## Appendix A. List of Notations

$\mathbf{x}$ is a vector of the MS general state

$\mathbf{u}$ is a vector of the generalized control

${\mathbf{h}}_{0},\text{\hspace{0.17em}}{\mathbf{h}}_{1}$ are known vector function that are used for the state $\mathbf{x}$ and conditions

${\mathbf{q}}^{(1)},\text{\hspace{0.17em}}{\mathbf{q}}^{(2)}$ are known spatio-temporal, technical and technological constraints

${J}_{\vartheta}$ are indicators characterizing MS schedule quality

$\mathsf{\xi}$ is a known vector-function of perturbation influences

$\mathsf{\psi}$ is a vector of conjugate system state

$\mathsf{{\rm H}}$ 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,

${M}_{o}$ is a dynamic model of MS operational control,

M_{f} is a dynamic model of MS flow control,

M_{p} is a dynamic model of MS resource control,

${M}_{e}$ is a dynamic model of MS operation parameters control,

${M}_{c}$ is a dynamic model of MS structure dynamic control,

${M}_{\nu}$ 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,

${\psi}_{l}$ is a component of adjoint vector $\mathsf{\psi}(t)$,

${x}_{l}$ is an element of a general vector $\mathbf{x}(t)$,

${\lambda}_{\alpha}(t)$ is a dynamic Lagrange multiplier which corresponds to the components of vector ${\mathbf{q}}_{\alpha}^{(1)}$,

${\rho}_{\beta}(t)$ is a dynamic Lagrange multiplier which is corresponds to the components of vector ${\mathbf{q}}_{\alpha}^{(2)}$,

${z}_{{}_{ij\mu}}^{(o)}$ is an auxiliary variable which characterizes the execution of the operation

$\alpha $ is a current number of constraints ${\mathbf{q}}_{\alpha}^{(1)}$,

$\beta $ is a current number of constraints ${\mathbf{q}}_{\alpha}^{(2)}$,

${\psi}_{l}({T}_{0})$ is a component of adjoint vector $\mathsf{\psi}(t)$ at the moment $t={T}_{0}$,

${\psi}_{l}({T}_{f})$ is a component of adjoint vector $\mathsf{\psi}(t)$ at the moment $t={T}_{f}$,

$H\left(\mathbf{x}(t),\mathbf{u}(t),\mathsf{\psi}(t)\right)={\mathsf{\Psi}}^{\mathrm{T}}\mathbf{f}(\mathbf{x},\mathbf{u},t)$ is a Hamilton’s function,

$\mathsf{\Phi}\left(\mathsf{\psi}({T}_{0})\right)$ is an implicit function of boundary condition,

$\mathbf{a}$ is a vector given quantity value which corresponds to the vector $\mathbf{x}(t)$,

${\mathsf{\Delta}}_{u}$ is a function of boundary conditions,

$\mathsf{\rho}({T}_{f})$ is a discrepancy of boudary conditions,

${\mathsf{\psi}}_{(r)}({T}_{0})$ is a vector of adjoint system at the moment, $t={T}_{0}$

$r$ is a current number of iteration during schedule optimization,

$\tilde{\mathsf{\Pi}}$ is a derivative matrix,

${\epsilon}_{u}$ is a given accuracy of Newton’s method iterative procedure,

${C}_{i}$ is a penalty coefficient,

${\mathsf{\Delta}}_{<i,(r-1)>}$ is a component of the function ${\mathsf{\Delta}}_{(r-1)}$ gradient on the iteration $r-1$,

${\gamma}_{<i,r>}$ is a step of gradient (subgradient) method (algorithm) on the iteration $r$

${\tilde{\tilde{\mathsf{\psi}}}}_{(0)}({T}_{0})$ is a adjoint vector at the moment $t={T}_{0}$ on the iteration $\u20330\u2033$,

${\tilde{\mathsf{\psi}}}_{(r)}({T}_{0})$ is a adjoint vector at the moment $t={T}_{0}$ on the iteration $\u2033r\u2033$,

σ′ is a some part of the interval $\sigma =({T}_{0},{T}_{f}]$,

$\tilde{\tilde{N}}$ is a assumed operator of new approximation procedure,

${\tilde{t}}_{<e,(r+1)>}^{\prime}$, ${\tilde{t}}_{<e,(r+1)>}^{\u2033}$ are time moments of operation interruption,

${\mathbf{u}}_{p}^{*}(t)$ is a vector of generalized control in relaxed problem of MS OPC construction,

${\mathbf{x}}_{p}^{*}(t)$ is a vector of general state in relaxed problem of MS OPC construction,

${\mathbf{u}}_{g}(t)$ is a vector of an arbitrary allowable control (allowable schedule),

${D}_{\mathsf{\ae}}^{(i)}$ is a operation $\u201c\mathsf{\ae}\u201d$ with object $\u201ci\u201d$,

${D}_{\xi}^{(\omega )}$ is a operation $\u201c\xi \u201d$ with object $\u201c\omega \u201d$,

${\mathsf{{\rm P}}}_{\mathsf{\ae}}^{(i)}$,${\mathsf{{\rm P}}}_{\xi}^{(\omega )}$ is a current number of the branch and bound method subproblems,

${J}_{\mathrm{p}0}^{(1)}$ is a value of scalar form of MS vector quality measure for the first subproblem,

${J}_{\mathrm{p}1}^{(1)}$ is a value of scalar form of MS vector quality measure for the second subproblem,

${\mathbf{D}}_{x}$ is an attainable set,

$\vartheta $ 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,

$\nu $ 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,

Θ is a current number of model,

$\mathsf{\ae}$ is a number of operation $\u201c\mathsf{\ae}\u201d$, $\xi $ is a number of operation $\u201c\xi \u201d$, $\omega $ is a number of operation $\u201c\omega \u201d$,

i is a current number of external object (customer),

j is a current number of internal object resource,

${\epsilon}_{1}$, ${\epsilon}_{2}$ are known constants which characterize the accuracy of iterative solution of boundary-value problem.

$\tilde{\mathsf{\delta}}$ is a step of integration