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### Open Access Journals

DCDS-B

Lyapunov's second or direct method is one of the most widely used techniques for
investigating stability properties of dynamical systems. This technique makes use of an
auxiliary function, called a Lyapunov function, to ascertain stability properties for a specific
system without the need to generate system solutions. An important question is the
converse or reversability of Lyapunov's second method; i.e., given a specific stability property
does there exist an appropriate Lyapunov function? We survey some of the available answers to this
question.

DCDS

We present a novel method to compute Lyapunov functions for continuous-time systems with multiple local attractors. In the proposed method one first computes an outer approximation of the local attractors using a graph-theoretic approach.
Then a candidate Lyapunov function is computed using a Massera-like construction adapted to multiple local attractors. In the final step this candidate Lyapunov function is interpolated over the simplices of a simplicial
complex and, by checking certain inequalities at the vertices of the complex, we can identify the region in which the Lyapunov function is decreasing along system trajectories. The resulting Lyapunov function
gives information on the qualitative behavior of the dynamics, including lower bounds on the basins of attraction of the individual local attractors.
We develop the theory in detail and present numerical examples demonstrating the applicability of our method.

JCD

We present a numerical technique for the computation of a Lyapunov function
for nonlinear systems with an asymptotically stable equilibrium point. The proposed approach
constructs a partition of the state space, called a triangulation, and then computes values at
the vertices of the triangulation using a Lyapunov function from a classical converse Lyapunov
theorem due to Yoshizawa. A simple interpolation of the vertex values then yields a Continuous
and Piecewise Affine (CPA) function.
Verification that the obtained CPA function is a Lyapunov function is shown
to be equivalent to verification of several simple inequalities.
Numerical examples are presented demonstrating different aspects of the proposed method.

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