# American Institute of Mathematical Sciences

June  2018, 10(2): 139-172. doi: 10.3934/jgm.2018005

## A note on time-optimal paths on perturbed spheroid

 1 Jagiellonian University, Faculty of Mathematics and Computer Science, ul. Prof. St. Lojasiewicza 6, 30 - 348 Kraków, Poland 2 Gdynia Maritime University, Faculty of Navigation, Al. Jana Pawla Ⅱ 3, 81-345 Gdynia, Poland

Received  January 2016 Revised  March 2018 Published  May 2018

Fund Project: The author is supported by a grant from the Polish National Science Center under research project number 2013/09/N/ST10/02537.

We consider Zermelo's problem of navigation on a spheroid in the presence of space-dependent perturbation $W$ determined by a weak velocity vector field, $|W|_h<1$. The approach is purely geometric with application of Finsler metric of Randers type making use of the corresponding optimal control represented by a time-minimal ship's heading $\varphi(t)$ (a steering direction). A detailed exposition including investigation of the navigational quantities is provided under a rotational vector field. This demonstrates, in particular, a preservation of the optimal control $\varphi(t)$ of the time-efficient trajectories in the presence and absence of acting perturbation. Such navigational treatment of the problem leads to some simple relations between the background Riemannian and the resulting Finsler geodesics, thought of the deformed Riemannian paths. Also, we show some connections with Clairaut's relation and a collision problem. The study is illustrated with an example considered on an oblate ellipsoid.

Citation: Piotr Kopacz. A note on time-optimal paths on perturbed spheroid. Journal of Geometric Mechanics, 2018, 10 (2) : 139-172. doi: 10.3934/jgm.2018005
##### References:
 [1] The International Maritime Organization (IMO), COLREG: Convention on the International Regulations for Preventing Collisions at Sea, London, United Kingdom, 2004. Google Scholar [2] N. Aldea and P. Kopacz, Generalized Zermelo navigation on Hermitian manifolds under mild wind, Diff. Geom. Appl., 54 (2017), 325-343.  doi: 10.1016/j.difgeo.2017.05.007.  Google Scholar [3] N. Aldea and P. Kopacz, Generalized Zermelo navigation on Hermitian manifolds with a critical wind, Results Math., 72 (2017), 2165-2180.  doi: 10.1007/s00025-017-0757-6.  Google Scholar [4] K. J. Arrow, On the use of winds in flight planning, J. Meteor., 6 (1949), 150-159.   Google Scholar [5] D. Bao and C. Robles, Ricci and flag curvatures in Finsler geometry, in A sampler of Riemann-Finsler geometry (eds. D. Bao et al.), Math. Sci. Res. Inst. Publ., 50 (2004), Cambridge Univ. Press, 197-259.  Google Scholar [6] D. Bao, C. Robles and Z. Shen, Zermelo navigation on Riemannian manifolds, J. Diff. Geom., 66 (2004), 377-435.   Google Scholar [7] D. C. Brody, G. W. Gibbons and D. M. Meier, Time-optimal navigation through quantum wind, New J. Phys., 17 (2015), 033048, 8pp. doi: 10.1088/1367-2630/17/3/033048.  Google Scholar [8] D. C. Brody, G. W. Gibbons and D. M. Meier, A Riemannian approach to Randers geodesics, J. Geom. Phys., 106 (2016), 98-101.  doi: 10.1016/j.geomphys.2016.03.019.  Google Scholar [9] D. C. Brody and D. M. Meier, Solution to the quantum Zermelo navigation problem, Phys. Rev. Lett., 114 (2015), 100502. Google Scholar [10] E. Caponio, M. A. Javaloyes and M. Sànchez, Wind Finslerian structures: from Zermelo's navigation to the causality of spacetimes, preprint, arXiv: 1407.5494. Google Scholar [11] C. Carathéodory, Calculus of Variations and Partial Differential Equations of the First Order, San Francisco-London-Amsterdam, 1965.  