No abstract

The CPA method uses linear programming to compute Continuous and Piecewise Affine Lyapunov function for nonlinear systems with asymptotically stable equilibria. In [14] it was shown that the method always succeeds in computing a CPA Lyapunov function for such a system. The size of the domain of the computed CPA Lyapunov function is only limited by the equilibrium's basin of attraction. However, for some systems, an arbitrary small neighborhood of the equilibrium had to be excluded from the domain a priori. This is necessary, if the equilibrium is not exponentially stable, because the existence of a CPA Lyapunov function in a neighborhood of the equilibrium is equivalent to its exponential stability as shown in [11]. However, if the equilibrium is exponentially stable, then this was an artifact of the method. In this paper we overcome this artifact by developing a revised CPA method. We show that this revised method is always able to compute a CPA Lyapunov function for a system with an exponentially stable equilibrium. The only conditions on the system are that it is C 2 and autonomous. The domain of the CPA Lyapunov function can be any a priori given compact neighborhood of the equilibrium which is contained in its basin of attraction. Whereas in a previous paper [10] we have shown these results for planar systems, in this paper we cover general n-dimensional systems.

No abstract

Given a nonautonomous discrete system with an equilibrium at the origin and a hypercube D containing the origin, we state a linear programming problem, of which any feasible solution parameterizes a continuous and piecewise affine (CPA) Lyapunov function V : D → R for the system. The linear programming problem depends on a triangulation of the hypercube. We prove that if the equilibrium at the origin is exponentially stable, the hypercube is a subset of its basin of attraction, and the triangulation fulfills certain properties, then such a linear programming problem possesses a feasible solution. We suggest an algorithm that generates such linear programming problems for a system, using more and more refined triangulations of the hypercube. In each step the algorithm checks the feasibility of the linear programming problem. This results in an algorithm that is always able to compute a Lyapunov function for a discrete system with an exponentially stable equilibrium. The domain of the Lyapunov function is only limited by the size of the equilibrium s domain of attraction. The system is assumed to have a C 2 right-hand side, but is otherwise arbitrary. Especially, it is not assumed to be of any specific algebraic type like linear, piecewise affine, etc. Our approach is a non-trivial adaption of the CPA method to compute Lyapunov functions for continuous systems to discrete systems.

A Riemannian metric with a local contraction property can be used to prove existence and uniqueness of a periodic orbit and determine a subset of its basin of attraction. While the existence of such a contraction metric is equivalent to the existence of an exponentially stable periodic orbit, the explicit construction of the metric is a difficult problem.In this paper, the construction of such a contraction metric is achieved by formulating it as an equivalent problem, namely a feasibility problem in semidefinite optimization. The contraction metric, a matrix-valued function, is constructed as a continuous piecewise affine (CPA) function, which is affine on each simplex of a triangulation of the phase space. The contraction conditions are formulated as conditions on the values at the vertices.The paper states a semidefinite optimization problem. We prove on the one hand that a feasible solution of the optimization problem determines a CPA contraction metric and on the other hand that the optimization problem is always feasible if the system has an exponentially stable periodic orbit and the triangulation is fine enough. An objective function can be used to obtain a bound on the largest Floquet exponent of the periodic orbit.