# Optimal Control Algorithms and Their Analysis for Short-Term Scheduling in Manufacturing Systems
## 2 Optimal Control Applications to Scheduling in Manufacturing Systems  
Optimal control problems belong to the class of extremum optimization theory, i.e., the analysis of the minimization or maximization of some functions ([21; 22; 23; 24; 25; 26; 27; 28]. This theory evolved on the basis of calculus variation principles developed by [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], and [PERSON]. In the 20 th century, two computational fundamentals of optimal control theory, maximum principle [21] and dynamic programming method [29], were developed. These methods extend the classical calculus variation theory that is based on control variations of a continuous trajectory and the observation of the performance impact of these variations at the trajectory's terminus. Since control systems in the middle of the 20 th century were increasingly characterized by continuous functions (such as 0-1 switch automats), the development of both the maximum principle and dynamic programming was necessary in order to solve the problem with complex constraints on control variables. Manufacturing managers are always interested in non-deterministic approaches to scheduling, particularly where scheduling is interconnected with the control function [30]. Studies by [31] and [11] demonstrated a wide range of advantages regarding the application of control theoretic models in combination with other techniques for production and logistics. Among others, they include a non-stationary process view and accuracy of continuous time. In addition, a wide range of analysis tools from control theory regarding stability, controllability, adaptability, etc. may be used if a schedule is described in terms of control. However, the calculation of the optimal program control (OPC) with the direct methods of the continuous maximum principle has not been proved efficient [32]. Accordingly, the application of OPC to scheduling is important for two reasons. First, a conceptual problem consists of the continuous values of the control variables. Second, a computational problem with a direct method of the maximum principle exists. These shortcomings set limitations on the application of OPC to purely combinatorial problems. To date, various dynamic models, methods and algorithms have been proposed to solve planning and scheduling problems in different application areas ([13; 19; 33; 34; 35; 36; 37; 38]). In these papers, transformation procedures from classical scheduling models to their dynamic analogue have been developed. For example, the following models of OPC were proposed:  
\(M_{g}\)--dynamic model of MS elements and subsystems motion control;  
\(M_{k}\)--dynamic model of MS channel control;  
\(M_{o}\)--dynamic model of MS operations control;  
\(M_{f}\)--dynamic model of MS flow control;  
\(M_{p}\)--dynamic model of MS resource control;  
\(M_{e}\)--dynamic model of MS operation parameters control;\(M_{c}\)--dynamic model of MS structure dynamic control;  
\(M_{v}\)--dynamic model of MS auxiliary operation control. The detailed mathematical formulation of these models was presented by [36; 37] as well as [38]. We provide the generalized dynamic model of MS control processes (\(M\) model) below:  
\[ \begin{gathered} M=\left\{\mathbf{u}(t)\big{|}\dot{\mathbf{x}}= \mathbf{f}(\mathbf{x},\mathbf{u},t);\mathbf{h}_{0}(\mathbf{x}(T_{0}))\leq \mathbf{O},\\ \mathbf{h}_{1}(\mathbf{x}(T_{f}))\leq\mathbf{O},\,\mathbf{q}^{(1)} (\mathbf{x},\mathbf{u})=\mathbf{O},\,\mathbf{q}^{(2)}(\mathbf{x},\mathbf{u})< \mathbf{O}\right\},\\ J_{\theta}=J_{\theta}(\mathbf{x}(t),\mathbf{u}(t),t)=\varphi_{ \theta}\Big{(}\mathbf{x}(t_{f})\Big{)}+\int\limits_{T_{0}}^{T_{f}}f_{\theta}( \mathbf{x}(\tau),\mathbf{u}(\tau),\tau)d\tau,\\ \vartheta\in\{g,\,k,\,o,\,f,\,p,\,e,\,c,\
u\},\end{gathered}