Google Scholar [12] S.-S. Chern and Z. Shen, Riemann-Finsler Geometry, Nankai tracts in mathematics, World Scientific, River Edge (N. J.), London, Singapore, 2005. doi: 10.1142/5263.  Google Scholar [13] M. A. Earle, Sphere to spheroid comparisons, J. Navigation, 59 (2006), 491-496.   Google Scholar [14] A. de Mira Fernandes, Sul problema brachistocrono di Zermelo, Rendiconti della R. Acc. dei Lincei, XV (1932), 47-52.   Google Scholar [15] C. A. R. Herdeiro, Mira Fernandes and a generalised Zermelo problem: Purely geometric formulations, Bol. Soc. Port. Mat., 2010, Special Issue. Publication date estimated, 179-191.  Google Scholar [16] M. R. Jardin, Toward Real-Time en Route Air Traffic Control Optimization, Ph. D thesis, Stanford University, 2003. Google Scholar [17] A. B. Katok, Ergodic perturbations of degenerate integrable Hamiltonian systems, Math. USSR Izv., 37 (1973), 539-576.   Google Scholar [18] P. Kopacz, On generalization of Zermelo navigation problem on Riemannian manifolds, preprint, arXiv: 1604.06487. Google Scholar [19] P. Kopacz, Application of planar Randers geodesics with river-type perturbation in search models, Appl. Math. Model., 49 (2017), 531-553.  doi: 10.1016/j.apm.2017.05.007.  Google Scholar [20] P. Kopacz, A note on generalization of Zermelo navigation problem on Riemannian manifolds with strong perturbation, An. Sti. U. Ovid. Co.-Mat., 25 (2017), 107-123.   Google Scholar [21] P. Kopacz, Application of codimension one foliation in Zermelo's problem on Riemannian manifolds, Diff. Geom. Appl., 35 (2014), 334-349.  doi: 10.1016/j.difgeo.2014.04.007.  Google Scholar [22] P. Kopacz, Proposal on a Riemannian approach to path modeling in a navigational decision support system, in Activities of Transport Telematics. TST 2013. Communications in Computer and Information Science 395 (eds. J. Mikulski), Springer, Berlin, Heidelberg, (2013), 294-302. Google Scholar [23] T. Levi-Civita, Über Zermelo's Luftfahrtproblem, ZAMM-Z. Angew. Math. Me., 11 (1931), 314-322.   Google Scholar [24] B. Manià, Sopra un problema di navigazione di Zermelo, Math. Ann., 133 (1937), 584-599.  doi: 10.1007/BF01571651.  Google Scholar [25] A. Pallikaris and G. Latsas, New Algorithm for Great Elliptic Sailing (GES), J. Navigation, 62 (2012), 493-507.   Google Scholar [26] A. Pressley, Elementary Differential Geometry, Springer Undergraduate Mathematics Series, Springer-Verlag, London, 2010. doi: 10.1007/978-1-84882-891-9.  Google Scholar [27] C. Robles, Geodesics in Randers spaces of constant curvature, T. Am. Math. Soc., 359 (2007), 1633-1651.  doi: 10.1090/S0002-9947-06-04051-7.  Google Scholar [28] B. Russell and S. Stepney, Zermelo navigation in the quantum brachistochrone, J. Phys. A - Math. Theor., 48 (2015), 115303, 29pp. doi: 10.1088/1751-8113/48/11/115303.  Google Scholar [29] Z. Shen, Finsler Metrics with ${\bf{K}} = 0$ and ${\bf{S}} = 0$, Can. J. Math., 55 (2003), 112-132.  doi: 10.4153/CJM-2003-005-6.  Google Scholar [30] W.-K. Tseng, M. A. Earle and J.-L. Guo, Direct and inverse solutions with geodetic latitude in terms of longitude for rhumb line sailing, J. Navigation, 65 (2012), 549-559.   