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Second order necessary conditions for quantum stochastic optimal control problems

Penghui Wang, Shan Wang

Abstract

This paper aims to establish second order necessary conditions for optimal control in quantum stochastic systems. We employ a variational approach, analogous to methods in classical stochastic control, to analyze systems governed by quantum stochastic differential equations driven by fermionic Brownian motion, where the control enters both the drift and diffusion terms. This result provides a theoretical foundation for further exploration of optimization problems and their practical applications in the field of quantum stochastic control.

Second order necessary conditions for quantum stochastic optimal control problems

Abstract

This paper aims to establish second order necessary conditions for optimal control in quantum stochastic systems. We employ a variational approach, analogous to methods in classical stochastic control, to analyze systems governed by quantum stochastic differential equations driven by fermionic Brownian motion, where the control enters both the drift and diffusion terms. This result provides a theoretical foundation for further exploration of optimization problems and their practical applications in the field of quantum stochastic control.
Paper Structure (3 sections, 4 theorems, 74 equations)

This paper contains 3 sections, 4 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main result

Key Result

Lemma 2.1

B.S.W.2 Assuming that $\textbf{(H1)}$ holds, the equation FSDE-control admits a unique solution $x(\cdot)\in C_\mathbb{A}([t_0,T];L^2(\mathscr{C}))$, and it holds that

Theorems & Definitions (8)

  • Definition 2.1
  • Lemma 2.1
  • Lemma 2.2
  • Definition 2.2
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof