Optimal control problems with oscillations, concentrations and discontinuities

Abstract

Optimal control problems with oscillations (chattering controls) and concentrations (impulsive controls) can have integral performance criteria such that concentration of the control signal occurs at a discontinuity of the state signal. Techniques from functional analysis (anisotropic parametrized measures) are applied to give a precise meaning of the integral cost and to allow for the sound application of numerical methods. We show how this can be combined with the Lasserre hierarchy of semidefinite programming relaxations. ContributionThe contribution of our paper with respect to previous work can be summarized as follows:• we propose a unified approach for handling the simulatenous presence of oscillations, concentrations and discontinuities, where previous work considered either oscillations without concentrations (see [16,15,9,10] and references therein), or concentrations without oscillations (see [4] and references therein), or oscillations and concentrations without discontinuities (see [5] and references therein);• we remove the technical assumptions of [5] to allow for the simultaneous presence of concentration (of the control) and discontinuity (of the trajectory);• as in [5], our approach allows for a constructive solution via the Lasserre hierarchy [13]; this now provides a unified numerical scheme to deal with oscillations, concentrations and discontinuities;

A sequential quadratic Hamiltonian scheme to compute optimal relaxed controls

“…Further, we obtain that the value of the state variable y and of the expected value of the control u mean (x) := K ad v ν(v, x) dv are in absolute value less than 10 −11 , that is close to zero machine precision. This experiment illustrates a control problem with oscillations [17].…”

“…Next, we consider a problem that involves concentrations effects. See [17] for a similar problem. We have…”

“…Therefore we would like to contribute to the field of numerical methods to solve relaxed optimal control problems where only few results are already available [10,11,16,17,23,32]. Specifically, in [10,23] the moments approach is proposed focusing on a linear governing model and a L 1 cost functional.…”

“…In this case, the linear model is approximated with Chebychev polynomials, and the resulting system of trajectories comes up to a simplified linear matrix inequality (LMI), where the unknown are the moments of the measures. In [11,17], the focus is on problems with concentration, oscillations, and discontinuities, and a generalized DiPerna-Majda relaxed measure, i.e. anisotropic parametrized measure, that can be decomposed into a measure in time and a Young measure is used.…”

Relaxation of functionals with linear growth: interactions of emerging measures and free discontinuities

“…Besides, interactions of oscillations, concentrations and discontinuities lead to interesting questions and challenges for mathematical research; see e.g. [26,27] for some recent results. For suitable examples, the relaxation results presented below allow us to observe the natural formation of all the three phenomena just described in minimizing sequences.…”