{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,25]],"date-time":"2025-06-25T05:52:24Z","timestamp":1750830744005,"version":"3.37.3"},"reference-count":42,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2021,8,20]],"date-time":"2021-08-20T00:00:00Z","timestamp":1629417600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2021,8,20]],"date-time":"2021-08-20T00:00:00Z","timestamp":1629417600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100008769","name":"Julius-Maximilians-Universit\u00e4t W\u00fcrzburg","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100008769","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer. Math."],"published-print":{"date-parts":[[2021,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>This paper is devoted to the numerical analysis of non-smooth ensemble optimal control problems governed by the Liouville (continuity) equation that have been originally proposed by R.W. Brockett with the purpose of determining an efficient and robust control strategy for dynamical systems. A numerical methodology for solving these problems is presented that is based on a non-smooth Lagrange optimization framework where the optimal controls are characterized as solutions to the related optimality systems. For this purpose, approximation and solution schemes are developed and analysed. Specifically, for the approximation of the Liouville model and its optimization adjoint, a combination of a Kurganov\u2013Tadmor method, a Runge\u2013Kutta scheme, and a Strang splitting method are discussed. The resulting optimality system is solved by a projected semi-smooth Krylov\u2013Newton method. Results of numerical experiments are presented that successfully validate the proposed framework.<\/jats:p>","DOI":"10.1007\/s00211-021-01223-6","type":"journal-article","created":{"date-parts":[[2021,8,20]],"date-time":"2021-08-20T19:18:43Z","timestamp":1629487123000},"page":"1-42","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["A numerical investigation of Brockett\u2019s ensemble optimal control problems"],"prefix":"10.1007","volume":"149","author":[{"given":"Jan","family":"Bartsch","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alfio","family":"Borz\u00ec","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Francesco","family":"Fanelli","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Souvik","family":"Roy","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,8,20]]},"reference":[{"key":"1223_CR1","doi-asserted-by":"crossref","unstructured":"Ak\u00e7elik, V., Biros, G., Ghattas, O., Hill, J., Keyes, D.B.: In: Heroux, M.A., Raghavan, P., Simon, H.D. (eds.) van Bloemen Waanders, Parallel algorithms for PDE-constrained optimization. In: Parallel Processing for Scientific Computing, vol. 16, pp. 291\u2013322. Society for Industrial and Applied Mathematics (2006)","DOI":"10.1137\/1.9780898718133.ch16"},{"key":"1223_CR2","doi-asserted-by":"publisher","first-page":"1191","DOI":"10.1017\/S0308210513000085","volume":"144","author":"L Ambrosio","year":"2014","unstructured":"Ambrosio, L., Crippa, G.: Continuity equations and ODE flows with non-smooth velocity. Proc. R. Soc. Edinb. Sect. A 144, 1191\u20131244 (2014)","journal-title":"Proc. R. Soc. Edinb. Sect. A"},{"key":"1223_CR3","doi-asserted-by":"crossref","unstructured":"Bartsch, J., Borz\u00ec, A., Fanelli, F., Roy, S.: A theoretical investigation of Brockett\u2019s ensemble optimal control problems. Calc. Var. Partial Differ. Equ. 58, 34 (2019)","DOI":"10.1007\/s00526-019-1604-2"},{"key":"1223_CR4","volume-title":"Computer Science and Applied Mathematics","author":"DP Bertsekas","year":"1982","unstructured":"Bertsekas, D.P.: Constrained optimization and Lagrange multiplier methods. In: Jovanovich, H.B. (ed.) Computer Science and Applied Mathematics. Academic Press, New York (1982)"},{"key":"1223_CR5","doi-asserted-by":"crossref","unstructured":"Borz\u00ec, A., Ciaramella, G., Sprengel, M.: Formulation and Numerical Solution of Quantum Control Problems. Computational Science & Engineering, vol. 16. