Nonlinear optimal control: approximations via moments and LMI-relaxations

Abstract

We consider the class of nonlinear optimal control problems with all data (differential equation, state and control constraints, cost) being polynomials. We provide a simple hierarchy of LMI-relaxations whose optimal values form a nondecreasing sequence of lower bounds on the optimal value. Preliminary results show that good approximations are obtained with few moments.

Nonlinear optimal control synthesis via occupation measures

“…In the current paper, we propose some techniques to constructively derive a control law from the solution of the convex linear matrix inequality (LMI) relaxations of the OCP. So our contribution can be seen as an extension to synthesis of the performance analysis results of [4,5].…”

“…In this paper we consider the class of OCPs for which all problem data are polynomial. The approach we deploy (which was introduced in [4]) is based on moment theory and consists in deriving a hierarchy of convex linear matrix inequality (LMI) relaxations of the OCP which give an increasing sequence of lower bounds on the optimal value. These LMI problems can be solved using off-the-shelf semidefinite programming (SDP) solvers.…”

“…The contribution with respect to [4] and its extended version [5] is twofold. First, the derivation of the relaxation is obtained in a simpler way, starting from basic concepts.…”

GloptiPoly 3: moments, optimization and semidefinite programming

“…For example, consider the GPM arising when solving polynomial optimal control problems as detailed in [7]. We are seeking two occupation measures dµ 1 (x, u) and dµ 2 (x) of a state vector x(t) and input vector u(t) whose time variation are governed by the differential…”

“…As explained in [7], a lower bound on the For the initial condition x 0 = [1 1] the exact minimum time is equal to 3.5. In Table 1 …”