AARJMD VOLUME 1 ISSUE 20 (APRIL 2014) ISSN : 2319 - 2801 Asian Academic Research Journal of Multidisciplinary www.asianacademicresearch.org 550 A Peer Reviewed International Journal of Asian Academic Research Associates AARJMD ASIAN ACADEMIC RESEARCH JOURNAL OF MULTIDISCIPLINARY COMMON LYAPUNOV FUNCTION FOR QUADRATIC STABILIZATION OF A FINITE COLLECTION OF UNCERTAIN NON-LINEAR DYNAMICAL SYSTEMS DR.SALAH EL -DIN AHMED MOHAMMED EL-TAHIR*; DR.RADHI ALI ZABOON** *Omdurman Ahlia University Sudan **AlmustansiriyahUniversity Iraq ABSTRACT The main theme of this work is oriented toward the stabilizability of a, collection of uncertain linear dynamical systems which satisfy some matching conditions and in which the nominal parts representation are not restricted to be in block companion form Tahir and Zaboon(2013)[1] the stabilization is based on a construction of a quadratic Lyapunov function. The theorems which are stated and proved in this work are based on the construction of such common function and a single state feedback controller for guarantee of stability of the collection of the systems. firstly the uncertainties in the state matrix keeping the input matrix fixed. Is considered. It is shown that the uncertain linear dynamical systems are stabilizable via a common Lyapunov function and a single state feedback controller. Then uncertainties in both state matrix and input matrix are dealt with and it is proved that these systems are stabilizable through the same conditions via the same common Lyapunov function as when only uncertainties in the state matrix are considered Then the approaches are extended for uncertain non-linear dynamical systems with uncertainties in the state matrix and it is proved that these systems are stabilized via a common Lyapunov function and single state feedback controller. Also, an algorithm based on the theorem is represented for computational purposes. Key words: common Lyapunov function ,state feedback controller ,.uncertain systems ,matching conditions ,stability