A linear relation approach to port-Hamiltonian differential-algebraic equations. (English) Zbl 07365308
Summary: We consider linear port-Hamiltonian differential-algebraic equations. Inspired by the geometric approach of A. van der Schaft and B. Maschke [Syst. Control Lett. 121, 31–37 (2018; Zbl 1408.93019)] and the linear algebraic approach of C. Mehl et al. [SIAM J. Matrix Anal. Appl. 39, No. 3, 1489–1519 (2018; Zbl 1403.15009)], we present another view by using the theory of linear relations. We show that this allows us to elaborate the differences and mutualities of the geometric and linear algebraic views, and we introduce a class of DAEs which comprises these two approaches. We further study the properties of matrix pencils arising from our approach via linear relations.
MSC:
| 34A09 | Implicit ordinary differential equations, differential-algebraic equations |
| 47A06 | Linear relations (multivalued linear operators) |
| 15A22 | Matrix pencils |
| 34A30 | Linear ordinary differential equations and systems |
Keywords:
linear relations; port-Hamiltonian systems; differential-algebraic equations; matrix pencilsReferences:
| [1] | T. Azizov, J. Behrndt, P. Jonas, and C. Trunk, Compact and finite rank perturbations of closed linear operators and relations in Hilbert spaces, Integral Equations Operator Theory, 63 (2009), pp. 151-163. · Zbl 1194.47016 |
| [2] | T. Azizov, A. Dijksma, and G. Wanjala, Compressions of maximal dissipative and self-adjoint linear relations and of dilations, Linear Algebra Appl., 439 (2013), pp. 771-792. · Zbl 1300.47009 |
| [3] | C. Beattie, V. Mehrmann, H. Xu, and H. Zwart, Port-Hamiltonian descriptor systems, Math. Control Signals Systems, 30 (2017), 17. · Zbl 1401.37070 |
| [4] | J. Behrndt, S. Hassi, and H. de Snoo, Boundary Value Problems, Weyl Functions, and Differential Operators, Monogr. Math. 108, Birkhäuser, Basel, 2020. · Zbl 1457.47001 |
| [5] | T. Berger and T. Reis, Structural properties of positive real and reciprocal rational matrices, in Proceedings of the MTNS 2014, Groningen, Netherlands, Birkhäuser, Boston, 2014, pp. 402-409. |
| [6] | T. Berger and T. Reis, Zero dynamics and funnel control for linear electrical circuits, J. Franklin Inst., 351 (2014), pp. 5099-5132. · Zbl 1307.93339 |
| [7] | T. Berger, C. Trunk, and H. Winkler, Linear relations and the Kronecker canonical form, Linear Algebra Appl., 488 (2016), pp. 13-44. · Zbl 1327.15021 |
| [8] | R. Cross, Multivalued Linear Operators, Marcel Dekker, New York, 1998. · Zbl 0911.47002 |
| [9] | F. Gantmacher, The Theory of Matrices, Vol. II, Chelsea, New York, 1959. · Zbl 0085.01001 |
| [10] | R. Horn and C. Johnson, Matrix Analysis, Cambridge University Press, New York, 2013. · Zbl 1267.15001 |
| [11] | B. Jacob and H. Zwart, Linear Port-Hamiltonian Systems on Infinite-Dimensional Spaces, Oper. Theory Adv. Appl. 223, Birkhäuser, Basel, 2012. · Zbl 1254.93002 |
| [12] | C. Mehl, V. Mehrmann, and M. Wojtylak, Linear algebra properties of dissipative Hamiltonian descriptor systems, SIAM J. Matrix Anal. Appl., 39 (2018), pp. 1489-1519. · Zbl 1403.15009 |
| [13] | C. Mehl, V. Mehrmann, and M. Wojtylak, Distance problems for dissipative Hamiltonian systems and related matrix polynomials, Linear Algebra Appl., 623 (2021), pp. 335-366, https://doi.org/10.1016/j.laa.2020.05.026. · Zbl 1466.15010 |
| [14] | T. Reis and T. Stykel, Lyapunov balancing for passivity-preserving model reduction of RC circuits, SIAM J. Appl. Dyn. Syst., 10 (2011), pp. 1-34. · Zbl 1213.78038 |
| [15] | A. van der Schaft, Port-Hamiltonian differential-algebraic systems, in Surveys in Differential-Algebraic Equations I, A. Ilchmann and T. Reis, eds., Diff.-Algebra. Equ. Forum, Springer, Berlin, 2013, pp. 173-226. · Zbl 1275.34002 |
| [16] | A. van der Schaft and B. Maschke, Generalized port-Hamiltonian DAE systems, Systems Control Lett., 121 (2018), pp. 31-37. · Zbl 1408.93019 |
| [17] | A. van der Schaft and D. Jeltsema, Port-Hamiltonian systems theory: An introductory Overview, NOW Foundations and Trends in Systems and Control, 1 (2014), pp. 173-378. · Zbl 1496.93055 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
