TY - GEN

T1 - Converse results on existence of sum of squares Lyapunov functions

AU - Ahmadi, Amir Ali

AU - Parrilo, Pablo A.

PY - 2011/12/1

Y1 - 2011/12/1

N2 - Despite the pervasiveness of sum of squares (sos) techniques in Lyapunov analysis of dynamical systems, the converse question of whether sos Lyapunov functions exist whenever polynomial Lyapunov functions exist has remained elusive. In this paper, we first show via an explicit counterexample that if the degree of the polynomial Lyapunov function is fixed, then sos programming can fail to find a valid Lyapunov function even though one exists. On the other hand, if the degree is allowed to increase, we prove that existence of a polynomial Lyapunov function for a homogeneous polynomial vector field implies existence of a polynomial Lyapunov function that is sos and that the negative of its derivative is also sos. The latter result is extended to develop a converse sos Lyapunov theorem for robust stability of switched linear systems.

AB - Despite the pervasiveness of sum of squares (sos) techniques in Lyapunov analysis of dynamical systems, the converse question of whether sos Lyapunov functions exist whenever polynomial Lyapunov functions exist has remained elusive. In this paper, we first show via an explicit counterexample that if the degree of the polynomial Lyapunov function is fixed, then sos programming can fail to find a valid Lyapunov function even though one exists. On the other hand, if the degree is allowed to increase, we prove that existence of a polynomial Lyapunov function for a homogeneous polynomial vector field implies existence of a polynomial Lyapunov function that is sos and that the negative of its derivative is also sos. The latter result is extended to develop a converse sos Lyapunov theorem for robust stability of switched linear systems.

UR - http://www.scopus.com/inward/record.url?scp=84860701101&partnerID=8YFLogxK

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U2 - 10.1109/CDC.2011.6161493

DO - 10.1109/CDC.2011.6161493

M3 - Conference contribution

AN - SCOPUS:84860701101

SN - 9781612848006

T3 - Proceedings of the IEEE Conference on Decision and Control

SP - 6516

EP - 6521

BT - 2011 50th IEEE Conference on Decision and Control and European Control Conference, CDC-ECC 2011

T2 - 2011 50th IEEE Conference on Decision and Control and European Control Conference, CDC-ECC 2011

Y2 - 12 December 2011 through 15 December 2011

ER -