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Complexity of ten decision problems in continuous time dynamical systems
"... Abstract — We show that for continuous time dynamical systems described by polynomial differential equations of modest degree (typically equal to three), the following decision problems which arise in numerous areas of systems and control theory cannot have a polynomial time (or even pseudopolynomi ..."
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Abstract — We show that for continuous time dynamical systems described by polynomial differential equations of modest degree (typically equal to three), the following decision problems which arise in numerous areas of systems and control theory cannot have a polynomial time (or even pseudopolynomial time) algorithm unless P=NP: local attractivity of an equilibrium point, stability of an equilibrium point in the sense of Lyapunov, boundedness of trajectories, convergence of all trajectories in a ball to a given equilibrium point, existence of a quadratic Lyapunov function, invariance of a ball, invariance of a quartic semialgebraic set under linear dynamics, local collision avoidance, and existence of a stabilizing control law. We also extend our earlier NPhardness proof of testing local asymptotic stability for polynomial vector fields to the case of trigonometric differential equations of degree four. I.
Stability of polynomial differential equations: Complexity and converse lyapunov questions
 CoRR
"... 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 ..."
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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 NPhard. Simple variations of our proof are shown to imply strong NPhardness 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, conver
Lower Bounds on Complexity of Lyapunov Functions for Switched Linear Systems
"... We show that for any positive integer d, there are families of switched linear systems— in fixed dimension and defined by two matrices only—that are stable under arbitrary switching but do not admit (i) a polynomial Lyapunov function of degree ≤ d, or (ii) a polytopic Lyapunov function with ≤ d face ..."
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We show that for any positive integer d, there are families of switched linear systems— in fixed dimension and defined by two matrices only—that are stable under arbitrary switching but do not admit (i) a polynomial Lyapunov function of degree ≤ d, or (ii) a polytopic Lyapunov function with ≤ d facets, or (iii) a piecewise quadratic Lyapunov function with ≤ d pieces. This implies that there cannot be an upper bound on the size of the linear and semidefinite programs that search for such stability certificates. Several constructive and nonconstructive arguments are presented which connect our problem to known (and rather classical) results in the literature regarding the finiteness conjecture, undecidability, and nonalgebraicity of the joint spectral radius. In particular, we show that existence of an extremal piecewise algebraic Lyapunov function implies the finiteness property of the optimal product, generalizing a result of Lagarias and Wang. As a corollary, we prove that the finiteness property holds for sets of matrices with an extremal Lyapunov function belonging to some of the most popular function classes in controls. Index terms: stability of switched systems, linear difference inclusions, the finiteness conjecture of the joint spectral radius, convex optimization for Lyapunov analysis. 1