{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,14]],"date-time":"2026-07-14T17:28:32Z","timestamp":1784050112688,"version":"3.55.0"},"reference-count":86,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,10,17]],"date-time":"2022-10-17T00:00:00Z","timestamp":1665964800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"FONDECYT","award":["1191544"],"award-info":[{"award-number":["1191544"]}]},{"name":"FONDECYT","award":["FB0008"],"award-info":[{"award-number":["FB0008"]}]},{"name":"FONDECYT","award":["ANR-21-CE48-0018"],"award-info":[{"award-number":["ANR-21-CE48-0018"]}]},{"name":"BASAL","award":["1191544"],"award-info":[{"award-number":["1191544"]}]},{"name":"BASAL","award":["FB0008"],"award-info":[{"award-number":["FB0008"]}]},{"name":"BASAL","award":["ANR-21-CE48-0018"],"award-info":[{"award-number":["ANR-21-CE48-0018"]}]},{"name":"IMPACTS","award":["1191544"],"award-info":[{"award-number":["1191544"]}]},{"name":"IMPACTS","award":["FB0008"],"award-info":[{"award-number":["FB0008"]}]},{"name":"IMPACTS","award":["ANR-21-CE48-0018"],"award-info":[{"award-number":["ANR-21-CE48-0018"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>A comprehensive overview of the irreversible port-Hamiltonian system\u2019s formulation for finite and infinite dimensional systems defined on 1D spatial domains is provided in a unified manner. The irreversible port-Hamiltonian system formulation shows the extension of classical port-Hamiltonian system formulations to cope with irreversible thermodynamic systems for finite and infinite dimensional systems. This is achieved by including, in an explicit manner, the coupling between irreversible mechanical and thermal phenomena with the thermal domain as an energy-preserving and entropy-increasing operator. Similarly to Hamiltonian systems, this operator is skew-symmetric, guaranteeing energy conservation. To distinguish from Hamiltonian systems, the operator depends on co-state variables and is, hence, a nonlinear-function in the gradient of the total energy. This is what allows encoding the second law as a structural property of irreversible port-Hamiltonian systems. The formalism encompasses coupled thermo-mechanical systems and purely reversible or conservative systems as a particular case. This appears clearly when splitting the state space such that the entropy coordinate is separated from other state variables. Several examples have been used to illustrate the formalism, both for finite and infinite dimensional systems, and a discussion on ongoing and future studies is provided.<\/jats:p>","DOI":"10.3390\/e24101478","type":"journal-article","created":{"date-parts":[[2022,10,17]],"date-time":"2022-10-17T21:11:51Z","timestamp":1666041111000},"page":"1478","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":21,"title":["An Overview on Irreversible Port-Hamiltonian Systems"],"prefix":"10.3390","volume":"24","author":[{"given":"Hector","family":"Ramirez","sequence":"first","affiliation":[{"name":"Departamento de Electr\u00f3nica, Universidad T\u00e9cnica Federico Santa Mar\u00eda, 2390123 Valparaiso, Chile"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yann","family":"Le Gorrec","sequence":"additional","affiliation":[{"name":"D\u00e9partement d\u2019Automatique et Syst\u00e8mes Micro-M\u00e9catroniques, FEMTO-ST UMR CNRS 6174, Universit\u00e9 de Bourgogne Franche Comt\u00e9, 25030 Besan\u00e7on, France"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,10,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1016\/j.ces.2012.12.002","article-title":"Irreversible port-Hamiltonian systems: A general formulation of irreversible processes with application to the CSTR","volume":"89","author":"Ramirez","year":"2013","journal-title":"Chem. Eng. Sci."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"513","DOI":"10.1016\/j.ejcon.2013.09.009","article-title":"Modelling and control of multi-energy systems: An irreversible port-Hamiltonian approach","volume":"19","author":"Ramirez","year":"2013","journal-title":"Eur. J. Control"},{"key":"ref_3","unstructured":"Maschke, B., and van der Schaft, A. (1992, January 24\u201326). Port controlled Hamiltonian systems: Modeling origins and system theoretic properties. Proceedings of the 3rd IFAC Symposium on Nonlinear Control Systems, NOLCOS\u201992, Bordeaux, France."