Multilevel control by duality

We discuss the multilevel control problem for linear dynamical systems, consisting in designing a piecewise constant control function taking values in a finite set. In particular, we provide a complete characterization of multilevel controls through a duality approach, based on the minimization of a suitable cost functional. In this manner, we devise an effective methodology to systematically build optimal multilevel controls that, depending on the nature of the dynamics considered, have different structural properties. In more detail, in the case of conservative or dissipative dynamics, our technique allows us to define multilevel controls where the levels are fixed a priori, provided that the time horizon is large enough. For more general dynamics satisfying the Kalman rank condition, instead, we are able to build multilevel controls which are effective in any positive time T > 0, whose levels blow-up as T -> 0+. Moreover, this method leads to efficient numerical algorithms for computing multilevel controls. (c) 2023 Elsevier B.V. All rights reserved.