Dissipative stochastic dynamical systems

Nonlinear Optimal Control for Stochastic Dynamical Systems

“…In this section, we recall several key results from [28] on stochastic dissipativity that are necessary for several results of this paper. For the dynamical system G given by ( 1) and (2), a function r : R m × R l → R is called a supply rate if, for all t ≥ 0 and u(•) ∈ U ,…”

“…Theorem 4 gives an equivalent characterization for stochastic dissipativity as defined by the energetic (i.e., supermartingale) Definition 2 using the power balance inequality (28). The energetic (i.e., supermartingale) definition of dissipativity requires the verification of ( 26) which is sample path dependent and can be difficult to verify in practice, whereas (28) is an algebraic condition for dissipativity involving a local power balance inequality using the system drift and diffusion functions of the stochastic dynamical system. This equivalence holds under the regularity conditions stated in Theorem 4.…”

“…Furthermore, the paper establishes connections between stochastic stability margins, stochastic meaningful inverse optimality, and stochastic dissipativity [27,28], showcasing the equivalence between stochastic dissipativity and optimality for stochastic dynamical systems. Specifically, utilizing extended Kalman-Yakubovich-Popov conditions characterizing stochastic dissipativity, our optimal feedback control law satisfies a return difference inequality predicated on the infinitesimal generator of a controlled Markov diffusion process, connecting optimality to stochastic dissipativity with a specific quadratic supply rate.…”

Stochastic thermodynamics: dissipativity, accumulativity, energy storage and entropy production

“…In this paper, we use a dissipative stochastic dynamical systems framework to develop a stochastic thermodynamic framework that is predicated on an alternative model from the model presented in [5][6][7][8][9]13]. Specifically, building on the results in [18,19], we define the notions of stochastic dissipativity, stochastic losslessness and stochastic accumulativity for an open Itô diffusion model driven by a Markov input process generating a unique solution satisfying the Markov property. Specifically, dissipativity, losslessness and accumulativity are defined in terms of dissipation, conservation and accumulation (in)equalities that hold in expectation and involve the difference between the stored and supplied system energy being a supermartingale, martingale and submartingale with respect to the system filtration.…”