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1109\/81.372847","article-title":"An intrinsic Hamiltonian formulation of the dynamics of LC-circuits","volume":"42","author":"Maschke","year":"1995","journal-title":"IEEE Trans. Circuits Syst. I Fundam. Theory Appl."},{"key":"ref_5","first-page":"362","article-title":"The Hamiltonian Formulation of Energy Conserving Physical Systems with External Ports","volume":"49","author":"Maschke","year":"1995","journal-title":"Arch. Elektron. \u00dcbertrag."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Duindam, V., Macchelli, A., Stramigioli, S., and Bruyninckx, H. (2009). (Eds.) Modeling and Control of Complex Physical Systems\u2014The Port-Hamiltonian Approach, Springer.","DOI":"10.1007\/978-3-642-03196-0"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Arnold, V.I. (1989). Mathematical Methods of Classical Mechanics, Springer. [2nd ed.]. Graduate Texts in Mathematics.","DOI":"10.1007\/978-1-4757-2063-1"},{"key":"ref_8","unstructured":"Martin, C., and Hermann, R. (1977). Control Theory and Analytical Mechanics. Geometric Control Theory, Math Sci Press."},{"key":"ref_9","unstructured":"Byrnes, C.I., and Lindquist, A. (1986). On Feedback Control of Hamiltonian Systems. Theory and Applications of Nonlinear Control Systems, Elsevier North-Holland."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Marsden, J. (1992). Lectures on Mechanics, Cambridge University Press.","DOI":"10.1017\/CBO9780511624001"},{"key":"ref_11","unstructured":"Van den Hof, P.M.J., Scherer, C., and Heuberger, P.S.C. (2009). Conservation Laws and Lumped System Dynamics. Model Based Control: Bridging Rigorous Theory and Advanced Technology, Springer."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Irschik, H., and Schlacher, K. (2004). Port-Hamiltonian systems: Network modeling and control of nonlinear physical systems. Advanced Dynamics and Control of Structures and Machines, Springer.","DOI":"10.1007\/978-3-7091-2774-2"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.geomphys.2004.01.002","article-title":"Dynamics on Leibniz manifolds","volume":"52","author":"Ortega","year":"2004","journal-title":"J. Geom. Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2135","DOI":"10.1002\/cjce.24505","article-title":"Quo Vadis Advanced Chemical Process Control","volume":"100","author":"Dubljevic","year":"2022","journal-title":"Can. J. Chem. Eng."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1007\/BF00276493","article-title":"Dissipative dynamical systems part I: General theory","volume":"45","author":"Willems","year":"1972","journal-title":"Arch. Ration. Mech. Anal."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"van der Schaft, A.J. (2000). L2-Gain and Passivity Techniques in Nonlinear Control, Springer.","DOI":"10.1007\/978-1-4471-0507-7"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Brogliato, B., Lozano, R., Maschke, B., and Egeland, O. (2020). Dissipative Systems Analysis and Control, Springer. [3rd ed.].","DOI":"10.1007\/978-3-030-19420-8"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1016\/S0009-2509(97)87571-9","article-title":"Robust control of hyperbolic PDE systems","volume":"53","author":"Christofides","year":"1998","journal-title":"Chem. Eng. Sci."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Christofides, P.D. (2001). Nonlinear and Robust Control of PDE Systems: Methods and Applications to Transport-Reaction Processes, Springer. [1st ed.].","DOI":"10.1007\/978-1-4612-0185-4"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1119","DOI":"10.1016\/0098-1354(96)00194-9","article-title":"Process Systems, Passivity and the Second Law of Thermodynamics","volume":"20","author":"Alonso","year":"1996","journal-title":"Comput. Chem. Eng."