Linear quadratic dynamic optimal control with Boltzmann machines

This paper develops a promising alternative for solving linear quadratic (LQ) optimal control problem of discrete-time system, by which the LQ performance index is transformed into the energy function of Boltzmann machines, and the control sequence into the state vector of the neurons of Boltzmann machines. As a result, solving LQ dynamic optimization problem is equivalent to operating associated Boltzmann machines from its initial state to the terminal state that represents the optimal control sequence. The theoretical study indicates that we are able to find a relevant Boltzmann machines of which energy function is corresponding to the LQ performance index. Through the experimental study we obtain that on the basis of this method, we proposed, Boltzmann machines can implement linear quadratic optimal control of any multivariable time-variant system.