26] |
A. Figalli and L. Rifford,
Mass transportation on sub-riemannian manifolds, Geometric And Functional Analysis, 20 (2010), 124-159.
doi: 10.1007/s00039-010-0053-z. |
[27] |
M. D. Finn and J.-L. Thiffeault,
Topological optimization of rod-stirring devices, SIAM Review, 53 (2011), 723-743.
doi: 10.1137/100791828. |
[28] |
G. Froyland and M. Dellnitz,
Detecting and locating near-optimal almost-invariant sets and cycles, SIAM Journal on Scientific Computing, 24 (2003), 1839-1863.
doi: 10.1137/S106482750238911X. |
[29] |
G. Froyland and K. Padberg,
Almost-invariant sets and invariant manifolds - connecting probabilistic and geometric descriptions of coherent structures in flows, Physica D, 238 (2009), 1507-1523.
doi: 10.1016/j.physd.2009.03.002. |
[30] |
G. Froyland, N. Santitissadeekorn and A. Monahan, Transport in time-dependent dynamical systems: Finite-time coherent sets, Chaos, 20 (2010), 043116, 10pp.
doi: 10.1063/1.3502450. |
[31] |
G. Froyland,
Dynamic isoperimetry and the geometry of Lagrangian coherent structures, Nonlinearity, 28 (2015), 3587-3622.
doi: 10.1088/0951-7715/28/10/3587. |
[32] |
G. Froyland, C. González-Tokman and T. M. Watson,
Optimal mixing enhancement by local perturbation, SIAM Review, 58 (2016), 494-513.
doi: 10.1137/15M1023221. |
[33] |
G. Froyland, O. Junge and P. Koltai,
Estimating long-term behavior of flows without trajectory integration: The infinitesimal generator approach, SIAM Journal on Numerical Analysis, 51 (2013), 223-247.
doi: 10.1137/110819986. |
[34] |
G. Froyland and N. Santitissadeekorn, Optimal mixing enhancement, SIAM J. Appl. Math., 77 (2017), 1444-1470, arXiv: 1610.01651.
doi: 10.1137/16M1091496. |
[35] |
A. Ghosh and P. Fischer,
Controlled propulsion of artificial magnetic nanostructured propellers, Nano letters, 9 (2009), 2243-2245.
doi: 10.1021/nl900186w. |
[36] |
N. Gigli and J. Maas,
Gromov-hausdorff convergence of discrete transportation metrics, SIAM Journal on Mathematical Analysis, 45 (2013), 879-899.
doi: 10.1137/120886315. |
[37] |
R. Gilmore,
Topological analysis of chaotic dynamical systems, Reviews of Modern Physics, 70 (1998), 1455-1529.
doi: 10.1103/RevModPhys.70.1455. |
[38] |
M. Grant, S. Boyd and Y. Ye, CVX: Matlab software for disciplined convex programming, Global Optimization, Nonconvex Optim. Appl., Springer, New York, 84 (2006), 155-210.
doi: 10.1007/0-387-30528-9_7. |
[39] |
P. Grover and K. Elamvazhuthi,
Optimal perturbations for nonlinear systems using graph-based optimal transport, Communications in Nonlinear Science and Numerical Simulation, 59 (2018), 197-215.
doi: 10.1016/j.cnsns.2017.09.020. |
[40] |
P. Grover, S. D. Ross, M. A. Stremler and P. Kumar, Topological chaos, braiding and bifurcation of almost-cyclic sets, Chaos: An Interdisciplinary Journal of Nonlinear Science, 22 (2012), 043135, 16pp.
doi: 10.1063/1.4768666. |
[41] |
G. Haller,
Lagrangian coherent structures, Annual Review of Fluid Mechanics, 47 (2015), 137-162.
|
[42] |
D. Damir Harabor, A. Grastien, et al., Online graph pruning for pathfinding on grid maps, AAAI, 2011. Google Scholar |
[43] |
A. Hindawi, J.-B. Pomet and L. Rifford,
Mass transportation with LQ cost functions, Acta Applicandae Mathematicae, 113 (2011), 215-229.
doi: 10.1007/s10440-010-9595-1. |
[44] |
S. Jerg, O. Junge and M. Post,
Global optimal feedbacks for stochastic quantized nonlinear event systems, AIMS, 1 (2014), 163-176.
