Homogeneous Lp Stability for Homogeneous Systems

The motivation of this paper comes from the fact that
$\mathcal {L}_{p}-$
stability and
$\mathcal {L}_{p}-$
gain, using the classical signal norms, is not well-defined for arbitrary continuous weighted homogeneous systems. However, using homogeneous signal norms it is possible to show that every internally stable homogeneous system has a globally defined finite homogeneous
$\mathcal {L}_{p}-$
gain, for
$p$
sufficiently large. If the system has a homogeneous approximation, the homogeneous
$\mathcal {L}_{p}-$
gain is inherited locally. Homogeneous
$\mathcal {L}_{p}-$
stability can be characterized by a homogeneous dissipation inequality, which in the input affine case can be transformed to a homogeneous Hamilton-Jacobi inequality. An estimation of an upper bound for the homogeneous
$\mathcal {L}_{p}-$
gain can be derived from these inequalities. Homogeneous
$\mathcal {L}_{\infty }-$
stability is also considered and its strong relationship to Input-to-State stability is studied. These results are extensions to arbitrary homogeneous systems of the well-known situation for linear time-invariant systems, where the Hamilton-Jacobi inequality reduces to an algebraic Riccati inequality. A natural application of finite-gain homogeneous
$\mathcal {L}_{p}-$
stability is in the study of stability for interconnected systems. An extension of the small-gain theorem for negative feedback systems and results for systems in cascade are derived for different homogeneous norms. Previous results in the literature use classical signal norms, hence, they can only be applied to a restricted class of homogeneous systems. The results are illustrated by several examples.