On existence of neural network suboptimal feedback controllers

Approximation of optimal feedback policies by multilayer neural networks is investigated from the existence point of view. The optimal control problem is considered in discrete time for a class of summable quality criterions with the target set being the origin. An associated suboptimal control problem is introduced to simplify analysis through relaxation of the final condition and of the optimality requirement. The controller structure is that of static state feedback realized by feedforward multilayered networks with Heaviside neuron activation functions. Under mild assumptions about the system, the obtained results guarantee existence of an appropriate neural network controller, such that the closed-loop system closely follows the open-loop optimal trajectories