Optimal Control of ensembles of dynamical systems

TL;DR

This paper develops a theoretical framework and numerical algorithms for optimal control of ensembles of affine systems, using $mma$-convergence and maximum principles to approximate and solve the control problems.

Contribution

It introduces a $mma$-convergence approach to approximate infinite ensembles with finite ones and derives a maximum principle for ensemble optimal control problems.

Findings

Proves well-posedness of the ensemble control problem.

Develops an algorithm based on subspace projection for numerical solutions.

Tests algorithms on linear systems in -dimensional space.

Abstract

In this paper we consider the problem of the optimal control of an ensemble of affine-control systems. After proving the well-posedness of the minimization problem under examination, we establish a $\Gamma$-convergence result that allows us to substitute the original (and usually infinite) ensemble with a sequence of finite increasing-in-size sub-ensembles. The solutions of the optimal control problems involving these sub-ensembles provide approximations in the $L^2$-strong topology of the minimizers of the original problem. Using again a $\Gamma$-convergence argument, we manage to derive a Maximum Principle for ensemble optimal control problems with end-point cost. Moreover, in the case of finite sub-ensembles, we can address the minimization of the related cost through numerical schemes. In particular, we propose an algorithm that consists of a subspace projection of the gradient field induced on the space of admissible controls by the approximating cost functional. In addition, we consider an iterative method based on the Pontryagin Maximum Principle. Finally, we test the algorithms on an ensemble of linear systems in $\mathbb{R}^2$.