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2017)","DOI":"10.1137\/1.9781611974843"},{"key":"1223_CR6","doi-asserted-by":"publisher","first-page":"1477","DOI":"10.1137\/S0363012901393432","volume":"41","author":"A Borz\u00ec","year":"2002","unstructured":"Borz\u00ec, A., Kunisch, K., Kwak, D.Y.: Accuracy and convergence properties of the finite difference multigrid solution of an optimal control optimality system. SIAM J. Control Optim. 41, 1477\u20131497 (2002)","journal-title":"SIAM J. Control Optim."},{"key":"1223_CR7","doi-asserted-by":"crossref","unstructured":"Brockett, R.: Notes on the control of the Liouville equation. In: Control of Partial Differential Equations of Lecture Notes in Mathematics, vol. 2048, pp. 101\u2013129. Springer, Heidelberg (2012)","DOI":"10.1007\/978-3-642-27893-8_2"},{"key":"1223_CR8","unstructured":"Brockett, R.W.: Optimal control of the Liouville equation. In: Proceedings of the International Conference on Complex Geometry and Related Fields, Volume 39 of AMS\/IP Stud. Adv. Math., Amer. Math. Soc., Providence, pp.\u00a023\u201335 (2007)"},{"key":"1223_CR9","doi-asserted-by":"crossref","unstructured":"Brockett, W.: Minimum attention control. In: Proceedings of the 36th IEEE Conference on Decision and Control, vol. 3, pp. 2628\u20132632. IEEE (1997)","DOI":"10.1109\/CDC.1997.657776"},{"key":"1223_CR10","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-1039-9","volume-title":"The Boltzmann Equation and Its Applications. Applied Mathematical Sciences","author":"C Cercignani","year":"1988","unstructured":"Cercignani, C.: The Boltzmann Equation and Its Applications. Applied Mathematical Sciences, vol. 67. Springer, New York (1988)"},{"key":"1223_CR11","first-page":"303","volume":"24","author":"A Chertock","year":"2010","unstructured":"Chertock, A., Kurganov, A.: On splitting-based numerical methods for convection-diffusion equations. Numer. Methods Balance Laws 24, 303\u2013343 (2010)","journal-title":"Numer. Methods Balance Laws"},{"key":"1223_CR12","doi-asserted-by":"publisher","first-page":"309","DOI":"10.1002\/fld.1355","volume":"59","author":"A Chertock","year":"2009","unstructured":"Chertock, A., Kurganov, A., Petrova, G.: Fast explicit operator splitting method for convection-diffusion equations. Int. J. Numer. Methods Fluids 59, 309\u2013332 (2009)","journal-title":"Int. J. Numer. Methods Fluids"},{"key":"1223_CR13","doi-asserted-by":"publisher","first-page":"817","DOI":"10.1016\/j.jcp.2015.10.030","volume":"305","author":"H Cho","year":"2016","unstructured":"Cho, H., Venturi, D., Karniadakis, G.E.: Numerical methods for high-dimensional probability density function equations. J. Comput. Phys. 305, 817\u2013837 (2016)","journal-title":"J. Comput. Phys."},{"key":"1223_CR14","doi-asserted-by":"publisher","first-page":"312","DOI":"10.1016\/j.cpc.2015.10.028","volume":"200","author":"G Ciaramella","year":"2016","unstructured":"Ciaramella, G., Borz\u00ec, A.: A LONE code for the sparse control of quantum systems. Comput. Phys. Commun. 200, 312\u2013323 (2016)","journal-title":"Comput. Phys. Commun."},{"key":"1223_CR15","doi-asserted-by":"publisher","first-page":"938","DOI":"10.1080\/01630563.2016.1184166","volume":"37","author":"G Ciaramella","year":"2016","unstructured":"Ciaramella, G., Borz\u00ec, A.: Quantum optimal control problems with a sparsity cost functional. Numer. Funct. Anal. Optim. 37, 938\u2013965 (2016)","journal-title":"Numer. Funct. Anal. Optim."},{"key":"1223_CR16","volume-title":"Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts","author":"F Clarke","year":"1983","unstructured":"Clarke, F.: Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts. Wiley, New York (1983)"},{"key":"1223_CR17","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4471-4820-3","volume-title":"Functional Analysis, Calculus of Variations and Optimal Control, vol. 264 of Graduate Texts in Mathematics","author":"F Clarke","year":"2013","unstructured":"Clarke, F.: Functional Analysis, Calculus of Variations and Optimal Control, vol. 264 of Graduate Texts in Mathematics. Springer, New York (2013)"},{"key":"1223_CR18","doi-asserted-by":"publisher","first-page":"511","DOI":"10.1007\/BF01393835","volume":"98","author":"RJ DiPerna","year":"1989","unstructured":"DiPerna, R.J., Lions, P.-L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98, 511\u2013547 (1989)","journal-title":"Invent. Math."