doi: 10.3934/jcd.2014.1.163. |
[45] |
O. Junge and H. M. Osinga,
A set oriented approach to global optimal control, ESAIM: Control, Optimisation and Calculus of Variations, 10 (2004), 259-270.
doi: 10.1051/cocv:2004006. |
[46] |
E. W. Justh and P. S. Krishnaprasad, Optimality, reduction and collective motion, Proc. R. Soc. A, The Royal Society, 471 (2015), 20140606, 22pp.
doi: 10.1098/rspa.2014.0606. |
[47] |
E. Kirillova and K. Spindler,
Optimal control of a vehicle during parking, 2004 IFAC Symposium on Nonlinear Control Systems (NOLCOS), 37 (2004), 925-930.
doi: 10.1016/S1474-6670(17)31344-7. |
[48] |
W. S. Koon, M. Lo, J. E. Marsden and S. D. Ross, Dynamical Systems, The Three-Body Problem and Space Mission Design, Marsden Books, 2008. Google Scholar |
[49] |
P. S. Krishnaprasad, Optimal Control and Poisson Reduction, Tech. report, DTIC Document, 1993. Google Scholar |
[50] |
A. Lasota and M. Mackey, Chaos, Fractals and Noise, Springer-Verlag, New York, 1994.
doi: 10.1007/978-1-4612-4286-4. |
[51] |
J. B. Lasserre, D. Henrion, C. Prieur and E. Trélat,
Nonlinear optimal control via occupation measures and LMI-relaxations, SIAM Journal on Control and Optimization, 47 (2008), 1643-1666.
doi: 10.1137/070685051. |
[52] |
F. Lekien and C. Coulliette,
Chaotic stirring in quasi-turbulent flows, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 365 (2007), 3061-3084.
doi: 10.1098/rsta.2007.0020. |
[53] |
P. F. J. Lermusiaux, T. Lolla, P. J. Haley Jr, K. Yigit, M. P. Ueckermann, T. Sondergaard and W. G. Leslie,
Science of Autonomy: Time-optimal Path Planning and Adaptive Sampling for Swarms of Ocean Vehicles, Springer Handbook of Ocean Engineering: Autonomous Ocean Vehicles, Subsystems and Control, (2015), 481-498.
doi: 10.1007/978-3-319-16649-0_21. |
[54] |
J. Maas,
Gradient flows of the entropy for finite markov chains, Journal of Functional Analysis, 261 (2011), 2250-2292.
doi: 10.1016/j.jfa.2011.06.009. |
[55] |
J. D. Meiss,
Symplectic maps, variational principles, and transport, Rev. Mod. Phys., 64 (1992), 795-848.
doi: 10.1103/RevModPhys.64.795. |
[56] |
I. Mezic,
Analysis of fluid flows via spectral properties of the Koopman operator, Annual Review of Fluid Mechanics, 45 (2013), 357-378.
doi: 10.1146/annurev-fluid-011212-140652. |
[57] |
A. Mielke,
Geodesic convexity of the relative entropy in reversible markov chains, Calculus of Variations and Partial Differential Equations, 48 (2013), 1-31.
doi: 10.1007/s00526-012-0538-8. |
[58] |
R. M. Murray and S. S. Sastry,
Nonholonomic motion planning: Steering using sinusoids, IEEE Transactions on Automatic Control, 38 (1993), 700-716.
doi: 10.1109/9.277235. |
[59] |
B. O'Donoghue, E. Chu, N. Parikh and S. Boyd, Operator splitting for conic optimization via homogeneous self-dual embedding, Journal of Optimization Theory and Applications, 159 (2016), 1042-1068, arXiv: 1312.3039.
doi: 10.1007/s10957-016-0892-3. |
[60] |
J. M. Ottino and S. Wiggins,
Introduction: Mixing in microfluidics, Philosophical Transactions: Mathematical, Physical and Engineering Sciences, 362 (2004), 923-935.
doi: 10.1098/rsta.2003.1355. |
[61] |
N. Papadakis, G. Peyré and E. Oudet,
Optimal transport with proximal splitting, SIAM Journal on Imaging Sciences, 7 (2014), 212-238.
doi: 10.1137/130920058. |
[62] |
K. E. Peyer, L. Zhang and B. J. Nelson,
Bio-inspired magnetic swimming microrobots for biomedical applications, Nanoscale, 5 (2013), 1259-1272.
