The Hybrid Minimum Principle in the Presence of Switching Costs Ali Pakniyat and Peter E. Caines Department of Electrical and Computer Engineering (ECE) and the Center for Intelligent Machines (CIM), McGill University Abstract—Hybrid optimal control problems are studied for systems where, in addition to running costs, switching between discrete states incurs costs. A key aspect of the analysis is the relationship between the Hamiltonian and the adjoint process before and after the switching instants. In this paper, the analysis is performed for systems for which autonomous and controlled state jumps are not permitted. First the results are established in the hybrid Mayer optimal control problem setup using the needle variation technique, and then the results for the hybrid Bolza optimal control problem are established via the calculus of variations methodology. I. I NTRODUCTION There is now an extensive literature on the optimal control of hybrid systems (see e.g. [1], [2], [3], [4], [5], [6], [7], [8], [9]). With the exception of the variational inequality in [8] and the work in [9], [10], the results and methods in this body of work consist of generalizations of the Pontryagin Maximum Principle (PMP). A feature of special interest in these analyses is the boundary conditions on adjoint processes and the Hamiltonian function at au- tonomous and controlled switching times and states; these boundary conditions may be viewed as a generalization to the optimal control case of the Erdmann-Weierstrass conditions of the calculus of variations. As is well known, Dynamic Programming (DP) provides sufficient conditions for optimality based upon the Principle of Dynamic Pro- gramming, which in the standard non-hybrid case, and under the assumption of smoothness of the value function, results in the celebrated Hamilton-Jacobi-Bellman (HJB) equation [11]. In the case of non-smooth value functions, the so-called viscosity solutions [9] give a general class of solutions to the HJB equation. Those hybrid optimal con- trol problems (with autonomous or controlled switchings) where switching incurs costs constitute a class of problems which have been the subject of only limited study. In fact the value function for hybrid systems with switching costs will not in general be smooth at the switching instants, and hence viscosity solutions are studied in [9], [10], where switching costs are a function of switching state. In this paper, the relationship between the Hamiltonian and the adjoint process before and after switching instants with costs is determined for hybrid systems which are general except for the restriction that autonomous and controlled state jumps are not permitted. These results are expressed in the Hybrid Minimum Principle (HMP) framework in this paper and a consecutive work will be studied in the Dynamic Programming framework. II. PROBLEM FORMULATION To simplify the analysis, necessary optimality conditions are only presented in this paper for trajectories with a single switching event; however the results may be gen- eralized (as for instance in [3]) to optimal trajectories with several switching events by iterating the proof procedure backwards in time along the trajectory from the terminal instant. A. Basic Assumptions Consider a hybrid system (structure) H H = {H := Q × R n ,I, Γ, A, F, M} (1) with the following properties: Q = {1, 2,..., |Q|} ≡ {q j } j∈Q is the finite set of discrete states (components). H = Q × R n is the (hybrid) state space of the hybrid system H. I =Σ × U is the set of system input values with Σ being the set of autonomous and controlled transition labels extended with the identity element such that σ i,j ∈ Σ for i ∈ Q only if j ∈ A (i) U ⊂ R m is the set of admissible input control values, where U is an open bounded set in R m . The set of admissible input control functions is taken to be U (U ) := L ∞ ([t 0 ,T * ) ,U ), which is the set of all measurable functions that are bounded up to a set of measure zero on [t 0 ,T * ) ,T * < ∞. The boundedness property necessarily holds since admissible input functions take values in the open bounded set U which has compact closure ¯ U . Γ: H × Σ → H is a time independent (partially defined) discrete (state) transition map which is the identity on the second (R n ) component of H. A : Q × Σ → Q is such that A (q i ,σ i,j )= q j . F = {f j } j∈Q is the collection of vector fields such that f j ∈ C k (R n × U → R n ) ,k ≥ 1 satisfies a uniform (in 52nd IEEE Conference on Decision and Control December 10-13, 2013. Florence, Italy 978-1-4673-5717-3/13/$31.00 ©2013 IEEE 3831