doi: 10.3934/jcd.2018001

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

* Corresponding author

Received  June 2017 Revised  March 2018

Fund Project: This work was funded by Mitsubishi Electric Research Labs

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.

Citation: Karthik Elamvazhuthi, Piyush Grover. Optimal transport over nonlinear systems via infinitesimal generators on graphs. Journal of Computational Dynamics, doi: 10.3934/jcd.2018001
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show all references

References:
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[3]

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M. AicardiG. CasalinoA. 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.

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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. BermanÁ. HalászM. 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.

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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.

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E. M. Bollt and N. Santitissadeekorn, Applied and Computational Measurable Dynamics, vol. 18, SIAM, 2013. doi: 10.1137/1.9781611972641.

[11]

P. BoylandH. 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. ChenT. 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. DellnitzO. JungeW. S. KoonF. LekienM. W. LoJ. E. MarsdenK. PadbergR. PreisS. 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).

[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).

[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. FroylandC. González-Tokman and T. M. Watson, Optimal mixing enhancement by local perturbation, SIAM Review, 58 (2016), 494-513. doi: 10.1137/15M1023221.

[33]

G. FroylandO. 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]

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Figure 1.  Approximation of infinitesimal generator $A$. The entry $A_{ij}$ is proportional to flux across $B_i\cap B_j$ from $B_i$ to $B_j$, due to vector field $f$.
Figure 2.  (A) Some minimizing geodesics to the origin in the Grushin plane. (B) Analytically computed optimal transport solution between a uniform measure whose support is the disk $\Omega = \{(x, y)|x^2+(y-.8)^2<.15^2\}$, and a measure concentrated at the origin.
Figure 3.  (A)-(E) The optimal transport solution in the Grushin plane using graph based algorithm between a measure whose support is the disk $\Omega = \{(x, y)|x^2+(y-.8)^2<.15^2\}$, and delta measure at the origin. The parameters are $m = 10^4, k = 75$. (F) Convergence of optimal transport cost with number of time discretization steps $k$ and grid size $m$.
Figure 4.  Particle trajectories with feedback control computed using Eq. (47) from the optimal transport solution in the Grushin plane. Each box contained in the support of uniform initial measure $\mu_0$ is initially populated with 4 particles.
Figure 5.  nvariant manifolds and lobe-dynamics in the double-gyre system (reproduced from Ref. [39]).
Figure 6.  Optimal transport in the periodic double gyre system (Eqs.(57a-57b)) between measures at $t = 0$ and $t_f = 1$. The parameters are $m = 1800, \Delta t = \dfrac{1}{40}$.
Figure 7.  Optimal transport in the periodic double gyre system between measures at $t = 0$ and $t_f = 10, \Delta t = \dfrac{1}{40}$. The optimal transport solution shows a quantization phenomenon. Ten 'packets' are transported via lobe-dynamics from the left side to the right side of the domain. During $2<t<5$, the third packet is transported to the right side via the sequence $F^{-1}(A)\rightarrow A\rightarrow F(A) \rightarrow F^2(A)$, where set $A$ is defined in Fig. 5. The last packet gets transported to the right side over $9<t<10$. Animation available at: https://www.youtube.com/watch?v=Pu7sCkpm4RY
Figure 8.  The cost of optimal transport between two measures supported on two AIS for the double-gyre system, as a function of time-horizon of the problem.
Figure 9.  Initial and final measures shown on $(x, y)$ plane for two cases of optimal transport in the unicycle model. The green arrows indicate the third coordinate $\theta$. (A) $\mu_0$ is supported on $(0, 0.5, 0)$, $\mu_1$ is supported on $(1, 0, 0)$ and $(1, 1, 0)$. (B) $\mu_0$ is supported on $(0, 0.5, 0)$, $\mu_1$ is supported on $(1, 0, \frac{3\pi}{2})$ and $(1, 1, \frac{\pi}{2}).$
Figure 10.  The optimal transport solution of unicycle model shown in the $x-y$ plane for case 1. The grid size is $m = 25^3$, and $k = 20$.
Figure 11.  The optimal transport solution of unicycle model shown in the $x-y$ plane for case 2. The grid size is $m = 25^3$, and $k$ = 20.
Figure 12.  Illustration of the proof of Theorem 3.7. The existence of an trajectory $f$ approximating a curve $\gamma$ connecting $B_v$ to $B_w$, obtained by piecewise constant control, is guaranteed by the small time local controllability. By continuity, this leads to non-zero transition rates, and hence strong connectivity of the control graph $\mathcal{G}_c$.
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