Lyapunov functions

dynax.ConvexLyapunov

Convex Lyapunov normalization.

This module normalizes a convex function by ensuring it is a valid Lyapunov function suitable for showing global stability. It performs the following transformation on a given function \(f:\mathbb{R}^n \rightarrow \mathbb{R}\): $$ F(x) = f(x) - f(x^\ast) - \frac{\partial f}{\partial x} \bigg\vert_{x^\ast}\cdot (x-x^\ast) + \epsilon\left\lVert x-x^\ast \right\rVert^2. $$ This ensures the resulting function \(F\) has a unique minimum at \(x^\ast\) and is positive definite due to the is a small regularization term \(\epsilon\).

__init__(func, state_size, minimum_init=<function zeros>, minimum_learnable=False, epsilon=1e-06, dtype=None, *, key)

Initialize the Lyapunov normalization.

PARAMETER	DESCRIPTION
func	Convex function \(f:\mathbb{R}^n \rightarrow \mathbb{R}\) TYPE: Callable[[Array], Shaped[Array, '']]
state_size	State size \(n\) needed to determine the size for the minimum \(x^\ast\). TYPE: int
minimum_init	Initializer for the minimum location. Can be any function with signature (key, shape, dtype) -> Array but typically is a JAX initializer. TYPE: jax.nn.initializers.Initializer DEFAULT: <function zeros>
minimum_learnable	If True, the minimum location \(x^\ast\) is learnable. Otherwise, its gradients are stopped, preventing updates during optimization. TYPE: bool DEFAULT: False
epsilon	Small value to ensure the Lyapunov function is positive definite. TYPE: float DEFAULT: 1e-06
dtype	The dtype to use for the minimum. Defaults to either jax.numpy.float32 or jax.numpy.float64 depending on whether JAX is in 64-bit mode. TYPE: type | None DEFAULT: None
key	PRNG key for random initialization of the minimum. TYPE: PRNGKeyArray

__call__(x)

Evaluate the Lyapunov function at a point x.

PARAMETER	DESCRIPTION
x	Evaluation point for the Lyapunov function. This should be a vector of shape (n,). TYPE: Array

RETURNS	DESCRIPTION
Shaped[Array, '']	Value of the Lyapunov function at x.