Quadratic Optimal Control of Linear Complementarity Systems: First-Order Necessary Conditions and Numerical Analysis

Abstract

This article is dedicated to the analysis of quadratic optimal control of linear complementarity systems (LCS), which are a class of strongly nonlinear and nonsmooth dynamical systems. Necessary first-order conditions are derived, that take the form of an LCS with inequality constraints, hence are numerically tractable. Then two numerical solvers are proposed, for the direct and the indirect approaches. They take advantage of MPEC solvers for computations. Numerical examples illustrate the theoretical developments and demonstrate the efficiency of the approach.

Dynamical Systems Coupled with Monotone Set-Valued Operators: Formalisms, Applications, Well-Posedness, and Stability

“…One important issue lies in the fact that in the non convex case, the biconjugate (ϕ ⋆ ) ⋆ may not equal ϕ, so that manipulations used to invert the set-valued part may no longer be valid (one path could be to use the results in [31]). Other extensions which have not been tackled, could be: consider a mixed LCP instead of an LCP in (2.22) (b), yielding the MLCS: [107] for preliminary results), time-varying LCS with A(t), B(t), C(t), D(t) (see Section 3.4 for a possible path), and control-related issues (optimal control has been tackled in [287,555,554], yielding MPEC problems, with applications in process control [484,96], trajectory tracking, etc). It could be of interest to study the relationships between optimal control of FOSwP tackled in [54,160,162,161,200,202,201,205,316,217,536], which rely on the convergence of a suitable time-discretization or of a regularization of the normal cone, and the results in [555,554], which rely on [287].…”

“…This allows us to state that the optimal control problem formulated for DIs as (2.16) and with F (x) = N S (x) and S finitely represented [206,205,217], can be interpreted as optimal control of a class of complementarity systems. Whether or not this covers [555] is an open question.…”

“…Feedback control and its many branches (optimal control, controllability, observability, state observer design and separation principle, stabilization, tracking control, classification of systems into subclasses depending on their controllability/observability properties, robust control) has not been treated in detail. While these problems have received some attention in the community [287,309,526,555], there are many related questions which need to be investigated in depth, as they lead to some challenging questions at the intersection of functional analysis, optimization algorithms, numerical implementation, and control theory.…”

Optimization of Controlled Free-Time Sweeping Processes with Applications to Marine Surface Vehicle Modeling

“…where C is taken from (16), the control constraints reduce to (10), and the dynamic nonoverlapping condition x i (t) − x j (t) ≥ R i + R j is equivalent to the pointwise state constraints…”

“…Nevertheless, within a rather short period of time, many important results have been obtained on necessary optimality conditions for controlled sweeping processes with valuable applications to friction and plasticity, robotics, traffic equilibria, ferromagnetism, hysteresis, economics, and other fields of engineering and applied sciences; see, e.g., [1], [3], [7], [8], [17] with more references and discussions. Let us mention to this end the recent papers [15], [16], where optimal control problems for linear complementarity systems have been studied and applied to practical models that are highly important in the area of Automatic Control. Such problems can be written in a form of controlled sweeping processes, where C is an orthant in R n .…”