The quantum dark side of the optimal control theory

In this paper, a closer and more detailed examination of the quantization problem is carried out, by considering three possible quantization procedures: right quantization, left quantization, and Feynman's path integral approach. The Bellman theory turns out to be the classical limit ħ→0 of these three different quantum theories. Also, the exact relation of the phase S(x,t) of the wave function Ψ(x,t)=eiħS(x,t) of the quantum theory with Bellman's cost function J+(x,t) is obtained. In fact, S(x,t) satisfies a 'conjugate' form of the Hamilton-Jacobi-Bellman equation, which implies that the cost functional J+(x,t) must necessarily satisfy the usual Hamilton-Jacobi-Bellman equation. Thus, the Bellman theory effectively corresponds to a quantum view of the optimal control problem.