  Physics-informed neural networks (PINNs) impose known physical laws into the
learning of deep neural networks, making sure they respect the physics of the
process while decreasing the demand of labeled data. For systems represented by
Ordinary Differential Equations (ODEs), the conventional PINN has a continuous
time input variable and outputs the solution of the corresponding ODE. In their
original form, PINNs do not allow control inputs neither can they simulate for
long-range intervals without serious degradation in their predictions. In this
context, this work presents a new framework called Physics-Informed Neural
Nets-based Control (PINC), which proposes a novel PINN-based architecture that
is amenable to control problems and able to simulate for longer-range time
horizons that are not fixed beforehand. First, the network is augmented with
new inputs to account for the initial state of the system and the control
action. Then, the response over the complete time horizon is split such that
each smaller interval constitutes a solution of the ODE conditioned on the
fixed values of initial state and control action. The complete response is
formed by setting the initial state of the next interval to the terminal state
of the previous one. The new methodology enables the optimal control of dynamic
systems, making feasible to integrate a priori knowledge from experts and data
collected from plants in control applications. We showcase our method in the
control of two nonlinear dynamic systems: the Van der Pol oscillator and the
four-tank system.
