MIXED-INTEGER DAE OPTIMAL CONTROL PROBLEMS: NECESSARY CONDITIONS AND BOUNDS MATTHIAS GERDTS * AND SEBASTIAN SAGER † Abstract. We are interested in the optimal control of dynamic processes that can be described by Differential Algebraic Equations (DAEs) and that include integer restrictions on some or all of the control functions. We assume the DAE system to be of index 1. In our study we consider necessary conditions of optimality for this specific case of a hybrid system and results on lower bounds that are important in an algorithmic setting. Both results generalize previous work for the case of Ordinary Differential Equations (ODE). Interestingly, the proofs for both analytical results are based on constructive elements to obtain integer controls from reformulations or relaxations to purely continuous control functions. These constructive elements can also be used for an efficient numerical calculation of optimal solutions. We illustrate the theoretical results by means of a mixed–integer nonlinear optimal control benchmark problem with algebraic variables. Key words. mixed integer programming, nonlinear programming, DAE, switched systems 1. Introduction. Technical or economical processes often involve discrete con- trol variables, which are used to model finitely many decisions, discrete resources, or switching structures like gear shifts in a car or operating modes of a device. This leads to optimal control problems with non-convex and partly discrete control set U . More specifically, some of the control variables may still assume any real value within a given convex set with non-empty interior, those are called continuous-valued control variables in the sequel, while other control variables are restricted to a finite set of values, those are called discrete control variables in the sequel. An optimal control problem involving continuous-valued and discrete control vari- ables is called mixed-integer optimal control problem (MIOCP). Mixed-integer optimal control is a field of increasing importance and practical applications can be found in [15, 11, 13, 34, 19]. For a web-site of further benchmark problems please refer to [27] and the corresponding paper [30]. An approach to solve mixed-integer optimal control problems is by exploiting necessary optimality conditions. A proof for index-one DAEs will be provided in Section 3. The proof exploits an idea of Dubovitskii and Milyutin, see [8, 7], [14, p. 95], [16, p. 148], who used a time transformation to transform the mixed-integer optimal control problem into an equivalent optimal control problem without discrete control variables. Necessary conditions are then obtained by applying suitable local minimum principles to the transformed problem. The result are necessary conditions in terms of global minimum principles. A global minimum principle for disjoint control sets and (noncontinuous) ordi- nary differential equations (ODEs) has been formulated and solved numerically via the newly developed method of Competing Hamiltonians in the work of Bock and Longman, [2, 3, 20]. To our knowledge this was the first time that a global minimum principle has been applied to solve a MIOCP. Global minimum principles for DAE optimal control problems can be found in [26] for Hessenberg DAE optimal control problems, in [5] for semi-explicit index one DAEs, in [6] for implicit control systems, in [1] for quasilinear DAEs, in [21] for * Institute of Mathematics and Applied Computing, Department of Aerospace Engineering, Uni- versit¨at der Bundeswehr, M¨ unchen, Germany, matthias.gerdts@unibw.de † Interdisciplinary Center for Scientific Computing (IWR), Ruprecht–Karls Universit¨at, Heidel- berg, Germany, corresponding author: sebastian.sager@iwr.uni-heidelberg.de 1