In this work, we address the optimal control of parameter-dependent systems. We introduce the notion of averaged control in which the quantity of interest is the average of the states with respect to the parameter family .
with the space of admissible controls and
the average state target. The optimization problem is subject to the finite dimensional linear control system
We use the classical gradient descent method based on the adjoint methodology, and obtain the corresponding adjoint system,
The functional is minimized by taking the steepest descent direction given by
and the new control reads as
for some small enough.
We have also used the conjugate gradient method in order to reach faster the optimal control. In order to be able to apply this method the state vector has been split as
where is the solution to the controlled system with zero initial condition,
and solves the free dynamics problem,
The functional can be expressed as
We introduce the linear operator
and its dual counterpart,
where is solution to
By doing this we can write the directional derivative of the functional as
After having defined and
we can apply the conjugate gradient method to solve the control problem.
Both steepest descent and conjugate gradient methods have been implemented in Matlab. The can be run by typing in the Command Window
- for the steepest descent method:
AveragedControlSD- for the conjugate gradient method:
AveragedControlCG
- E. Zuazua (2014). Averaged Control. Automatica, 50 (12), p. 3077-3087.