A Polynomial Algorithm solving a Special Class of Hybrid Optimal Control Problems Dario Bauso and Raffaele Pesenti Abstract— Hybrid optimal control problems are, in general, difficult to solve. A current research goal is to isolate those problems that lead to tractable solutions [5]. In this paper, we identify a special class of hybrid optimal control problems which are easy to solve. We do this by using a paradigm borrowed from the Operations Research field. As main re- sult, we present a solution algorithm that converges to the exact solution in polynomial time. Our approach consists in approximating the hybrid optimal control problem via an integer-linear programming reformulation. The integer-linear programming problem is a Set-covering one with a totally unimodular constraint matrix and therefore solving the Set- covering problem is equivalent to solving its linear relaxation. It turns out that any solution of the linear relaxation is a feasible solution for the hybrid optimal control problem. Then, given the feasible solution, obtained solving the linear relaxation, we find the optimal solution via local search. I. INTRODUCTION Hybrid optimal control problems are, in general, difficult to solve (see, e.g., [5], [7], [14] and references therein). A current research goal is to isolate those problems that lead to tractable solutions [5]. To the author’s knowledge, the drawback is that no theoretical approaches are available to study the difficulty of the problems and the computational complexity of the available solution algorithms. In this paper, we identify a special class of hybrid optimal control problems which are easy to solve. We do this by using a paradigm borrowed from the Operations Research field. As main result, we present a solution algorithm that converges to the exact solution in polynomial time. The hybrid system considered is a continuous-time system subject to controlled impulses [5], i.e., the state jumps in response to a control command with an associated cost. Examples can be found in the manufacturing and inventory management literature [12] (see also the Wagner and Within formulation and solution methods surveyed in [13]). For most of the aforementioned problems, the standard solution approach based on Dynamic Programming has a high computational cost due to the curse of dimensionality [4], [12]. Our approach consists in approximating the hybrid optimal control problem via an integer-linear programming reformu- lation. Actually, the application of integer or mixed-integer linear programming to hybrid optimal control problems This work was not supported by any organization D. Bauso is with Dipartimento di Ingegneria Informatica, Uni- versit` a di Palermo, Viale Delle Scienze, I-90139 Palermo, Italy, dario.bauso@unipa.it R. Pesenti is with with Dipartimento di Ingegneria Informatica, Universit` a di Palermo, Viale Delle Scienze, I-90139 Palermo, Italy, pesenti@unipa.it has been successfully dealt with in [2], [3]. The integer linear programming problem is a Set-covering problem [6], [11] whose constraints are described by an interval matrix, which is totally unimodular [11]. Therefore, solving the Set- covering problem is equivalent to solving its linear relaxation (see, e.g., a previous efficient solution approach based on linear programming in [8]). It turns out that any solution of the linear relaxation is a feasible solution for the hybrid optimal control problem. Given the feasible solution returned by the linear relaxation, we can find the optimal solution in polynomial time via local search [9]. This paper is organized as follows. In Section II, we introduce the hybrid optimal control problem. In Section III, we describe the Set-covering reformulation and discuss its linear relaxation. In Section IV, we derive the local search algorithm. In Section V, we present a simulation example. Finally, in Section VI, we draw some conclusions and discuss future works. II. HYBRID OPTIMAL CONTROL UNDER STATE CONSTRAINTS Consider the special class of hybrid systems subject to controlled impulses, i.e., the state jumps in response to a control command with an associated cost [5]. As a reference example, consider the following inventory application. Let x(τ ) ∈ R be the continuous-time state variable indicating the inventory level, d(τ ) ∈ R the demand faced by the retailer, u t ∈ {0, 1} at each discrete-time t =1, 2,... a binary decision describing the choice of the retailer of reordering, u t =1, in which case the inventory is restored at level S, or not reordering, u t =0. Then, over a finite horizon of length N , the inventory evolves according to the following hybrid dynamics consisting in a continuous- time dynamics subject to discrete-time inputs, ˙ x(τ )= −d(τ ) + N  t=1 (S − x(τ ))δ(τ − t)u t    , reset at S where δ(τ − t) is the Dirac impulse at time t. For each time t, each term (S − x(τ ))δ(τ − t)u t in the right hand side basically resets the state at S if u t =1. For each time t let w t :=  t+1 t d(τ ). If we sample the continuous dynamics at discrete times t =1, 2,... we have the following difference equation x t+1 = x t − w t +(S − x t )u t . (1) Let K t be a time-varying transportation cost incurred by the retailer when reordering. Then, for each time t, we can Proceedings of the 2006 IEEE International Conference on Control Applications Munich, Germany, October 4-6, 2006 WeA11.2 0-7803-9796-7/06/$20.00 ©2006 IEEE 349