Google Scholar [31] A. Weintrit and P. Kopacz, Computational algorithms implemented in marine navigation electronic systems, in Telematics in the Transport Environment. TST 2012. Communications in Computer and Information Science 329 (eds. J. Mikulski), Springer, Berlin, Heidelberg, (2012), 148-158. Google Scholar [32] A. Weintrit and P. Kopacz, A Novel Approach to Loxodrome (Rhumb Line), Orthodrome (Great Circle) and Geodesic Line in ECDIS and Navigation in General, in Methods and Algorithms in Navigation (eds. A. Weintrit and T. Neumann), CRC Press, (2011), 123-132. Google Scholar [33] R. Yoshikawa and S. V. Sabau, Kropina metrics and Zermelo navigation on Riemannian manifolds, Geom. Dedicata, 171 (2014), 119-148.  doi: 10.1007/s10711-013-9892-8.  Google Scholar [34] E. Zermelo, Über die Navigation in der Luft als Problem der Variationsrechnung, Jahresber. Deutsch. Math.-Verein., 89 (1930), 44-48.   Google Scholar [35] E. Zermelo, Über das Navigationsproblem bei ruhender oder veränderlicher Windverteilung, ZAMM-Z. Angew. Math. Me., 11 (1931), 114-124.   Google Scholar [36] W. Ziller, Geometry of the Katok examples, Ergodic Theory Dyn. Syst., 3 (1983), 135-157.  doi: 10.1017/S0143385700001851.  Google Scholar

show all references

##### References:
 [1] The International Maritime Organization (IMO), COLREG: Convention on the International Regulations for Preventing Collisions at Sea, London, United Kingdom, 2004. Google Scholar [2] N. Aldea and P. Kopacz, Generalized Zermelo navigation on Hermitian manifolds under mild wind, Diff. Geom. Appl., 54 (2017), 325-343.  doi: 10.1016/j.difgeo.2017.05.007.  Google Scholar [3] N. Aldea and P. Kopacz, Generalized Zermelo navigation on Hermitian manifolds with a critical wind, Results Math., 72 (2017), 2165-2180.  doi: 10.1007/s00025-017-0757-6.  Google Scholar [4] K. J. Arrow, On the use of winds in flight planning, J. Meteor., 6 (1949), 150-159.   Google Scholar [5] D. Bao and C. Robles, Ricci and flag curvatures in Finsler geometry, in A sampler of Riemann-Finsler geometry (eds. D. Bao et al.), Math. Sci. Res. Inst. Publ., 50 (2004), Cambridge Univ. Press, 197-259.  Google Scholar [6] D. Bao, C. Robles and Z. Shen, Zermelo navigation on Riemannian manifolds, J. Diff. Geom., 66 (2004), 377-435.   Google Scholar [7] D. C. Brody, G. W. Gibbons and D. M. Meier, Time-optimal navigation through quantum wind, New J. Phys., 17 (2015), 033048, 8pp. doi: 10.1088/1367-2630/17/3/033048.  Google Scholar [8] D. C. Brody, G. W. Gibbons and D. M. Meier, A Riemannian approach to Randers geodesics, J. Geom. Phys., 106 (2016), 98-101.  doi: 10.1016/j.geomphys.2016.03.019.  Google Scholar [9] D. C. Brody and D. M. Meier, Solution to the quantum Zermelo navigation problem, Phys. Rev. Lett., 114 (2015), 100502. Google Scholar [10] E. Caponio, M. A. Javaloyes and M. Sànchez, Wind Finslerian structures: from Zermelo's navigation to the causality of spacetimes, preprint, arXiv: 1407.5494. Google Scholar [11] C. Carathéodory, Calculus of Variations and Partial Differential Equations of the First Order, San Francisco-London-Amsterdam, 1965.  Google Scholar [12] S.-S. Chern and Z. Shen, Riemann-Finsler Geometry, Nankai tracts in mathematics, World Scientific, River Edge (N. J.), London, Singapore, 2005. doi: 10.1142/5263.  Google Scholar [13] M. A. Earle, Sphere to spheroid comparisons, J. Navigation, 59 (2006), 491-496.   Google Scholar [14] A. de Mira Fernandes, Sul problema brachistocrono di Zermelo, Rendiconti della R. Acc. dei Lincei, XV (1932), 47-52.   