},{"key":"1223_CR19","volume-title":"Transport Theory. A Wiley-Interscience Publication","author":"JJ Duderstadt","year":"1979","unstructured":"Duderstadt, J.J., Martin, W.R.: Transport Theory. A Wiley-Interscience Publication. Wiley, New York (1979)"},{"key":"1223_CR20","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-7643-8843-0","volume-title":"Singular Limits in Thermodynamics of Viscous Fluids, Advances in Mathematical Fluid Mechanics","author":"E Feireisl","year":"2009","unstructured":"Feireisl, E., Novotn\u00fd, A.: Singular Limits in Thermodynamics of Viscous Fluids, Advances in Mathematical Fluid Mechanics. Birkh\u00e4user Verlag, Basel (2009)"},{"key":"1223_CR21","doi-asserted-by":"crossref","unstructured":"Gerya, T.: Numerical solutions of the momentum and continuity equations. In: Introduction to Numerical Geodynamic Modelling, pp. 82\u2013104. Cambridge University Press (2019)","DOI":"10.1017\/9781316534243.008"},{"key":"1223_CR22","unstructured":"Godlewski, E., Raviart, P.-A.: Hyperbolic Systems of Conservation Laws, vol. 3\/4 of Math\u00e9matiques & Applications (Paris). Ellipses, Paris (1991)"},{"key":"1223_CR23","doi-asserted-by":"publisher","first-page":"73","DOI":"10.1090\/S0025-5718-98-00913-2","volume":"67","author":"S Gottlieb","year":"1998","unstructured":"Gottlieb, S., Shu, C.-W.: Total variation diminishing Runge\u2013Kutta schemes. Math. Comput. 67, 73\u201385 (1998)","journal-title":"Math. Comput."},{"key":"1223_CR24","doi-asserted-by":"publisher","first-page":"43","DOI":"10.1016\/j.procs.2015.11.007","volume":"66","author":"M Kraposhin","year":"2015","unstructured":"Kraposhin, M., Bovtrikova, A., Strijhak, S.: Adaptation of Kurganov\u2013Tadmor numerical scheme for applying in combination with the PISO method in numerical simulation of flows in a wide range of Mach numbers. Procedia Comput. Sci. 66, 43\u201352 (2015)","journal-title":"Procedia Comput. Sci."},{"key":"1223_CR25","doi-asserted-by":"publisher","first-page":"133","DOI":"10.4310\/CMS.2007.v5.n1.a6","volume":"5","author":"A Kurganov","year":"2007","unstructured":"Kurganov, A., Petrova, G.: A second-order well-balanced positivity preserving central-upwind scheme for the Saint\u2013Venant system. Commun. Math. Sci. 5, 133\u2013160 (2007)","journal-title":"Commun. Math. Sci."},{"key":"1223_CR26","doi-asserted-by":"publisher","first-page":"241","DOI":"10.1006\/jcph.2000.6459","volume":"160","author":"A Kurganov","year":"2000","unstructured":"Kurganov, A., Tadmor, E.: New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations. J. Comput. Phys. 160, 241\u2013282 (2000)","journal-title":"J. Comput. Phys."},{"key":"1223_CR27","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-65024-6","volume-title":"Optimal Control of Systems Governed by Partial Differential Equations","author":"J-L Lions","year":"1971","unstructured":"Lions, J.-L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, New York (1971)"},{"key":"1223_CR28","doi-asserted-by":"crossref","unstructured":"Nishikawa, H.: A truncation error analysis of third-order MUSCL scheme for nonlinear conservation laws. Int. J. Numer. Methods Fluids (2020)","DOI":"10.1002\/fld.4918"},{"key":"1223_CR29","doi-asserted-by":"publisher","first-page":"947","DOI":"10.1137\/0722057","volume":"22","author":"S Osher","year":"1985","unstructured":"Osher, S.: Convergence of generalized MUSCL schemes. SIAM J. Numer. Anal. 22, 947\u2013961 (1985)","journal-title":"SIAM J. Numer. Anal."},{"key":"1223_CR30","doi-asserted-by":"publisher","first-page":"955","DOI":"10.1137\/0721060","volume":"21","author":"S Osher","year":"1984","unstructured":"Osher, S., Chakravarthy, S.: High resolution schemes and the entropy condition. SIAM J. Numer. Anal. 21, 955\u2013984 (1984)","journal-title":"SIAM J. Numer. Anal."},{"key":"1223_CR31","doi-asserted-by":"crossref","unstructured":"Pogodaev, N.: Optimal control of continuity equations. Nonlinear Differ. Equ. Appl. 23, pp. Art. 21, 24 (2016)","DOI":"10.1007\/s00030-016-0357-2"},{"key":"1223_CR32","doi-asserted-by":"crossref","unstructured":"Qi, L., Sun, D.: A nonsmooth version of Newton\u2019s methods. Math. Programm. 