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1739","DOI":"10.1016\/S0005-1098(01)00140-6","article-title":"Stabilization of distributed systems using irreversible thermodynamics","volume":"37","author":"Alonso","year":"2001","journal-title":"Automatica"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"507","DOI":"10.1016\/S0959-1524(01)00017-8","article-title":"From irreversible thermodynamics to a robust control theory for distributed process systems","volume":"12","author":"Alonso","year":"2002","journal-title":"J. Process. Control"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1016\/j.automatica.2018.04.041","article-title":"Dissipative observers for coupled diffusion\u2013convection\u2013reaction systems","volume":"94","author":"Schaum","year":"2018","journal-title":"Automatica"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"117117","DOI":"10.1063\/5.0067784","article-title":"On port-Hamiltonian formulations of 3-dimensional compressible Newtonian fluids","volume":"33","author":"Mora","year":"2021","journal-title":"Phys. Fluids"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1877","DOI":"10.1016\/j.automatica.2010.07.011","article-title":"Power-shaping control of reaction systems: The CSTR case","volume":"46","author":"Favache","year":"2010","journal-title":"Automatica"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"618","DOI":"10.1016\/j.sysconle.2011.04.021","article-title":"Power-shaping control: Writing the system dynamics into the Brayton\u2013Moser form","volume":"60","author":"Favache","year":"2011","journal-title":"Syst. Control Lett."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1090\/qam\/169746","article-title":"A theory of nonlinear networks. I","volume":"22","author":"Brayton","year":"1964","journal-title":"Q. Appl. Math."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"193","DOI":"10.4310\/jdg\/1214430827","article-title":"On the Mathematical Foundations of Electrical Circuit Theory","volume":"7","author":"Smale","year":"1972","journal-title":"J. Differ. Geom."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"6620","DOI":"10.1103\/PhysRevE.56.6620","article-title":"Dynamics and thermodynamics of complex fluids. I. Development of a general formalism","volume":"56","author":"Grmela","year":"1997","journal-title":"Phys. Rev. E"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"6633","DOI":"10.1103\/PhysRevE.56.6620","article-title":"Dynamics and thermodynamics of complex fluids. II. Illustrations of a general formalism","volume":"56","author":"Grmela","year":"1997","journal-title":"Phys. Rev. E"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"448","DOI":"10.1016\/S0378-4371(00)00252-1","article-title":"A simple example for comparing GENERIC with rational non-equilibrium Thermodynamics","volume":"285","author":"Mushik","year":"2000","journal-title":"Physica A"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1016\/j.jprocont.2008.06.018","article-title":"On the control of non-linear processes: An IDA\u2013PBC approach","volume":"19","author":"Ramirez","year":"2009","journal-title":"J. Process Control"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1449","DOI":"10.1016\/j.jprocont.2011.06.014","article-title":"The port Hamiltonian approach to modelling and control of Continuous Stirred Tank Reactors","volume":"21","author":"Hoang","year":"2011","journal-title":"J. Process Control"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"412","DOI":"10.1016\/j.jprocont.2011.12.007","article-title":"Lyapunov-based control of non isothermal continuous stirred tank reactors using irreversible thermodynamics","volume":"22","author":"Hoang","year":"2012","journal-title":"J. Process Control"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Gay-Balmaz, F., and Yoshimura, H. (2018). A Variational Formulation of Nonequilibrium Thermodynamics for Discrete Open Systems with Mass and Heat Transfer. Entropy, 20.","DOI":"10.3390\/e20030163"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1016\/0034-4877(91)90017-H","article-title":"Contact structure in thermodynamic theory","volume":"29","author":"Mrugala","year":"1991","journal-title":"Rep. Math. Phys."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1016\/S0034-4877(07)00024-9","article-title":"An extension of Hamiltonian systems to the thermodynamic phase space: Towards a geometry of nonreversible processes","volume":"60","author":"Eberard","year":"2007","journal-title":"Rep. Math. Phys."