# Optimal Control Algorithms and Their Analysis for Short-Term Scheduling in Manufacturing Systems
## 6 Conclusions  
Optimal controls as functions of the system and control state allow for the generation of optimal decisions in consideration of a system's evolution in time in the presence of perturbations which result in different system states. Optimal control approaches take another perspective as mathematical programming methods which represent schedules as trajectories. Computational algorithms with regard to state, control, and conjunctive variable spaces exist in the literature. We have demonstrated that the advantages of optimal control methods are applicable to the treatment of large scale problems with complex constraints, the consideration of non-stationary process execution dynamics, the representation in differential equations of complex interrelations between process execution, capacity evolution, and machine setups. In addition, the advantages of optimal control also include accuracy of continuous time and accurate presentation of continuous flows (e.g., in process industry or energy systems) with the help of continuous state variables. An important observation is that schedule presentation in terms of optimal control makes it possible to incorporate the rich variety of control theoretic axioms with regard to feedback adaptive control (mostly applied in the framework of production-inventory control models) as well as the use of control tools of qualitative performance analysis, such as attainable (reachable) sets. Limitations of control applications include conceptual and algorithmic restrictions such as continuous process applications and specific (i.e., non-generalized) forms of constructing algorithms with necessary requirements concerning optimality, convergence, and numerical stability. In this study, we exemplified an application of optimal control to manufacturing scheduling. Fundamentally, this application dynamically decomposes the assignment matrix in time using differential equations, and then solves (by tendency) polynomial problems of small dimensionality at each point of time, with a subsequent integration of these partial solutions using the maximum principle by integrating main and adjoint equation systems. The small dimensionality at each point of time results from dynamic decomposition of job execution described by precedence relation constraints, i.e., at each point of time, we consider only operations that can be assigned to machines at this point of time, excluding those operations that already have been completed as well as those that cannot start because the predecessors have not yet been completed. Algorithmically, we solved these dimensionally small problems at subsequent point of times, integrate main and adjoint systems by the maximum principle, and considered how a particular assignment decision changes the schedule performance (e.g., tardiness). If an improvement is observed, the algorithm takes this assignment and moves further to next point of time and continues in this manner until T\({}_{f}\). With regard to the limitations of this study, the schedule algorithm analysis by attainable sets was used to analyse the schedule outputs at the end of the planning horizon. At the same time, attainable sets (more precisely, their geometric approximations) can also be applied prior to schedule optimization to analyse if any feasible schedule exists for the given problem settings (resource capacities, etc.). This issue has not yet been considered. In light of the revealed methodical shortcomings and application limitations of optimal control methods based on the maximum principle, the following future research avenues can be stated. First, concrete application cases need to be considered for which specific control models and algorithms will be developed. The construction of models and computational procedures within proved axioms of control theory is important. Second, application of qualitative performance analysis methods for control policy dynamic investigation under uncertainty, such as attainable sets, must be named.