Google Scholar [15] C. A. R. Herdeiro, Mira Fernandes and a generalised Zermelo problem: Purely geometric formulations, Bol. Soc. Port. Mat., 2010, Special Issue. Publication date estimated, 179-191.  Google Scholar [16] M. R. Jardin, Toward Real-Time en Route Air Traffic Control Optimization, Ph. D thesis, Stanford University, 2003. Google Scholar [17] A. B. Katok, Ergodic perturbations of degenerate integrable Hamiltonian systems, Math. USSR Izv., 37 (1973), 539-576.   Google Scholar [18] P. Kopacz, On generalization of Zermelo navigation problem on Riemannian manifolds, preprint, arXiv: 1604.06487. Google Scholar [19] P. Kopacz, Application of planar Randers geodesics with river-type perturbation in search models, Appl. Math. Model., 49 (2017), 531-553.  doi: 10.1016/j.apm.2017.05.007.  Google Scholar [20] P. Kopacz, A note on generalization of Zermelo navigation problem on Riemannian manifolds with strong perturbation, An. Sti. U. Ovid. Co.-Mat., 25 (2017), 107-123.   Google Scholar [21] P. Kopacz, Application of codimension one foliation in Zermelo's problem on Riemannian manifolds, Diff. Geom. Appl., 35 (2014), 334-349.  doi: 10.1016/j.difgeo.2014.04.007.  Google Scholar [22] P. Kopacz, Proposal on a Riemannian approach to path modeling in a navigational decision support system, in Activities of Transport Telematics. TST 2013. Communications in Computer and Information Science 395 (eds. J. Mikulski), Springer, Berlin, Heidelberg, (2013), 294-302. Google Scholar [23] T. Levi-Civita, Über Zermelo's Luftfahrtproblem, ZAMM-Z. Angew. Math. Me., 11 (1931), 314-322.   Google Scholar [24] B. Manià, Sopra un problema di navigazione di Zermelo, Math. Ann., 133 (1937), 584-599.  doi: 10.1007/BF01571651.  Google Scholar [25] A. Pallikaris and G. Latsas, New Algorithm for Great Elliptic Sailing (GES), J. Navigation, 62 (2012), 493-507.   Google Scholar [26] A. Pressley, Elementary Differential Geometry, Springer Undergraduate Mathematics Series, Springer-Verlag, London, 2010. doi: 10.1007/978-1-84882-891-9.  Google Scholar [27] C. Robles, Geodesics in Randers spaces of constant curvature, T. Am. Math. Soc., 359 (2007), 1633-1651.  doi: 10.1090/S0002-9947-06-04051-7.  Google Scholar [28] B. Russell and S. Stepney, Zermelo navigation in the quantum brachistochrone, J. Phys. A - Math. Theor., 48 (2015), 115303, 29pp. doi: 10.1088/1751-8113/48/11/115303.  Google Scholar [29] Z. Shen, Finsler Metrics with ${\bf{K}} = 0$ and ${\bf{S}} = 0$, Can. J. Math., 55 (2003), 112-132.  doi: 10.4153/CJM-2003-005-6.  Google Scholar [30] W.-K. Tseng, M. A. Earle and J.-L. Guo, Direct and inverse solutions with geodetic latitude in terms of longitude for rhumb line sailing, J. Navigation, 65 (2012), 549-559.   Google Scholar [31] A. Weintrit and P. Kopacz, Computational algorithms implemented in marine navigation electronic systems, in Telematics in the Transport Environment. TST 2012. Communications in Computer and Information Science 329 (eds. J. Mikulski), Springer, Berlin, Heidelberg, (2012), 148-158. Google Scholar [32] A. Weintrit and P. Kopacz, A Novel Approach to Loxodrome (Rhumb Line), Orthodrome (Great Circle) and Geodesic Line in ECDIS and Navigation in General, in Methods and Algorithms in Navigation (eds. A. Weintrit and T. Neumann), CRC Press, (2011), 123-132. Google Scholar [33] R. Yoshikawa and S. V. Sabau, Kropina metrics and Zermelo navigation on Riemannian manifolds, Geom. Dedicata, 171 (2014), 119-148.  doi: 10.1007/s10711-013-9892-8.  Google Scholar [34] E. Zermelo, Über die Navigation in der Luft als Problem der Variationsrechnung, Jahresber. Deutsch. Math.-Verein., 89 (1930), 44-48.   Google Scholar [35] E. Zermelo, Über das Navigationsproblem bei ruhender oder veränderlicher Windverteilung, ZAMM-Z. Angew. Math. Me., 11 (1931), 114-124.   Google Scholar [36] W. Ziller, Geometry of the Katok examples, Ergodic Theory Dyn. Syst., 3 (1983), 135-157.  doi: 10.1017/S0143385700001851.  Google Scholar