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"5204","DOI":"10.1016\/j.ces.2010.06.019","article-title":"An entropy-based formulation of irreversible processes based on contact structures","volume":"65","author":"Favache","year":"2010","journal-title":"Chem. Eng. Sci."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"475","DOI":"10.1016\/j.sysconle.2013.02.008","article-title":"Feedback equivalence of input\u2013output contact systems","volume":"62","author":"Ramirez","year":"2013","journal-title":"Syst. Control Lett."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1431","DOI":"10.1109\/TAC.2016.2572403","article-title":"Partial Stabilization of Input-Output Contact Systems on a Legendre Submanifold","volume":"62","author":"Ramirez","year":"2017","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"779","DOI":"10.1007\/s00161-012-0277-2","article-title":"On a variational principle in thermodynamics","volume":"25","author":"Merker","year":"2013","journal-title":"Contin. Mech. Thermodyn."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"van der Schaft, A., and Maschke, B. (2018). Geometry of Thermodynamic Processes. Entropy, 20.","DOI":"10.3390\/e20120925"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"117107","DOI":"10.1016\/j.ces.2021.117107","article-title":"Boundary controlled irreversible port-Hamiltonian systems","volume":"248","author":"Ramirez","year":"2022","journal-title":"Chem. Eng. Sci."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"1864","DOI":"10.1137\/040611677","article-title":"Dirac structures and Boundary Control Systems associated with Skew-Symmetric Differential Operators","volume":"44","author":"Zwart","year":"2005","journal-title":"SIAM J. Control Optim."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"436","DOI":"10.1016\/j.ifacol.2021.10.393","article-title":"An irreversible port-Hamiltonian model for a class of piezoelectric actuators","volume":"54","author":"Caballeria","year":"2021","journal-title":"IFAC-PapersOnLine"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1016\/j.ifacol.2019.07.004","article-title":"Irreversible Port-Hamiltonian Formulation of some Non-isothermal Electrochemical Processes","volume":"52","author":"Ramirez","year":"2019","journal-title":"IFAC-PapersOnLine"},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Mora, L., Le Gorrec, Y., and Ramirez, H. (2022, January 8\u201310). Available Energy-Based Interconnection and Entropy Assignment (ABI-EA) Boundary Control of the Heat Equation: An Irreversible Port Hamiltonian Approach. Proceedings of the 2022 American Control Conference (ACC), Atlanta, GA, USA.","DOI":"10.23919\/ACC53348.2022.9867213"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"11539","DOI":"10.1016\/j.ifacol.2020.12.630","article-title":"Energy shaping plus Damping injection of Irreversible Port Hamiltonian Systems","volume":"53","author":"Villalobos","year":"2020","journal-title":"IFAC-PapersOnLine"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1016\/j.automatica.2015.07.002","article-title":"On the passivity based control of irreversible processes: A port-Hamiltonian approach","volume":"64","author":"Ramirez","year":"2016","journal-title":"Automatica"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"2527","DOI":"10.1109\/TAC.2008.2006930","article-title":"Control by Interconnection and Standard Passivity-Based Control of Port-Hamiltonian Systems","volume":"53","author":"Ortega","year":"2008","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Libermann, P., and Marle, C.M. (1987). Symplectic Geometry and Analytical Mechanics, D. Reidel Publishing Company.","DOI":"10.1007\/978-94-009-3807-6"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"426","DOI":"10.1007\/BFb0008472","article-title":"System theory and mechanics","volume":"Volume 135","author":"Nijmeijer","year":"1989","journal-title":"Three Decades of Mathematical System Theory"},{"key":"ref_53","unstructured":"Maschke, B., and van der Schaft, A. (1997). Interconnected Mechanical systems. Part 1 and 2. Modelling and Control of Mechanical Systems, Imperial College Press."