# Optimal Control Algorithms and Their Analysis for Short-Term Scheduling in Manufacturing Systems
## Appendix A List of Notations  
* **x** is a vector of the MS general state
* **u** is a vector of the generalized control
* **b\({}_{0}\)**, **h\({}_{1}\)** are known vector function that are used for the state **x** and conditions
* **q\({}^{(1)}\)**, **q\({}^{(2)}\)** are known spatio-temporal, technical and technological constraints
* **j\({}_{\theta}\)** are indicators characterizing MS schedule quality
* **k** is a known vector-function of perturbation influences
* **b** is a vector of conjugate system state
* **H** is a Hamilton's function
* **_j\({}_{ob}\)_ is a scalar form of the vector quality measure
* **_M\({}_{g}\)_ is a dynamic model of MS motion control,
* **_M\({}_{k}\)_ is a dynamic model of MS channel control,
* **_M\({}_{b}\)_ is a dynamic model of MS operational control,
* **_M\({}_{j}\)_ is a dynamic model of MS flow control,
* **_M\({}_{g}\)_ is a dynamic model of MS resource control,
* **_M\({}_{c}\)_ is a dynamic model of MS operation parameters control,
* **_M\({}_{c}\)_ is a dynamic model of MS structure dynamic control,
* **_M\({}_{v}\)_ is a dynamic model of MS auxiliary operation control,
* **_t_ is a current value of time,
* **_T\({}_{0}\)_ is start instant of time of the planning (scheduling) horizon,
* **_T\({}_{f}\)_ is end instant of time of the planning (scheduling) horizon,
* **_t\({}_{f}\)_ is a component of adjoint vector \(\psi(t)\),
* **_x\({}_{l}\)_ is an element of a general vector \(\textbf{x}(t)\),
* \(\lambda_{\textbf{a}}(t)\) is a dynamic Lagrange multiplier which corresponds to the components of vector \(\textbf{q}_{\textbf{a}}^{(1)}\),
* \(\rho_{\beta}(t)\) is a dynamic Lagrange multiplier which is corresponds to the components of vector \(\textbf{q}_{\textbf{a}}^{(2)}\),
* **_z\({}_{\mu_{\beta}}^{(o)}\)_ is an auxiliary variable which characterizes the execution of the operation
* **a** is a current number of constraints \(\textbf{q}_{\textbf{a}}^{(1)}\),
* **_\(\beta\)_ is a current number of constraints \(\textbf{q}_{\textbf{a}}^{(2)}\),
* **_\(\psi_{l}(T_{0})\)_ is a component of adjoint vector \(\psi(t)\) at the moment \(t=T_{0}\),
* **_\(\psi_{l}(T_{f})\)_ is a component of adjoint vector \(\psi(t)\) at the moment \(t=T_{f}\),
* **_H\((\textbf{x}(t),\textbf{u}(t),\textbf{\psi}(t))=\textbf{v}^{\text{T}}\textbf{ f}(\textbf{x},\textbf{u},t)\)_ is a [PERSON]'s function,
* **_\(\Phi(\psi(T_{0}))\)_ is an implicit function of boundary condition,
* **a** is a vector given quantity value which corresponds to the vector \(\textbf{x}(t)\),
* **_\(\Delta_{u}\)_ is a function of boundary conditions,
* **_\(\rho(T_{f})\)_ is a discrepancy of boundary conditions,_
* **_\(\psi_{(r)}(T_{0})\)_ is a vector of adjoint system at the moment,_ \(t=T_{0}\)_
* **_r_ is a current number of iteration during schedule optimization,
* **_\(\widetilde{\Pi}\)_ is a derivative matrix,\(\varepsilon_{u}\) is a given accuracy of [PERSON]'s method iterative procedure,  
\(C_{i}\) is a penalty coefficient,  
\(\Delta_{c,i(r-1)>}\) is a component of the function \(\Delta_{(r-1)}\) gradient on the iteration \(r-1\),  
\(\gamma_{c(i\gamma)^{\prime}}\) is a step of gradient (subgradient) method (algorithm) on the iteration \(r\)  
\(\widetilde{\Psi}_{(0)}(T_{0})\) is a adjoint vector at the moment \(t=T_{0}\) on the iteration \({}^{\prime\prime}0^{\prime\prime}\),  
\(\widetilde{\Psi}_{(r)}(T_{0})\) is a adjoint vector at the moment \(t=T_{0}\) on the iteration \({}^{\prime\prime}r^{\prime\prime}\),  
\(\sigma^{\prime}\) is a some part of the interval \(\sigma=(T_{0},T_{f})\),  
\(\widetilde{\widetilde{N}}\) is a assumed operator of new approximation procedure,  
\(\widetilde{T}_{c_{c},(r+1)>}^{\prime}\), \(\widetilde{T}_{c_{c},(r+1)>}^{\prime}\) are time moments of operation interruption,  
\(\mathbf{u}_{p}^{\prime}(t)\) is a vector of generalized control in relaxed problem of MS OPC construction,  
\(\mathbf{x}_{p}^{\prime}(t)\) is a vector of general state in relaxed problem of MS OPC construction,  
\(\mathbf{u}_{g}^{\prime}(t)\) is a vector of an arbitrary allowable control (allowable schedule),  
\(D_{\mathbf{x}}^{(i)}\) is a operation \"ae\" with object \"\(i\)\",  
\(D_{\tilde{\zeta}}^{(\omega)}\) is a operation \"\(\zeta\)\" with object \"\(\omega\)\",  
\(\mathbf{P}_{p}^{(i)}\), \(\mathbf{P}_{\tilde{\zeta}}^{(\omega)}\) is a current number of the branch and bound method subproblems,  
\(f_{p0}^{(1)}\) is a value of scalar form of MS vector quality measure for the first subproblem,  
\(f_{p1}^{(1)}\) is a value of scalar form of MS vector quality measure for the second subproblem,  
\(\mathbf{D}_{\mathbf{x}}\) is an attainable set,  
\(\theta\) is a current index of MS control model,  
\(g\) is an index of MS motion control model,  
\(k\) is an index of MS channel control model,  
\(o\) is an index of MS operation control model,  
\(f\) is an index of MS flow control model,  
\(p\) is an index of MS resource control model,  
\(e\) is an index of MS operation parameters control model,  
\(c\) is an index of MS structure dynamic control model,  
\(v\) is an index of MS auxiliary operation control model,  
\(l\) is a current number of MS elements and subsystems,  
\(u\) is a scalar allowable control input,  
\(\Theta\) is a current number of model,  
ae is a number of operation \"ae\", \(\zeta\) is a number of operation \"\(\zeta\)\",  
\(i\) is a current number of external object (customer),  
\(j\) is a current number of internal object resource,  
\(\varepsilon_{1}\), \(\varepsilon_{2}\) are known constants which characterize the accuracy of iterative solution of boundary-value problem.