The Riemannian geodesics on the spheroid $\Sigma^2$ with $a: = \frac{3}{4}$ starting from $(0, \frac{\pi}{2})$, with the increments $\Delta \varphi_0 = \frac{\pi}{8}$. On the right, the corresponding solutions $x(t), y(t), z(t)$ for the point $(1, 0, 0)\in\mathbb{R}^3$; $t\leq3$
The contour plot and the graph of the norm $|W|_h$ in the case of the infinitesimal rotation as the acting mild perturbation, with $c: = \frac{5}{7}$
The geodesics of the new Riemannian metric $\alpha$ starting from $(0, \frac{\pi}{2})$, with the increments $\Delta \varphi_0 = \frac{\pi}{8}$ (16 curves); $t\leq3$
The time-efficient paths on the spheroid with $a: = \frac{3}{4}$ starting from $(0, \frac{\pi}{2})$, with the increments $\Delta \varphi_0 = \frac{\pi}{8}$, $t\leq3$ (left) and divided into time segments with $t\in[0, 1) \text{- red}, t\in[1, 2) \text{- blue}, t\in[2, 3] \text{ - purple}$. Right: "top view"
The transpolar (blue) and circumpolar (red) Randers geodesics starting from $(\phi_0, \theta_0) = (0, \frac{\pi}{2})$ in the presence of the rotational perturbation, with $c: = \frac{5}{7}$, the increments $\Delta \varphi_0 = \frac{\pi}{4}$, $t\leq50$. Right: "top" view
The $F$-isochrones in the presence of the perturbing infinitesimal rotation, with $c: = \frac{5}{7}$ for $t = 1$ (blue), $t = 2$ (red), $t = 3$ (purple), $t = 4$ (magenta) and the starting point in $(\phi_0, \theta_0): = (0, \frac{\pi}{2})$
The $F$-isochrones on the spheroid $\Sigma^2$ under the perturbing infinitesimal rotation (7), with $c: = \frac{5}{7}$, $t = \left\{1\ \text{(blue)}, 2\ \text{(red)}, 3 \ \text{(purple)}, 4\ \text{(magenta)}\right\}$ and the starting point $(0, \frac{\pi}{2})$. Right: "top" view
Comparing the solutions $x(t)$ - blue, $y(t)$ - red, $z(t)$ - black in the absence (dashed) and the presence (solid) of the wind (7), with $\Delta \varphi_0 = \frac{\pi}{8}$ and the starting point $(1, 0, 0)\in\mathbb{R}^3$; $t\leq3$
The comparison of the perturbed (red) and unperturbed (blue) time-efficient paths starting from $(0, \frac{\pi}{2})$, with $\Delta \varphi_0 = \frac{\pi}{4}$, $t\leq1$ (left) and $\Delta \varphi_0 = \frac{\pi}{8}$, $t\leq3$ (middle and "top" view on the right)
The corresponding background $h$-Riemannian (blue), new $\alpha$-Riemannian (green) and $F$-Randers (red) geodesics starting from $(0, \frac{\pi}{2})$, with $\Delta \varphi_0 = \frac{\pi}{4}$, $t\leq1$ (left) and with $\Delta \varphi_0 = \frac{\pi}{8}$, $t\leq3$ (middle: "side" view and right: "top" view)
The initial resulting speed $|{\bf{v}}_0|_h$ (black) as the function of the initial control angle $\varphi_0\in[0, 2\pi )$
The linear (on the left) and angular speeds (on the right) as the functions of time, in the absence (dashed) and in the presence (solid) of the perturbation (7); the resulting linear speed is shown in black; $t\leq7$
The polar plot of the solutions $\phi(t)$ (blue), $\theta(t)$ (red) in the absence (dashed) and presence (solid) of the perturbation (7); $t\leq20$
