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Optimal control of a production-inventory system with state constraints and a quadratic cost criterion. (English) Zbl 0586.90048

For the production inventory model of the form \(\dot x(\)t)\(=Ax(t)+bu(t)\), \(A=\left[ \begin{matrix} 0\\ 0\end{matrix} \begin{matrix} 1\\ -\alpha \end{matrix} \right]\), \(b=[0,\alpha]^ T\) where \(x_ 1\) denotes the difference between the actual and the final inventory, \(x_ 2\) represents the actual production rate and the control variable u is the desired production rate, the quadratic cost criterion \[ J(u)=\int^{t}_{0}x^ T(t)Cx(t)dt, \] \(t\in [0,T]\) is minimized with respect to the constraints \(| u| \leq 1\) and \(S(x):=x_ 2\leq \beta\). The optimal controls for the regular, the singular and the state constrained cases were derived applying minimum principle. Concerning the synthesis, the global structure of the optimal trajectory as well as the behavior of the control at switching points are given.

MSC:

90B30 Production models
90B05 Inventory, storage, reservoirs
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