An invariance kernel representation of ISDS Lyapunov functions

We apply set valued analysis techniques in order to characterize the input-to-state dynamical stability (ISDS) property, a variant of the well known input-to-state stability (ISS) property. Using a suitable augmented differential inclusion we are able to characterize the epigraphs of minimal ISDS Lyapunov functions as invariance kernels. This characterization gives new insight into local ISDS properties and provides a basis for a numerical approximation of ISDS and ISS Lyapunov functions via set oriented numerical methods.ou

Nonlinear Iwasawa Decomposition Of Control Flows

Let θ(t, ·,u) be the flow of a control system on a Riemannian manifold M of constant curvature. For a given initial orthonormal frame k in the tangent space Tx0M for some x0 ∈ M, there exists a unique decomposition θt = ⊖t o pt where ⊖t is a control flow in the group of isometries of M and the remainder component pt fixes x0 with derivative Dpt{k) = k · stSt where St are upper triangular matrices. Moreover, if M is flat, an affine component can be extracted from the remainder.1802/03/15339354Arnold, L., (1998) Random Dynamical Systems, , Springer-VerlagArnold, L., Imkeller, P., Rotation numbers for linear stochastic differential equations (1999) Ann. Probab, 27, pp. 130-149Aubin, J., Frankowska, H., (1990) Set-Valued Analysis, , BirkhäuserColonius, F., Kliemann, W., (2000) The Dynamics of Control, , BirkhäuserColonius, F., Kliemann, W., Limits of Input-to-State Stability (2003) System and Control Letters, 49, pp. 111-120K.D. Elworthy, Geometric Aspects of Diffusions on Manifolds, in Ecole d'Eté de Probabilités de Saint-Flour(ed. P.L. Hennequin), XV, XVII, 1985, 1987, 276 - 425. Lecture Notes Math. 1362, Springer-Verlag, 1987Johnson, R.A., Palmer, K.R., Sell, G.R., Ergodic properties of linear dynamical systems (1987) SIAM J. Math. Anal. Appl, 18, pp. 1-33Klingenberg, W., (1982) Riemannian Geometry, , Walter de GruyterKobayashi, S., Nomizu, K., (1963) Foundations of Differential Geometry, 1. , WileyInterscience PublicationH. Kunita, Stochastic differential equations and stochastic flows of diffeomorphisms, in Ecole d'Eté de Probabilités de Saint-Flour (ed. P.L. Hennequin), XII - 1982, 143-303, Lecture Notes Math. 1097, Springer-Verlag, 1984Kunita, H., (1988) Stochastic Flows and Stochastic Differential Equations, , Cambridge University PressLiao, M., Liapounov exponents of stochastic flows (1997) Ann. Probab, 25, pp. 1241-1256Liao, M., Decomposition of stochastic flows and Lyapunov exponents (2000) Probab. Theory Rel. Fields, 117, pp. 589-607Ratcliffe, J.G., The Mathematical Legacy of Wilhelm Magnus: Groups, Geometry and Special Functions (1992) On the isometry groups of hyperbolic manifolds, pp. 491-495. , Brooklyn, NY, Contemp. Math, 169, Amer. Math. Soc, Providence, RIRuffino, P.R.C., (1999) Matrix of rotation for stochastic dynamical systems, 18, pp. 213-226. , Computational and Applied Maths, SBMARuffino, P.R.C., Decomposition of stochastic flows and rotation matrix (2002) Stochastic and Dynamics, 2, pp. 93-108P. R. C. Ruffino, Non-linear Iwasawa decomposition of stochastic flows: geometrical characterization and examples, in Proceedings of Semigroup Operators: Theory and Applications II(eds. C. Kubrusly, N. Levan, M. da Silveira), (SOTA-2), Rio de Janeiro, 10-14 Sep. 2001, 213-226, Optimization Software, Los Angeles, 2002San Martin, L.A.B., Tonelli, P.A., Semigroup actions on homogeneous spaces (1995) Semigroup Forum, 50, pp. 59-8