doi: 10.1039/C2NR32554C. |
[63] |
H. Poincaré, Les Méthodes Nouvelles de la Mécanique Céleste: Méthodes de mm. Newcomb, Glydén, Lindstedt et Bohlin, (French) Dover Publications, Inc., New York, N. Y., 1957. |
[64] |
A. C. Poje and G. Haller,
Geometry of cross-stream mixing in a double-gyre ocean model, Journal of Physical Oceanography, 29 (1999), 1649-1665.
doi: 10.1175/1520-0485(1999)029<1649:GOCSMI>2.0.CO;2. |
[65] |
A. Raghunathan and U. Vaidya,
Optimal stabilization using Lyapunov measures, IEEE Transactions on Automatic Control, 59 (2014), 1316-1321.
doi: 10.1109/TAC.2013.2289707. |
[66] |
L. Rifford, Sub-riemannian Geometry and Optimal Transport, Springer, 2014.
doi: 10.1007/978-3-319-04804-8. |
[67] |
S. D Ross, S. Jerg and O. Junge,
Optimal capture trajectories using multiple gravity assists, Communications in Nonlinear Science and Numerical Simulations, 14 (2009), 4168-4175.
doi: 10.1016/j.cnsns.2008.12.009. |
[68] |
J. Sabuco, M. A. F. Sanjuán and J. A. Yorke, Dynamics of partial control, Chaos, 22 (2012), 047507, 9pp.
doi: 10.1063/1.4754874. |
[69] |
C. Senatore and S. D. Ross,
Fuel-efficient navigation in complex flows, 2008 American Control Conference, IEEE, (2008), 1244-1248.
doi: 10.1109/ACC.2008.4586663. |
[70] |
J. Solomon, R. Rustamov, L. Guibas and A. Butscher, Continuous-flow graph transportation distances, arXiv preprint, arXiv: 1603.06927, (2016). Google Scholar |
[71] |
M. A. Stremler, S. D. Ross, P. Grover and P. Kumar, Topological chaos and periodic braiding of almost-cyclic sets, Physical Review Letters, 106 (2011), 114101. Google Scholar |
[72] |
H. J Sussmann,
A general theorem on local controllability, SIAM Journal on Control and Optimization, 25 (1987), 158-194.
doi: 10.1137/0325011. |
[73] |
P. Tallapragada and S. D. Ross,
A set oriented definition of finite-time Lyapunov exponents and coherent sets, Communications in Nonlinear Science and Numerical Simulation, 18 (2013), 1106-1126.
doi: 10.1016/j.cnsns.2012.09.017. |
[74] |
J.-L. Thiffeault and M. D. Finn,
Topology, braids and mixing in fluids, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 364 (2006), 3251-3266.
doi: 10.1098/rsta.2006.1899. |
[75] |
S. M. Ulam, Problems in Modern Mathematics, Science Editions John Wiley & Sons, Inc., New York, 1964. |
[76] |
U. Vaidya and P. G. Mehta,
Lyapunov measure for almost everywhere stability, IEEE Transactions on Automatic Control, 53 (2008), 307-323.
doi: 10.1109/TAC.2007.914955. |
[77] |
U. Vaidya, P. G. Mehta and U. V. Shanbhag,
Nonlinear stabilization via control Lyapunov measure, IEEE Transactions on Automatic Control, 55 (2010), 1314-1328.
doi: 10.1109/TAC.2010.2042226. |
[78] |
U. Vaidya and I. Mezić,
Controllability for a class of area-preserving twist maps, Physica D: Nonlinear Phenomena, 189 (2004), 234-246.
doi: 10.1016/j.physd.2003.10.008. |
[79] |
D. L. Vainchtein, A. I. Neishtadt and I. Mezic, On passage through resonances in volume-preserving systems, Chaos: An Interdisciplinary Journal of Nonlinear Science, 16 (2006), 043123, 11pp.
doi: 10.1063/1.2404585. |
[80] |
C. Villani, Topics in Optimal Transportation no. 58, American Mathematical Society, 2003.
doi: 10.1007/b12016. |
[81] |
S. Wiggins,
The dynamical systems approach to Lagrangian transport in oceanic flows, Annu. Rev. Fluid Mech., 37 (2005), 295-328.
doi: 10.1146/annurev.fluid.37.061903.175815. |
[82] |
____, Chaotic Transport in Dynamical Systems, vol. 2, Springer Science & Business Media, 2013. Google Scholar |
[83] |
S. Wiggins and J. M. Ottino,
Foundations of chaotic mixing, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 362 (2004), 937-970.
