Linear-quadratic optimal control for abstract differential-algebraic equations (2402.08762v2)
Abstract: In this paper, we extend a classical approach to linear quadratic (LQ) optimal control via Popov operators to abstract linear differential-algebraic equations (ADAEs) in Hilbert spaces. To ensure existence of solutions, we assume that the underlying differential-algebraic equation has index one in the pseudo-resolvent sense. This leads to the existence of a degenerate semigroup that can be used to define a Popov operator for our system. It is shown that under a suitable coercivity assumption for the Popov operator the optimal costs can be described by a bounded Riccati operator and that the optimal control input is of feedback form. Furthermore, we characterize exponential stability of ADAEs which is required to solve the infinite horizon LQ problem.
- Vector-valued Laplace Transforms and Cauchy Problems, 2nd edition, volume 96 of Monographs in Mathematics. Birkhäuser, Basel, 2011.
- W. Arendt. Approximation of degenerate semigroups. Taiwanese J. Math, 5(2):279–295, 2001.
- T. Berger and T. Reis. Controllability of linear differential-algebraic systems - a survey. In A. Ilchmann and T. Reis, editors, Surveys in Differential-Algebraic Equations I, Differential-Algebraic Equations Forum, pages 1–69. Springer, Berlin/Heidelberg, 2013.
- A. Favini and A. Yagi. Degenerate differential equations on Banach spaces. Marcel Dekker, New York, 1999.
- H. Gernandt and T. Reis. A pseudo-resolvent approach to abstract differential-algebraic equations, 2023. arXiv:2312.02303v1.
- B. Jacob and K. A. Morris. On solvability of dissipative partial differential-algebraic equations. IEEE Control Systems Letters, 6:3188–3193, 2022.
- B. Jacob and H. Zwart. Linear Port-Hamiltonian Systems on Infinite-dimensional Spaces, volume 223 of Operator Theory: Advances and Applications. Birkhäuser, 2012.
- T. Kato. Remarks on pseudo-resolvents and infinitesimal generators of semi-groups. Proceedings of the Japan Academy, 35(8):467 – 468, 1959.
- V. L. Mehrmann. The autonomous linear quadratic control problem: theory and numerical solution. Springer, 1991.
- V. Mehrmann and H. Zwart. Abstract dissipative Hamiltonian differential-algebraic equations are everywhere. arXiv:2311.03091, 2023.
- R. Rebarber. Conditions for the equivalence of internal and external stability of distributed parameter systems. IEEE Trans. Autom. Control, 38:994–998, 1993.
- T. Reis. Systems Theoretic Aspects of PDAEs and Applications to Electrical Circuits. Doctoral dissertation, Fachbereich Mathematik, Technische Universität Kaiserslautern, Kaiserslautern, 2006.
- T. Reis. Consistent initialization and perturbation analysis for abstract differential-algebraic equations. Math. Control Signals Systems, 19(3):255–281, 2007.
- T. Reis and C. Tischendorf. Frequency domain methods and decoupling of linear infinite dimensional differential algebraic systems. Journal of Evolution equation, 5(3):357–385, 2005.
- T. Reis and M. Voigt. Linear-quadratic optimal control of differential-algebraic systems: the infinite time horizon problem with zero terminal state. SIAM Journal on Control and Optimization, 57(3):1567–1596, 2019.
- M. Tucsnak and G. Weiss. Observation and control for operator semigroups. Springer Science & Business Media, 2009.
- G. Weiss. Representation of shift-invariant operators on L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT by H∞superscript𝐻H^{\infty}italic_H start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT transfer functions: An elementary proof, a generalization to Lpsuperscript𝐿𝑝L^{p}italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT, and a counterexample for L∞superscript𝐿L^{\infty}italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT. Math. Control Signal Systems, 4:193–203, 1991.
- M. Weiss and G. Weiss. Optimal control of stable weakly regular linear systems. Math. Control Signal Systems, 266(10):287–330, 1997.
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