},{"key":"ref_54","doi-asserted-by":"crossref","unstructured":"Ortega, R., Astolfi, A., Bastin, G., and Rodriguez, H. (2000, January 28\u201330). Stabilization of Food-Chain Systems Using a Port-Controlled Hamiltonian Description. Proceedings of the American Control Conference, Chicago, IL, USA.","DOI":"10.1109\/ACC.2000.878579"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"621","DOI":"10.1016\/j.jprocont.2007.01.005","article-title":"Averaging level control: An approach based on mass balance","volume":"17","author":"Sbarbaro","year":"2007","journal-title":"J. Process Control"},{"key":"ref_56","unstructured":"Bao, J., and Lee, P. (2007). Process Control, The Passive Systems Approach, Springer. [1st ed.]. Advances in Industrial Control."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"1413","DOI":"10.1016\/j.jprocont.2009.07.015","article-title":"An introduction to interconnection and damping assignment passivity-based control in process engineering","volume":"19","author":"Johnsen","year":"2009","journal-title":"J. Process Control"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"230","DOI":"10.1007\/BF00281751","article-title":"Chemical reaction dynamics. Part I: Geometrical structure","volume":"55","author":"Oster","year":"1974","journal-title":"Arch. Ration. Mech. Anal."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"554","DOI":"10.1016\/j.sysconle.2007.12.003","article-title":"Local dissipative Hamiltonian description of reversible reaction networks","volume":"57","author":"Alonso","year":"2008","journal-title":"Syst. Control Lett."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"923","DOI":"10.1016\/S0016-0032(92)90049-M","article-title":"An intrinsic Hamiltonian formulation of network dynamics: Non-standard Poisson structures and gyrators","volume":"329","author":"Maschke","year":"1992","journal-title":"J. Frankl. Inst."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"54","DOI":"10.1137\/S0363012996312039","article-title":"On representation and integrability of mathematical structures in energy-conserving physical systems","volume":"37","author":"Dalsmo","year":"1999","journal-title":"SIAM J. Control Optim."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"585","DOI":"10.1016\/S0005-1098(01)00278-3","article-title":"Interconnection and damping assignment passivity based control of port-controlled Hamiltonian systems","volume":"38","author":"Ortega","year":"2002","journal-title":"Automatica"},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1016\/S0375-9601(02)00190-1","article-title":"Lagrange hydrodynamics as extended Euler hydrodynamics: Hamiltonian and GENERIC structures","volume":"296","author":"Grmela","year":"2002","journal-title":"Phys. Lett. A"},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1016\/j.jnnfm.2003.11.008","article-title":"The mathematical representation of driven thermodynamical systems","volume":"120","author":"Jongschaap","year":"2004","journal-title":"J. Non-Newton. Fluid Mech."},{"key":"ref_65","unstructured":"Callen, H. (1985). Thermodynamics and an Introduction to Thermostatistics, Wiley."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1080\/13873950500068823","article-title":"Bond graph modelling for chemical reactors","volume":"12","author":"Couenne","year":"2006","journal-title":"Math. Comput. Model. Dyn. Syst."},{"key":"ref_67","doi-asserted-by":"crossref","unstructured":"Jacob, B., and Zwart, H. (2012). Linear Port-Hamiltonian Systems on Infinite-dimensional Spaces, Birkh\u00e4user. Operator Theory: Advances and Applications.","DOI":"10.1007\/978-3-0348-0399-1"},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"2849","DOI":"10.1109\/TAC.2014.2315754","article-title":"Exponential Stabilization of Boundary Controlled Port-Hamiltonian Systems with Dynamic Feedback","volume":"59","author":"Ramirez","year":"2014","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"1700","DOI":"10.1109\/TAC.2016.2595263","article-title":"On the Synthesis of Boundary Control Laws for Distributed Port-Hamiltonian Systems","volume":"62","author":"Macchelli","year":"2017","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1016\/j.automatica.2017.07.045","article-title":"Stabilization of infinite dimensional port-Hamiltonian systems by nonlinear dynamic boundary control","volume":"85","author":"Ramirez","year":"2017","journal-title":"Automatica"},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.sysconle.2016.02.002","article-title":"Building systems from simple hyperbolic ones","volume":"91","author":"Zwart","year":"2016","journal-title":"Syst. Control Lett."