doi: 10.1098/rsta.2003.1356. |
[84] |
R. Wood, R. Nagpal and G.-Y. Wei,
Flight of the robobees, Scientific American, 308 (2013), 60-65.
doi: 10.1038/scientificamerican0313-60. |











[1] |
Hong Niu, Zhijiang Feng, Qijin Xiao, Yajun Zhang. A PID control method based on optimal control strategy. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 117-126. doi: 10.3934/naco.2020019 |
[2] |
Simone Fiori. Error-based control systems on Riemannian state manifolds: Properties of the principal pushforward map associated to parallel transport. Mathematical Control & Related Fields, 2021, 11 (1) : 143-167. doi: 10.3934/mcrf.2020031 |
[3] |
Claudia Lederman, Noemi Wolanski. An optimization problem with volume constraint for an inhomogeneous operator with nonstandard growth. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020391 |
[4] |
Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020 doi: 10.3934/mcrf.2020046 |
[5] |
Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020107 |
[6] |
Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Fractional optimal control problems on a star graph: Optimality system and numerical solution. Mathematical Control & Related Fields, 2021, 11 (1) : 189-209. doi: 10.3934/mcrf.2020033 |
[7] |
Christian Clason, Vu Huu Nhu, Arnd Rösch. Optimal control of a non-smooth quasilinear elliptic equation. Mathematical Control & Related Fields, 2020 doi: 10.3934/mcrf.2020052 |
[8] |
Hongbo Guan, Yong Yang, Huiqing Zhu. A nonuniform anisotropic FEM for elliptic boundary layer optimal control problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1711-1722. doi: 10.3934/dcdsb.2020179 |
[9] |
Guo Zhou, Yongquan Zhou, Ruxin Zhao. Hybrid social spider optimization algorithm with differential mutation operator for the job-shop scheduling problem. Journal of Industrial & Management Optimization, 2021, 17 (2) : 533-548. doi: 10.3934/jimo.2019122 |
[10] |
Yuan Tan, Qingyuan Cao, Lan Li, Tianshi Hu, Min Su. A chance-constrained stochastic model predictive control problem with disturbance feedback. Journal of Industrial & Management Optimization, 2021, 17 (1) : 67-79. doi: 10.3934/jimo.2019099 |
[11] |
Xin Zhang, Jie Xiong, Shuaiqi Zhang. Optimal reinsurance-investment and dividends problem with fixed transaction costs. Journal of Industrial & Management Optimization, 2021, 17 (2) : 981-999. doi: 10.3934/jimo.2020008 |
[12] |
Youming Guo, Tingting Li. Optimal control strategies for an online game addiction model with low and high risk exposure. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020347 |
[13] |
Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels. Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 243-271. doi: 10.3934/dcdss.2020213 |
[14] |
Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020110 |
[15] |
Elimhan N. Mahmudov. Infimal convolution and duality in convex optimal control problems with second order evolution differential inclusions. Evolution Equations & Control Theory, 2021, 10 (1) : 37-59. doi: 10.3934/eect.2020051 |
[16] |
Lars Grüne, Roberto Guglielmi. On the relation between turnpike properties and dissipativity for continuous time linear quadratic optimal control problems. Mathematical Control & Related Fields, 2021, 11 (1) : 169-188. doi: 10.3934/mcrf.2020032 |
[17] |
Jingrui Sun, Hanxiao Wang. Mean-field stochastic linear-quadratic optimal control problems: Weak closed-loop solvability. Mathematical Control & Related Fields, 2021, 11 (1) : 47-71. doi: 10.3934/mcrf.2020026 |
[18] |
Arthur Fleig, Lars Grüne. Strict dissipativity analysis for classes of optimal control problems involving probability density functions. Mathematical Control & Related Fields, 2020 doi: 10.3934/mcrf.2020053 |
[19] |
Wen Li, Wei-Hui Liu, Seak Weng Vong. Perron vector analysis for irreducible nonnegative tensors and its applications. Journal of Industrial & Management Optimization, 2021, 17 (1) : 29-50. doi: 10.3934/jimo.2019097 |
[20] |
Bernard Bonnard, Jérémy Rouot. Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model. Journal of Geometric Mechanics, 2020 doi: 10.3934/jgm.2020032 |
Impact Factor:
Tools
Metrics
Other articles
by authors
[Back to Top]