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Author (up) Contreras, G.M.; Peña, J.P. pdf  doi
  Title The quantum dark side of the optimal control theory Type Journal Article
  Year 2019 Publication Physica A: Statistical Mechanics and its Applications Abbreviated Journal  
  Volume 515 Issue Pages 450-473  
  Keywords Optimal control theory; Pontryagin's equations; Hamilton-Jacobi-Bellman equation; Constrained systems; Dirac's method; Quantum mechanics  
  Abstract In a recent article, a generic optimal control problem was studied from a physicist's point of view (Contreras et al. 2017). Through this optic, the Pontryagin equations are equivalent to the Hamilton equations of a classical constrained system. By quantizing this constrained system, using the right ordering of the operators, the corresponding quantum dynamics given by the Schrödinger equation is equivalent to that given by the Hamilton-Jacobi-Bellman equation of Bellman's theory. The conclusion drawn there were based on certain analogies between the equations of motion of both theories. 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 h->0 of these three different quantum theories. Also, the exact relation of the phase S(x,t) of the wave function Ψ(x,t)=eihS(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.  
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  Series Volume Series Issue Edition  
  ISSN 0378-4371 ISBN Medium  
  Area Expedition Conference  
  Notes Approved no  
  Call Number UAI @ eduardo.moreno @ Serial 903  
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