},{"key":"ref_72","unstructured":"De Groot, S.R., and Mazur, P. (1962). Non-Equilibrium Thermodynamics, North-Holland Publishing Company."},{"key":"ref_73","unstructured":"Bird, R.B., Stewart, W.E., and Lightfoot, E.N. (2006). Transport Phenomena, John Wiley and Sons, Inc."},{"key":"ref_74","doi-asserted-by":"crossref","unstructured":"Kjelstrup, S., Bedeaux, D., Johannessen, E., and Gross, J. (2017). Non-Equilibrium Thermodynamics for Engineers, World Scientific. [2nd ed.].","DOI":"10.1142\/10286"},{"key":"ref_75","first-page":"115","article-title":"Compositional modelling of distributed-parameter systems","volume":"Volume 311","author":"Maschke","year":"2005","journal-title":"Advanced Topics in Control Systems Theory. Lecture Notes from FAP 2004"},{"key":"ref_76","doi-asserted-by":"crossref","unstructured":"Olver, P. (1993). Application of Lie Groups to Differential Equations, Springer.","DOI":"10.1007\/978-1-4612-4350-2"},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"166","DOI":"10.1016\/S0393-0440(01)00083-3","article-title":"Hamiltonian formulation of distributed-parameter systems with boundary energy flow","volume":"42","author":"Maschke","year":"2002","journal-title":"J. Geom. Phys."},{"key":"ref_78","unstructured":"Maschke, B., Philipp, F., Schaller, M., Worthhmann, K., and Faulwasser, T. (2022, January 12\u201316). Optimal control of thermodynamic port-Hamiltonian systems. Proceedings of the 25th International Symposium on Mathematical Theory of Networks and Systems (MTNS 2022), Bayreuth, Germany."},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1016\/j.ejcon.2021.06.017","article-title":"Control of port-Hamiltonian systems with minimal energy supply","volume":"62","author":"Schaller","year":"2021","journal-title":"Eur. J. Control"},{"key":"ref_80","doi-asserted-by":"crossref","first-page":"62","DOI":"10.1063\/1.2916405","article-title":"Thermodynamics in finite time","volume":"37","author":"Andresen","year":"1984","journal-title":"Phys. Today"},{"key":"ref_81","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1016\/S0360-5442(00)00059-1","article-title":"Principles of control thermodynamics","volume":"26","author":"Salamon","year":"2001","journal-title":"Energy"},{"key":"ref_82","doi-asserted-by":"crossref","unstructured":"Andresen, B., and Salamon, P. (2022). Future Perspectives of Finite-Time Thermodynamics. Entropy, 24.","DOI":"10.3390\/e24050690"},{"key":"ref_83","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1080\/13873954.2021.1979592","article-title":"Exergetic port-Hamiltonian systems: Modelling basics","volume":"27","author":"Lohmayer","year":"2021","journal-title":"Math. Comput. Model. Dyn. Syst."},{"key":"ref_84","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1016\/j.physd.2013.07.017","article-title":"Finite-time thermodynamics of port-Hamiltonian systems","volume":"267","author":"Delvenne","year":"2014","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_85","doi-asserted-by":"crossref","unstructured":"Parker, M.C., and Jeynes, C. (2021). A Relativistic Entropic Hamiltonian\u2013Lagrangian Approach to the Entropy Production of Spiral Galaxies in Hyperbolic Spacetime. Universe, 7.","DOI":"10.3390\/universe7090325"},{"key":"ref_86","doi-asserted-by":"crossref","unstructured":"Califano, F., Rashad, R., and Stramigioli, S. (2022). A differential geometric description of thermodynamics in continuum mechanics with application to Fourier-Navier-Stokes fluids. Phys. Fluids, in press.","DOI":"10.1063\/5.0119517"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/10\/1478\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:55:47Z","timestamp":1760144147000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/10\/1478"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,10,17]]},"references-count":86,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2022,10]]}},"alternative-id":["e24101478"],"URL":"https:\/\/doi.org\/10.3390\/e24101478","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,10,17]]}}}