American Institute of Mathematical Sciences

Advanced
September  2021, 16(3): 493-511. doi: 10.3934/nhm.2021014

Irrigable measures for weighted irrigation plans

Qing Sun

Department of Mathematics, Penn State University, University Park, PA 16803, USA

Received  January 2020 Revised  April 2021 Published  September 2021 Early access  July 2021

A model of irrigation network, where lower branches must be thicker in order to support the weight of the higher ones, was recently introduced in [7]. This leads to a countable family of ODEs, describing the thickness of every branch, solved by backward induction. The present paper determines what kind of measures can be irrigated with a finite weighted cost. Indeed, the boundedness of the cost depends on the dimension of the support of the irrigated measure, and also on the asymptotic properties of the ODE which determines the thickness of branches.

Keywords: Optimization, ordinary differential equation, calculus of variation.
Mathematics Subject Classification: Primary: 34A05, 34A36; Secondary: 92B05.
Citation: Qing Sun. Irrigable measures for weighted irrigation plans. Networks & Heterogeneous Media, 2021, 16 (3) : 493-511. doi: 10.3934/nhm.2021014
References:
[1]

M. Bernot, V. Caselles and J. -M. Morel, Optimal Transportation Networks. Models and Theory, Lecture Notes in Mathematics, 1955, Springer, Berlin, 2009.  Google Scholar

[2]

M. Bernot, V. Caselles and J.-M. Morel, Traffic plans, Publicacions Matemàtiques, 49 (2005), 417-451.  doi: 10.5565/PUBLMAT_49205_09.  Google Scholar

[3]

A. Brancolini and S. Solimini, Fractal regularity results on optimal irrigation patterns, J. Math. Pures Appl., 102 (2014), 854-890.  doi: 10.1016/j.matpur.2014.02.008.  Google Scholar

[4]

A. Bressan, M. Palladino and Q. Sun, Variational problems for tree roots and branches, Calc. Var. & Part. Diff. Equat., 59 (2020), Paper No. 7, 31 pp. doi: 10.1007/s00526-019-1666-1.  Google Scholar

[5]

A. Bressan and F. Rampazzo, On differential systems with vector-valued impulsive controls, Boll. Un. Matematica Italiana B, 2 (1988), 641-656.   Google Scholar

[6]

A. Bressan and Q. Sun, On the optimal shape of tree roots and branches, Math. Models & Methods Appl. Sci., 28 (2018), 2763-2801.  doi: 10.1142/S0218202518500604.  Google Scholar

[7]

A. Bressan and Q. Sun, Weighted irrigation plans, submitted, arXiv: 1906.02232. Google Scholar

[8]

G. Devillanova and S. Solimini, On the dimension of an irrigable measure, Rend. Sem. Mat. Univ. Padova., 117 (2007), 1-49.   Google Scholar

[9]

G. Devillanova and S. Solimini, Elementary properties of optimal irrigation patterns, Calc. Var. & Part. Diff. Equat., 28 (2007), 317-349.  doi: 10.1007/s00526-006-0046-9.  Google Scholar

[10]

G. Devillanova and S. Solimini, Some remarks on the fractal structure of irrigation balls, Advanced Nonlinear Studies, 19 (2019), 55-68.  doi: 10.1515/ans-2018-2035.  Google Scholar

[11] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Revised Edition. CRC Press, 2015.   Google Scholar
[12]

E. N. Gilbert, Minimum cost communication networks, Bell System Tech. J., 46 (1967), 2209-2227.  doi: 10.1002/j.1538-7305.1967.tb04250.x.  Google Scholar

[13]

P. Hartman, Ordinary Differential Equations, , Second Edition, Birkhäuser, Boston, Mass., 1982.  Google Scholar

[14]

F. Maddalena and S. Solimini, Synchronic and asynchronic descriptions of irrigation problems, Adv. Nonlinear Stud., 13 (2013), 583-623.  doi: 10.1515/ans-2013-0303.  Google Scholar

[15]

F. Maddalena, G. Taglialatela and J.-M. Morel, A variational model of irrigation patterns, Interfaces Free Bound., 5 (2003), 391-415.  doi: 10.4171/IFB/85.  Google Scholar

[16]

Q. Xia, Optimal paths related to transport problems, Comm. Contemp. Math., 5 (2003), 251-279.  doi: 10.1142/S021919970300094X.  Google Scholar

[17]

Q. Xia, Motivations, ideas and applications of ramified optimal transportation, ESAIM Math. Model. Numer. Anal., 49 (2015), 1791-1832.  doi: 10.1051/m2an/2015028.  Google Scholar

show all references

References:
[1]

M. Bernot, V. Caselles and J. -M. Morel, Optimal Transportation Networks. Models and Theory, Lecture Notes in Mathematics, 1955, Springer, Berlin, 2009.  Google Scholar