58, 353\u2013367 (1993)","DOI":"10.1007\/BF01581275"},{"key":"1223_CR33","volume-title":"Real and Complex Analysis","author":"W Rudin","year":"1987","unstructured":"Rudin, W.: Real and Complex Analysis, 3rd edn. McGraw-Hill, New York (1987)","edition":"3"},{"key":"1223_CR34","first-page":"267","volume":"1","author":"V Rusanov","year":"1961","unstructured":"Rusanov, V.: Calculation of intersection of non-steady shock waves with obstacles. J. Comput. Math. Phys. USSR 1, 267\u2013279 (1961)","journal-title":"J. Comput. Math. Phys. USSR"},{"key":"1223_CR35","doi-asserted-by":"publisher","first-page":"439","DOI":"10.1016\/0021-9991(88)90177-5","volume":"77","author":"C-W Shu","year":"1988","unstructured":"Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77, 439\u2013471 (1988)","journal-title":"J. Comput. Phys."},{"key":"1223_CR36","doi-asserted-by":"publisher","first-page":"3084","DOI":"10.1137\/120878641","volume":"51","author":"RL Speth","year":"2013","unstructured":"Speth, R.L., Green, W.H., MacNamara, S., Strang, G.: Balanced splitting and rebalanced splitting. SIAM J. Numer. Anal. 51, 3084\u20133105 (2013)","journal-title":"SIAM J. Numer. Anal."},{"key":"1223_CR37","doi-asserted-by":"publisher","first-page":"159","DOI":"10.1007\/s10589-007-9150-9","volume":"44","author":"G Stadler","year":"2009","unstructured":"Stadler, G.: Elliptic optimal control problems with $$L^1$$-control cost and applications for the placement of control devices. Comput. Optim. Appl. 44, 159\u2013181 (2009)","journal-title":"Comput. Optim. Appl."},{"key":"1223_CR38","doi-asserted-by":"publisher","first-page":"506","DOI":"10.1137\/0705041","volume":"5","author":"G Strang","year":"1968","unstructured":"Strang, G.: On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5, 506\u2013517 (1968)","journal-title":"SIAM J. Numer. Anal."},{"key":"1223_CR39","unstructured":"Tang, T., Teng, Z.: Monotone Difference Schemes for Two Dimensional Nonhomogeneous Conservation Laws. Pitman Research Notes in Mathematics Series, pp. 229\u2013243 (1998)"},{"key":"1223_CR40","volume-title":"Optimal Control of Partial Differential Equations. Graduate Studies in Mathematics","author":"F Tr\u00f6ltzsch","year":"2010","unstructured":"Tr\u00f6ltzsch, F.: Optimal Control of Partial Differential Equations. Graduate Studies in Mathematics, vol. 112. American Mathematical Society, Providence (2010)"},{"key":"1223_CR41","volume-title":"Semismooth Newton Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces. MOS-SIAM Series on Optimization","author":"M Ulbrich","year":"2011","unstructured":"Ulbrich, M.: Semismooth Newton Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces. MOS-SIAM Series on Optimization, vol. 11. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2011)"},{"key":"1223_CR42","first-page":"3","volume":"25","author":"M Urabe","year":"1961","unstructured":"Urabe, M.: Theory of errors in numerical integration of ordinary differential equations. J. Sci. Hiroshima Univ. Ser. A-I (Math.) 25, 3\u201362 (1961)","journal-title":"J. Sci. Hiroshima Univ. Ser. A-I (Math.)"}],"container-title":["Numerische Mathematik"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-021-01223-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00211-021-01223-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-021-01223-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,9,7]],"date-time":"2024-09-07T00:40:20Z","timestamp":1725669620000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00211-021-01223-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,8,20]]},"references-count":42,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2021,9]]}},"alternative-id":["1223"],"URL":"https:\/\/doi.org\/10.1007\/s00211-021-01223-6","relation":{},"ISSN":["0029-599X","0945-3245"],"issn-type":[{"type":"print","value":"0029-599X"},{"type":"electronic","value":"0945-3245"}],"subject":[],"published":{"date-parts":[[2021,8,20]]},"assertion":[{"value":"13 February 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 May 2021","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 July 2021","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 August 2021","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}