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Dissipative Quantum Mechanics of Superconducting Junctions
Published in Andrei D. Zaikin, Dmitry S. Golubev, Dissipative Quantum Mechanics of Nanostructures, 2019
Andrei D. Zaikin, Dmitry S. Golubev
Taking the limit of vanishing dissipation ZS(ω) → ∞ and switching from the path integral formulation of quantum mechanics to the Hamilton one, we conclude that in the low-energy limit, quantum dynamics of a Josephson junction is indeed described by the Hamilton operator (2.15), as we already anticipated in the beginning of this chapter.
Nonlinear stochastic receding horizon control: stability, robustness and Monte Carlo methods for control approximation
Published in International Journal of Control, 2018
This assumption7is standard in the path integral formulation of optimal control (Kappen, 2005b), but it also appears more generally in the stochastic optimal control literature (Fleming & Soner, 2006). This assumption allows us (Fleming & Soner, 2006; Kappen, 2005b) to write which is a linear partial differential equation on [0, T] with terminal condition ψ(T, x) = exp [−φ(x)/γ]. It now follows by the Feynman–Kac formula that the solution to the above PDE at (0, x) is given by where now is a nonlinear (uncontrolled) process satisfying with initial condition Zt, xt = x. Note that Now, given the solution for ψ(x) derived via the Feynman–Kac formula, it is informally straightforward to devise a Monte Carlo approximation for the control; e.g. one can first simulate sample paths of (11), then form a Monte Carlo approximation of the integral for ψ(x), and approximate the spatial derivative of ψ(x) via differencing. Going forward, we explore a more formal Monte Carlo approximation circumventing the need for crude numerical (spatial) differentiation. First, we need the following result.
A multilevel approach for stochastic nonlinear optimal control
Published in International Journal of Control, 2022
Ajay Jasra, Jeremy Heng, Yaxian Xu, Adrian N. Bishop
This article is structured as follows. In Section 2, we begin by detailing the stochastic optimal control problem of interest and its path integral formulation. We then describe our proposed methodology to compute the optimal control in Section 3, and state some theoretical results on its complexity in Section 4. In Section 5, we validate our theory on two examples, including a nonlinear stochastic compartmental model for an epidemic with cost-controlled vaccination. The appendix features the assumptions and proofs for our complexity theorem in Section 4.