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Nonlinear Filtering, Bellman Equations, and Schrodinger Equations

Akira Ohsumi

Published Sep 1, 1994• 0 citations
OpenAlex

Abstract

In this paper, the relations between theories of the quantum mechanics and the stochastic control are briefly surveyed based on the recent works of the author (Ohsumi, $1989a(Ohsumi, $ ,$ $1989b(Ohsumi, $ $ , 1992)).Concretely, two types of (nonlinear) Schr\"odinger eqations from the stochsatic control theory by introducing wavefunctions which connect two solutions of nonlinear filtering equation and stochastic optimal control problem.Furthermore, an inverse prob- lem is investigated to derive the optimal control problem from a given Schr\"odinger equation.

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In this paper, the relations between theories of the quantum mechanics and the stochastic control are briefly surveyed based on the recent works of the author (Ohsumi, 1989a(Ohsumi,1989a(Ohsumi, 1989a(Ohsumi, , 1989b(Ohsumi, , 1992)).Concretely, two types of (nonlinear) Schr"odinger eqations from the stochsatic control theory by introducing wavefunctions which connect two solutions of nonlinear filtering equation and stochastic optimal control problem.Furthermore, an inverse prob- lem is investigated to derive the optimal control problem from a given Schr"odinger equation.
Quantum mechanical systems exhibit an inherently probabilistic nature upon measurement. Using a quantum noise model to describe the stochastic evolution of the open quantum system and working in parallel with classical indeterministic control theory, we present the theory of nonlinear optimal quantum feedback control. The resulting quantum Bellman equation is then applied to the explicitly solvable quantum linear-quadratic-Gaussian (LQG) problem which emphasizes many similarities with the corresponding classical control problem.
We exploit the separation of the filtering and control aspects of quantum feedback control to consider the optimal control as a classical stochastic problem on the space of quantum states. We derive the corresponding Hamilton-Jacobi-Bellman equations using the elementary arguments of classical control theory and show that this is equivalent, in the Stratonovich calculus, to a stochastic Hamilton-Pontryagin setup. We show that, for cost functionals that are linear in the state, the theory yields the traditional Bellman equations treated so far in quantum feedback.
The quantum-mechanical formulation of a class of stochastic optimal control problems is investigated and two versions of non-linear Schrödinger equation are derived. First, a wave-function is introduced which connects the probability density function of a diffusion process with the minimum cost functional of the stochastic optimal control problem associated with the diffusion process. It is then demonstrated that this wave-function satisfies a kind of Schrödinger-Langevin equation in quantum physics. Finally, the probabilistic interpretation of the wave-function is discussed.
Non‐relativistic quantum systems are analyzed theoretically or by numerical approaches using the Schrödinger equation. Compared to the options available to treat classical mechanical systems this is limited, both in methods and in scope. However, based on Nelson's stochastic mechanics, the mathematical structure of quantum mechanics has in some aspects been developed into a form analogous to classical analytical mechanics. We show here that finding the Nash equilibrium for a stochastic optimal control problem, which is the quantum equivalent to Hamilton's principle of least action, allows to derive two things: i) the Schrödinger equation as the Hamilton‐Jacobi‐Bellman equation of this optimal control problem and ii) a set of quantum dynamical equations which are the generalization of Hamilton's equations of motion to the quantum world. We derive their general form for the non‐stationary and the stationary case. For the harmonic oscillator, the stationary equations lead to the coherent states, and we establish a numerical procedure to solve for the ground state properties without using the Schrödinger equation. image