Nonlinear Optimal Control via Occupation Measures and LMI-Relaxations

Abstract

We consider the class of nonlinear optimal control problems (OCP) with polynomial data, i.e., the differential equation, state and control constraints and cost are all described by polynomials, and more generally for OCPs with smooth data. In addition, state constraints as well as state and/or action constraints are allowed. We provide a simple hierarchy of LMI (linear matrix inequality)-relaxations whose optimal values form a nondecreasing sequence of lower bounds on the optimal value. Under some convexity assumptions, the sequence converges to the optimal value of the OCP. Preliminary results show that good approximations are obtained with few moments.

Optimal switching control design for polynomial systems: an LMI approach

“…As remarked in [28], this changes the interpretation of LP (10) to the minimization of the expected value of the cost given the initial distribution. See also [24] for the Liouville interpretation of the LP as transporting measure ξ 0 along the optimal flow.…”

“…Therefore, all that is left to show is the weak- * continuity of A, hence of each L j . Following [28], this can be shown by noticing that each L j is continuous for the strong topology of C 1 (K), hence for its associated weak topologies. Operators L j , hence A, are therefore weakly- * continuous, and each sequence ( x (n) p , c p , Ax (n) p ) n∈N converges in R, which concludes the proof.…”

“…Therefore, an LMI relaxation of an optimal switching control problem with n states and m controls will be solved in O(d 9 2 (n+m+1) ) operations when solved with the general formulation of [28] and O(m 3 2 d 9 2 (n+1) ) when solved with structured formulation (10). That is a n+1 n+m+1 reduction of the polynomial rate at which the CPU time grows with relaxation orders.…”

“…We know from [28] that the optimal sequence consists of starting with mode 1, i.e., u 1 (t) = 1, u 2 (t) = 0 for t ∈ [0, 2], then chattering with equal proportion between mode 1 and 2, i.e., u 1 (t) = u 2 (t) = 1 2 for t ∈ [2, 5 2 ] and then eventually driving the state to the origin with mode 2, i.e., u 1 (t) = 0, u 2 (t) = 1 for t ∈ [ 5 2 , 7 2 ]. Here too the infimum p * = 7 2 is not attained for problem (1), whereas it is attained with the above controls for problem (4).…”

“…Classically, we formulate the optimal control switching problem as an optimal control problem with artificial controls being functions of time, each valued in the discrete set {0, 1}, and we relax it into a control problem with such controls now valued in the interval [0,1]. In contrast with existing approaches following this relaxation strategy, see, e.g., [4,37], relying on the local Pontryagin maximum principle, our aim is to apply the global approach of [28]. We show that by specializing the convex objects for switched systems, one obtains a numerical method for which the number of modes driving the system enters linearly into the complexity of the problem, instead of exponentially.…”