The distance (Euclidean) between the "Riemannian ship" and "Finslerian ship", with $t\leq 15$ (left) and $t\leq 50$ (right). The red horizontal line indicates collision and the green one - maximum distance, so the ships are positioned in the antipodal points of the spheroid's equator. The time of the first collision is $t_{col_1} = \frac{14\pi}{5}\approx8.8$
The first collision (also, the seventh intersection) of the corresponding time-minimal trajectories, i.e., the background Riemannian (blue) and the Randers preserving the optimal control (red); $t\leq\frac{14\pi}{5}\approx8.8$
The first intersection (no collision) of the Riemannian (blue) and Randers (red) geodesics coming from the starting point in $(0, \frac{\pi}{2})$; with $t\leq1$ (solid), $t\leq 1.66$ (dashed). Right: "top" view
The spherical coordinates $\phi$ (blue), $\theta$ (red) of the corresponding Riemannian (dashed, $t\leq1$) and Randers (solid, $t\leq1.66$) geodesics coming from the starting point in $(0, \frac{\pi}{2})$ till the first intersection (no collision) in the Cartesian (left) and the polar plot (right; time is expressed by length of the radius). The azimuth (longitude) of the first intersection $\phi\approx 45^\circ$ is marked by the black line
The Riemannian background geodesic of the ellipsoid $\Sigma^2$ departing from $(0, \frac{\pi}{2})$ and its planar $xy$-projection; $t\leq25$
The background Riemannian geodesic (blue) versus the corresponding Randers geodesic (red), $t\leq7$, and their $xy$-projection (right), $t\leq20$
Comparing the corresponding geodesics in the absence (blue, $h$-Riemannian) and in the presence (red, $F$-Randers) of the perturbing vector field ($7$), with $c: = \frac{5}{7}$. Right: "top" view; $t\leq25$
The Cartesian solutions in the absence (dashed) and presence (solid) of the applied rotational perturbation (7) in the base $(x, y, z)$ (left) and the corresponding polar plot of the spherical solutions in the base $(\phi, \theta)$ (right); $t\leq7$
The Randers geodesic starting from $(0, \frac{\pi}{2})$ under the rotational perturbation (7), with $c: = \frac{5}{7}$, $t\leq25$, and its $xy$-projection, $t\leq20$
The parametric plots of the solutions $\phi, \theta$ (blue, $t\leq7$, left), their first derivatives (black, $t\leq3$, middle) and second derivatives (right, $t\leq5.35$, overlapped), in the absence (dashed) and presence (solid) of the rotational wind ($7$)
The resulting time-optimal steering angle $\Phi(t)$ ("course over ground") without (dashed blue) and with (solid red) the wind (7) in the Cartesian plot (left) and the corresponding polar plot (right); $t\leq7$
The drift angle $\Psi(t)$ (dashed blue), the optimal control $\varphi(t)$ (black) and the optimal resulting angle ("course over ground") $\Phi(t)$ (red) in the presence of the perturbation (7); $t\leq7$
The first collision (also, the first intersection) in $(\phi, \theta)\approx(45^\circ, 128^\circ)$ of the time-minimal trajectories generated from the same Riemannian geodesic (blue), i.e. the Randers geodesic with the preserved optimal control (red) and with the supplementary optimal control (black). On the right, "top view"; $t\leq3.3$
The first collision (also, the first intersection) in $(\phi, \theta)\approx(138^\circ, 53^\circ)$ of the Riemannian geodesic (blue) and the Randers geodesic with the supplementary optimal control (black). The corresponding Randers geodesic preserving the optimal control is presented in red. Right: "top" view; $t\leq2.06$
