# Optimal Control Algorithms and Their Analysis for Short-Term Scheduling in Manufacturing Systems
## 2 Optimal Control Applications to Scheduling in Manufacturing Systems
Additional transversality conditions for the two ends of the state trajectory should be added for a general case:  
\[\psi_{l}(T_{0})=-\left.\frac{\partial J_{ob}}{\partial x_{l}}\right|_{x_{l}( T_{0})=x_{l0}},\,\psi_{l}(T_{f})=-\left.\frac{\partial J_{ob}}{\partial x_{l}} \right|_{x_{l}(T_{f})=x_{lf}} \tag{5}\]  
Let us consider the algorithmic realization of the maximum principle. In accordance with this principle, two systems of differential equations should be solved: the main system (1) and the adjoint  
Figure 1: The scheme of optimal program control model interconnection. system (2). This will provide the optimal program control vector \(\mathbf{u}^{*}(t)\) (the indices \(\varphi\)\(l\)\(\triangleright\) are omitted) and the state trajectory \(\mathbf{x}^{*}(t)\). The vector \(\mathbf{u}^{*}(t)\) at time \(t=T_{0}\) under the conditions \(\mathbf{h}_{0}\left(\mathbf{x}(T_{0})\right)\leq\mathbf{0}\) and for the given value of \(\psi(T_{0})\) should return the maximum to the Hamilton's function:  
\[H(\mathbf{x}(t),\mathbf{u}(t),\mathbf{\psi}(t))=\mathbf{\overline{v}}^{\text {T}}\mathbf{f}(\mathbf{x},\mathbf{u},t) \tag{6}\]  
Here we assume that general functional of MS schedule quality is transformed to [PERSON]'s form [21]. The received control is used in the right members of (1), (2), and the first integration step for the main and for the adjoint system is made: \(t_{1}=T_{0}+\widetilde{\delta}\) (\(\widetilde{\delta}\) is a step of integration). The process of integration is continued until the end conditions \(\mathbf{h}_{1}\left(\mathbf{x}(T_{f})\right)\leq\overset{\rightarrow}{O}\) are satisfied and the convergence accuracy for the functional and for the alternatives is adequate. This terminates the construction of the optimal program control \(\mathbf{u}^{*}(t)\) and of the corresponding state trajectory \(\mathbf{x}^{*}(t)\). However, the only question that is not addressed within the described procedure is how to determine \(\psi(T_{0})\) for a given state vector \(\mathbf{x}(T_{0})\). The value of \(\psi(T_{0})\) depends on the end conditions of the MS schedule problem. Therefore, the maximum principle allows the transformation of a problem of non-classical calculus of variations to a boundary problem. In other words, the problem of MS OPC (in other words, the MS schedule) construction is reduced to the following problem of transcendental equations solving:  
\[\mathbf{\Phi}=\mathbf{\Phi}(\psi(T_{0}))=0 \tag{7}\]  
At this stage, we consider the most complicated form of the MS OPC construction with MS initial state at time \(t=T_{0}\). The end point of the trajectory and the time interval are fixed:  
\[\mathbf{x}(T_{f})=\mathbf{a} \tag{8}\]  
where \(\mathbf{a}=\)\(||a_{1},a_{2},\ldots,a_{\widetilde{n}}||^{\text{T}}\) is a given vector; \(\widetilde{n}\) is the dimensionality of the general MS state vector. Hence, the problem of optimal MS OPC construction is reduced to a search for the vector \(\psi(T_{0})\), such that:  
\[\mathbf{\Phi}=\mathbf{\Phi}(\psi(T_{0}))=\mathbf{x}(T_{f})-\mathbf{a} \tag{9}\]  
The problem of solving Equations (9) is equivalent to a minimization of the function:  
\[\Delta_{u}(\psi(T_{0}))=\frac{1}{2}\Big{(}\mathbf{a}-\mathbf{x}(T_{f})\Big{)} ^{\text{T}}\Big{(}\mathbf{a}-\mathbf{x}(T_{f})\Big{)} \tag{10}\]  
The main feature of the problems (9), (10) is that the interdependency of \(\psi(T_{0})\) and \(\boldsymbol{\rho}(T_{f})=\Big{(}\mathbf{a}-\mathbf{x}(T_{f})\Big{)}\) is defined indirectly via the system of differential Equation (1). This feature leads to the use of iterative methods. Classification and Analyses of Optimal Control Computational Algorithms for Short-Term Scheduling in MSs  
Real-world optimal control problems are characterized by high dimensionality, complex control constraints, and essential non-linearity. Prior to introducing particular algorithms and methods of optimal control computation, let us consider their general classification (Figure 2).