Stability of Polynomial Differential Equations: Complexity and Converse Lyapunov Questions

Stability analysis of polynomial differential equations is a central topic in nonlinear dynamics and control which in recent years has undergone major algorithmic developments due to advances in optimization theory. Notably, the last decade has seen a widespread interest in the use of sum of squares (sos) based semidefinite programs that can automatically find polynomial Lyapunov functions and produce explicit certificates of stability. However, despite their popularity, the converse question of whether such algebraic, efficiently constructable certificates of stability always exist has remained elusive. Naturally, an algorithmic question of this nature is closely intertwined with the fundamental computational complexity of proving stability. In this paper, we make a number of contributions to the questions of (i) complexity of deciding stability, (ii) existence of polynomial Lyapunov functions, and (iii) existence of sos Lyapunov functions. (i) We show that deciding local or global asymptotic stability of cubic vector fields is strongly NP-hard. Simple variations of our proof are shown to imply strong NP-hardness of several other decision problems: testing local attractivity of an equilibrium point, stability of an equilibrium point in the sense of Lyapunov, invariance of the unit ball, boundedness of trajectories, convergence of all trajectories in a ball to a given equilibrium point, existence of a quadratic Lyapunov function, local collision avoidance, and existence of a stabilizing control law. (ii) We present a simple, explicit example of a globally asymptotically stable quadratic vector field on the plane which does not admit a polynomial Lyapunov function (joint work with M. Krstic and presented here without proof). For the subclass of homogeneous vector fields, we conjecture that asymptotic stability implies existence of a polynomial Lyapunov function, but show that the minimum degree of such a Lyapunov function can be arbitrarily large even for vector fields in fixed dimension and degree. For the same class of vector fields, we further establish that there is no monotonicity in the degree of polynomial Lyapunov functions. (iii) We show via an explicit counterexample that if the degree of the polynomial Lyapunov function is fixed, then sos programming may 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 planar or a homogeneous vector field implies existence of a polynomial Lyapunov function that is sos and that the negative of its derivative is also sos. ∗Amir Ali Ahmadi is a Goldstine Fellow at the Department of Business Analytics and Mathematical Sciences of the IBM Watson Research Center. Pablo A. Parrilo is with the Laboratory for Information and Decision Systems, Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology. Email: {a a a, parrilo}@mit.edu. †This research was partially supported by the NSF Focused Research Group Grant on Semidefinite Optimization and Convex Algebraic Geometry DMS-0757207. 1 ar X iv :1 30 8. 68 33 v1 [ m at h. O C ] 3 0 A ug 2 01 3