[2]

M. Bernot, V. Caselles and J.-M. Morel, Traffic plans, Publicacions Matemàtiques, 49 (2005), 417-451.  doi: 10.5565/PUBLMAT_49205_09.  Google Scholar

[3]

A. Brancolini and S. Solimini, Fractal regularity results on optimal irrigation patterns, J. Math. Pures Appl., 102 (2014), 854-890.  doi: 10.1016/j.matpur.2014.02.008.  Google Scholar

[4]

A. Bressan, M. Palladino and Q. Sun, Variational problems for tree roots and branches, Calc. Var. & Part. Diff. Equat., 59 (2020), Paper No. 7, 31 pp. doi: 10.1007/s00526-019-1666-1.  Google Scholar

[5]

A. Bressan and F. Rampazzo, On differential systems with vector-valued impulsive controls, Boll. Un. Matematica Italiana B, 2 (1988), 641-656.   Google Scholar

[6]

A. Bressan and Q. Sun, On the optimal shape of tree roots and branches, Math. Models & Methods Appl. Sci., 28 (2018), 2763-2801.  doi: 10.1142/S0218202518500604.  Google Scholar

[7]

A. Bressan and Q. Sun, Weighted irrigation plans, submitted, arXiv: 1906.02232. Google Scholar

[8]

G. Devillanova and S. Solimini, On the dimension of an irrigable measure, Rend. Sem. Mat. Univ. Padova., 117 (2007), 1-49.   Google Scholar

[9]

G. Devillanova and S. Solimini, Elementary properties of optimal irrigation patterns, Calc. Var. & Part. Diff. Equat., 28 (2007), 317-349.  doi: 10.1007/s00526-006-0046-9.  Google Scholar

[10]

G. Devillanova and S. Solimini, Some remarks on the fractal structure of irrigation balls, Advanced Nonlinear Studies, 19 (2019), 55-68.  doi: 10.1515/ans-2018-2035.  Google Scholar

[11] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Revised Edition. CRC Press, 2015.   Google Scholar
[12]

E. N. Gilbert, Minimum cost communication networks, Bell System Tech. J., 46 (1967), 2209-2227.  doi: 10.1002/j.1538-7305.1967.tb04250.x.  Google Scholar

[13]

P. Hartman, Ordinary Differential Equations, , Second Edition, Birkhäuser, Boston, Mass., 1982.  Google Scholar

[14]

F. Maddalena and S. Solimini, Synchronic and asynchronic descriptions of irrigation problems, Adv. Nonlinear Stud., 13 (2013), 583-623.  doi: 10.1515/ans-2013-0303.  Google Scholar

[15]

F. Maddalena, G. Taglialatela and J.-M. Morel, A variational model of irrigation patterns, Interfaces Free Bound., 5 (2003), 391-415.  doi: 10.4171/IFB/85.  Google Scholar

[16]

Q. Xia, Optimal paths related to transport problems, Comm. Contemp. Math., 5 (2003), 251-279.  doi: 10.1142/S021919970300094X.  Google Scholar

[17]

Q. Xia, Motivations, ideas and applications of ramified optimal transportation, ESAIM Math. Model. Numer. Anal., 49 (2015), 1791-1832.  doi: 10.1051/m2an/2015028.  Google Scholar

Figure 1.  Left: A free standing tree with 5 branches. In this example, $ \mathcal{O}(1) = \{2,3 \},\mathcal{O}(3) = \{4,5\}, \mathcal{O}(2) = \mathcal{O}(4) = \mathcal{O}(5) = \emptyset $. Right: On each branch, the weight decreases as one moves from the lower portion to the tip
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Left: Two finite truncation plans, showing three maximal $ \varepsilon $-good paths (thick lines) and six maximal $ \varepsilon' $-good paths (thin lines), for $ 0<\varepsilon'<\varepsilon $. Right: The three maximal $ \varepsilon $-good paths can be partitioned into five elementary branches, by the Path Splitting Algorithm
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Left: The dyadic approxmiated measure $ \mu_1 $ is supported on the four centers $ x_1^1,\ldots,x^1_4 $ of small cubes. Right: Dyadic approximated measures corresponding to a family of partitions into dyadic cubes in $ \bf{R}^2 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  The dyadic irrigation plans in $ \bf{R}^2 $. Left: The dyadic irrigation plan $ \chi_1 $. The multiplicity on each branch equals to the mass on the terminal point. Right: The dyadic irrigation plan $ \chi_2 $. The particles are first transported to the 4 centers in $ \mathcal{P}_1 $, then on each center in $ \mathcal{P}_1 $, the particles are transported to the neighboring 4 centers in $ \mathcal{P}_2 $
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Abdulrazzaq T. Abed, Azzam S. Y. Aladool. Applying particle swarm optimization based on Padé approximant to solve ordinary differential equation. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021008
[2]

