A numerical investigation of Brockett’s ensemble optimal control problems

Abstract

This paper is devoted to the numerical analysis of non-smooth ensemble optimal control problems governed by the Liouville (continuity) equation that have been originally proposed by R.W. Brockett with the purpose of determining an efficient and robust control strategy for dynamical systems. A numerical methodology for solving these problems is presented that is based on a non-smooth Lagrange optimization framework where the optimal controls are characterized as solutions to the related optimality systems. For this purpose, approximation and solution schemes are developed and analysed. Specifically, for the approximation of the Liouville model and its optimization adjoint, a combination of a Kurganov–Tadmor method, a Runge–Kutta scheme, and a Strang splitting method are discussed. The resulting optimality system is solved by a projected semi-smooth Krylov–Newton method. Results of numerical experiments are presented that successfully validate the proposed framework.

Second-order analysis of Fokker–Planck ensemble optimal control problems

“…Recently, Brockett's research programme has received much impetus through novel theoretical and numerical work focusing on deterministic models with random initial conditions and the corresponding Liouville equation [5,6], and in the case of a linear Boltzmann equation [7]. The modelling and simulation of FP ensemble optimal control problems has been investigated in view of their large applicability [12,[33][34][35].…”

Dynamical optimal transport of nonlinear control-affine systems

“…Optimal transport of nonlinear systems have also been considered in the context of degenerate heat equations [42,30]. When there is no terminal constraint on the density at final time, similar problems have also been considered in the context of control of the Fokker-Planck equation arising from stochastic control problems [18], and control of the continuity equation where the vector-field is given by a nonlinear control system [10,11]. Similarly, a Pontryagin's maximum principle has been derived [15].…”