The Minkowski-Lyapunov equation for linear dynamics: theoretical foundations. (English) Zbl 1297.93124
Automatica 50, No. 8, 2015-2024 (2014); corrigendum ibid. 106, 411-412 (2019).
Summary: We consider the Lyapunov equation for the linear dynamics, which arises naturally when one seeks for a Lyapunov function with a uniform, exact decrease. In this setting, a solution to the Lyapunov equation has been characterized only for quadratic Lyapunov functions. We demonstrate that the Lyapunov equation is a well-posed equation for strictly stable dynamics and a much more general class of Lyapunov functions specified via Minkowski functions of proper \(C\)-sets, which include Euclidean and weighted Euclidean vector norms, polytopic and weighted polytopic (\(1, \infty\))-vector norms as well as vector semi-norms induced by the Minkowski functions of proper \(C\)-sets. Furthermore, we establish that the Lyapunov equation admits a basic solution, i.e., the unique solution within the class of Minkowski functions associated with proper \(C\)-sets. Finally, we provide a characterization of the lower and upper approximations of the basic solution that converge pointwise and compactly to it, while, in addition, the upper approximations satisfy the classical Lyapunov inequality.
MSC:
| 93D05 | Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory |
| 93D30 | Lyapunov and storage functions |
| 93C05 | Linear systems in control theory |
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