Left: the Riemannian geodesic (blue) and its two generated Randers geodesics: with the preserved optimal control (red, "Randers_1") and the supplementary optimal control (black, "Randers_2"). Right: the corresponding distances (Euclidean) between the ships: "Riemannian-Randers_1" (purple), "Riemannian-Randers_2" (dashed black) and "Randers_1-Randers_2" (blue); $t\leq 15$. The red horizontal line indicates collision and the green one the maximum distance, i.e., the ships are located in the antipodal points of the spheroid's equator
 [1] George W. Patrick. The geometry of convergence in numerical analysis. Journal of Computational Dynamics, 2021, 8 (1) : 33-58. doi: 10.3934/jcd.2021003 [2] Giulia Luise, Giuseppe Savaré. Contraction and regularizing properties of heat flows in metric measure spaces. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 273-297. doi: 10.3934/dcdss.2020327 [3] Vivina Barutello, Gian Marco Canneori, Susanna Terracini. Minimal collision arcs asymptotic to central configurations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 61-86. doi: 10.3934/dcds.2020218 [4] Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248 [5] Wei Ouyang, Li Li. Hölder strong metric subregularity and its applications to convergence analysis of inexact Newton methods. Journal of Industrial & Management Optimization, 2021, 17 (1) : 169-184. doi: 10.3934/jimo.2019105 [6] Sören Bartels, Jakob Keck. Adaptive time stepping in elastoplasticity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 71-88. doi: 10.3934/dcdss.2020323 [7] Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020076 [8] Min Chen, Olivier Goubet, Shenghao Li. Mathematical analysis of bump to bucket problem. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5567-5580. doi: 10.3934/cpaa.2020251 [9] Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253 [10] Luca Battaglia, Francesca Gladiali, Massimo Grossi. Asymptotic behavior of minimal solutions of $-\Delta u = \lambda f(u)$ as $\lambda\to-\infty$. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 681-700. doi: 10.3934/dcds.2020293 [11] Emre Esentürk, Juan Velazquez. Large time behavior of exchange-driven growth. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 747-775. doi: 10.3934/dcds.2020299 [12] Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384 [13] Kien Trung Nguyen, Vo Nguyen Minh Hieu, Van Huy Pham. Inverse group 1-median problem on trees. Journal of Industrial & Management Optimization, 2021, 17 (1) : 221-232. doi: 10.3934/jimo.2019108 [14] 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 [15] Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137 [16] Guido Cavallaro, Roberto Garra, Carlo Marchioro. Long time localization of modified surface quasi-geostrophic equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020336 [17] Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020339 [18] Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444 [19] Veena Goswami, Gopinath Panda. Optimal customer behavior in observable and unobservable discrete-time queues. Journal of Industrial & Management Optimization, 2021, 17 (1) : 299-316. doi: 10.3934/jimo.2019112 [20] Chongyang Liu, Meijia Han, Zhaohua Gong, Kok Lay Teo. Robust parameter estimation for constrained time-delay systems with inexact measurements. Journal of Industrial & Management Optimization, 2021, 17 (1) : 317-337. doi: 10.3934/jimo.2019113

2019 Impact Factor: 0.649