On the hyper-Lyapunov matrix inclusions. (English) Zbl 1541.15012
Author’s abstract: By adding a scalar parameter to the classical Lyapunov matrix inequality, the underlying structure turns to be richer: Partial order of stability sets is introduced. This is then used to improve estimates of trajectories associated with differential inclusions.
Technically, the spectrum of all matrices, satisfying a given Lyapunov inequality, lies within a special disk in the right-half plane: Under inversion such a disk is mapped onto itself. As a by-product, it is shown that these disks are a natural tool to understanding the Matrix-Sign-Function iteration scheme, used in matrix computations.
Hyper-Lyapunov inclusions are formulated through Matrix-Quadratic-Form Inequalities and so are the analogous Hyper-Stein sets of matrices whose spectrum lies within sub-unit disks.
Technically, the spectrum of all matrices, satisfying a given Lyapunov inequality, lies within a special disk in the right-half plane: Under inversion such a disk is mapped onto itself. As a by-product, it is shown that these disks are a natural tool to understanding the Matrix-Sign-Function iteration scheme, used in matrix computations.
Hyper-Lyapunov inclusions are formulated through Matrix-Quadratic-Form Inequalities and so are the analogous Hyper-Stein sets of matrices whose spectrum lies within sub-unit disks.
Reviewer: Costică Moroşanu (Iaşi)
MSC:
| 15A30 | Algebraic systems of matrices |
| 15A39 | Linear inequalities of matrices |
| 15A18 | Eigenvalues, singular values, and eigenvectors |
| 15B48 | Positive matrices and their generalizations; cones of matrices |
| 34H05 | Control problems involving ordinary differential equations |
| 47N70 | Applications of operator theory in systems, signals, circuits, and control theory |
| 93B20 | Minimal systems representations |
| 93C15 | Control/observation systems governed by ordinary differential equations |
Keywords:
invertible disks; matrix-convex sets; matrix-convex invertible set; matrix quadratic form; matrix sign function; Lyapunov inclusion; differential inclusionSoftware:
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