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

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.