Daniel G. Alfaro Vigo, Amaury C. Álvarez, Grigori Chapiro, Galina C. García, Carlos G. Moreira. Solving the inverse problem for an ordinary differential equation using conjugation. Journal of Computational Dynamics, 2020, 7 (2) : 183-208. doi: 10.3934/jcd.2020008
[3]

Richard A. Norton, G. R. W. Quispel. Discrete gradient methods for preserving a first integral of an ordinary differential equation. Discrete & Continuous Dynamical Systems, 2014, 34 (3) : 1147-1170. doi: 10.3934/dcds.2014.34.1147
[4]

Tomás Caraballo, Renato Colucci, Luca Guerrini. Bifurcation scenarios in an ordinary differential equation with constant and distributed delay: A case study. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2639-2655. doi: 10.3934/dcdsb.2018268
[5]

Lukáš Adam, Jiří Outrata. On optimal control of a sweeping process coupled with an ordinary differential equation. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2709-2738. doi: 10.3934/dcdsb.2014.19.2709
[6]

Michael Herty, Veronika Sachers. Adjoint calculus for optimization of gas networks. Networks & Heterogeneous Media, 2007, 2 (4) : 733-750. doi: 10.3934/nhm.2007.2.733
[7]

Ivar Ekeland. From Frank Ramsey to René Thom: A classical problem in the calculus of variations leading to an implicit differential equation. Discrete & Continuous Dynamical Systems, 2010, 28 (3) : 1101-1119. doi: 10.3934/dcds.2010.28.1101
[8]

Lijuan Wang, Yashan Xu. Admissible controls and controllable sets for a linear time-varying ordinary differential equation. Mathematical Control & Related Fields, 2018, 8 (3&4) : 1001-1019. doi: 10.3934/mcrf.2018043
[9]

Yuriy Golovaty, Anna Marciniak-Czochra, Mariya Ptashnyk. Stability of nonconstant stationary solutions in a reaction-diffusion equation coupled to the system of ordinary differential equations. Communications on Pure & Applied Analysis, 2012, 11 (1) : 229-241. doi: 10.3934/cpaa.2012.11.229
[10]

Qiong Meng, X. H. Tang. Multiple solutions of second-order ordinary differential equation via Morse theory. Communications on Pure & Applied Analysis, 2012, 11 (3) : 945-958. doi: 10.3934/cpaa.2012.11.945
[11]

Jan-Hendrik Webert, Philip E. Gill, Sven-Joachim Kimmerle, Matthias Gerdts. A study of structure-exploiting SQP algorithms for an optimal control problem with coupled hyperbolic and ordinary differential equation constraints. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1259-1282. doi: 10.3934/dcdss.2018071
[12]

Nguyen Thi Hoai. Asymptotic approximation to a solution of a singularly perturbed linear-quadratic optimal control problem with second-order linear ordinary differential equation of state variable. Numerical Algebra, Control & Optimization, 2021, 11 (4) : 495-512. doi: 10.3934/naco.2020040
[13]

Bernard Dacorogna, Alessandro Ferriero. Regularity and selecting principles for implicit ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 87-101. doi: 10.3934/dcdsb.2009.11.87
[14]

Janosch Rieger. The Euler scheme for state constrained ordinary differential inclusions. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2729-2744. doi: 10.3934/dcdsb.2016070
[15]

Zvi Artstein. Averaging of ordinary differential equations with slowly varying averages. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 353-365. doi: 10.3934/dcdsb.2010.14.353
[16]

Tomasz Kapela, Piotr Zgliczyński. A Lohner-type algorithm for control systems and ordinary differential inclusions. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 365-385. doi: 10.3934/dcdsb.2009.11.365
[17]

Stefano Maset. Conditioning and relative error propagation in linear autonomous ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2879-2909. doi: 10.3934/dcdsb.2018165
[18]

W. Sarlet, G. E. Prince, M. Crampin. Generalized submersiveness of second-order ordinary differential equations. Journal of Geometric Mechanics, 2009, 1 (2) : 209-221. doi: 10.3934/jgm.2009.1.209
[19]

Jacson Simsen, Mariza Stefanello Simsen, José Valero. Convergence of nonautonomous multivalued problems with large diffusion to ordinary differential inclusions. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2347-2368. doi: 10.3934/cpaa.2020102
[20]

Aeeman Fatima, F. M. Mahomed, Chaudry Masood Khalique. Conditional symmetries of nonlinear third-order ordinary differential equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 655-666. doi: 10.3934/dcdss.2018040

2020 Impact Factor: 1.213

Tools

Metrics

  • PDF downloads (58)
  • HTML views (187)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy