Optimal control of quantum stochastic systems in fermion fields: The Pontryagin-type maximum principle (II)

TL;DR

This paper develops a method to solve second-order adjoint equations in quantum stochastic control of fermion fields, enabling the application of Pontryagin's maximum principle in this complex quantum setting.

Contribution

It introduces a relaxed transposition method to solve second-order adjoint equations, advancing the theoretical framework for quantum stochastic optimal control in fermion fields.

Findings

Successfully solves second-order adjoint equations in W*-topology

Provides a foundation for applying Pontryagin's maximum principle in quantum control

Enhances mathematical tools for quantum stochastic differential equations

Abstract

In the present paper, by using the relaxed transposition method[29], we solve the second-order adjoint equations, corresponding to the optimal control of quantum stochastic systems in fermion fields, which plays the fundamental roles in the study of the Pointryagin-type maximum principle in quantum stochastic optimal control. The second-order adjoint equation is a backard operator valued quantum stochastic differential equation, which has no definition in the algebra of bounded operators, and the solution derived from the relaxed transposition method makes sense in W*-topology.