Continuous-Time Stochastic (ct-SS)

Functions for computing barrier certificates for continuous-time stochastic systems over a finite time horizon with Brownian and Poisson noise.

ct_SS

from src.functions.ct_SS import ct_SS

result = ct_SS(b_degree, dim, L_initial, U_initial, L_unsafe, U_unsafe,
               L_space, U_space, x, f, delta, rho, p_rate, t,
               optimize=False, solver="mosek", confidence=None, gam=None,
               lam=None, c_val=None, l_degree=None)

Parameters:

Parameter	Type	Description
b_degree	int	Degree of the barrier polynomial.
dim	int	Dimension of the state space.
L_initial	np.ndarray	Lower bounds of the initial set. Shape: (dim,).
U_initial	np.ndarray	Upper bounds of the initial set. Shape: (dim,).
L_unsafe	np.ndarray	Lower bounds of unsafe set(s). Shape: (n_regions, dim).
U_unsafe	np.ndarray	Upper bounds of unsafe set(s). Shape: (n_regions, dim).
L_space	np.ndarray	Lower bounds of the state space. Shape: (dim,).
U_space	np.ndarray	Upper bounds of the state space. Shape: (dim,).
x	tuple	SymPy symbols for state variables.
f	np.ndarray	Array of SymPy expressions defining the drift dynamics \(\dot{x} = f(x)\).
delta	np.ndarray	Diffusion coefficient(s) for Brownian motion. Shape: (dim,).
rho	np.ndarray	Reset map for Poisson jumps. Shape: (dim,).
p_rate	np.ndarray	Poisson rate values. Shape: (dim,).
t	int	Finite time horizon.
optimize	bool	If True, optimize the barrier certificate. Default: False.
solver	str	"mosek" (default) or "cvxopt".
confidence	float or None	Minimal confidence threshold.
gam	float or None	Fixed value for \(\gamma\).
lam	float or None	Fixed value for \(\lambda\).
c_val	float or None	Value for \(c\) (multiplied by time horizon in the confidence bound).
l_degree	int or None	Degree of Lagrangian multipliers. Defaults to b_degree.

Returns: dict with barrier certificate details, or None if infeasible.

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parallel_ct_SS

from src.functions.parallel_ct_SS import parallel_ct_SS

result = parallel_ct_SS(b_degree, dim, L_initial, U_initial, L_unsafe,
                        U_unsafe, L_space, U_space, x, f, delta, rho,
                        p_rate, t, optimize=False, solver="mosek",
                        confidence=None, gam=None, lam=None, c_val=None,
                        l_degree=None)

Searches barrier degrees 2, 4, …, b_degree in parallel. All parameters are identical to ct_SS except b_degree is the maximum degree.

Note

Must be called inside if __name__ == '__main__': due to Python multiprocessing.

Returns: The result dict from the first successful degree, or None.