Communications in Mathematical Sciences

Volume 10 (2012)

Number 4

Optimal input flows for a PDE–ODE model of supply chains

Pages: 1225 – 1240



Ciro D’Apice (Department of Electronic and Information Engineering, University of Salerno, Italy)

Rosanna Manzo (Department of Electronic and Information Engineering, University of Salerno, Italy)

Benedetto Piccoli (Department of Mathematical Sciences, Rutgers University)


In this paper we deal with a continuous model for supply chains, consisting of a PDE for the density of processed parts and an ODE for the queue buffer occupancy. We discuss the optimal control problem stated as the minimization of the queues and the quadratic difference between the effective outflow and a desired one. Here the input flow is the control and is assumed to have uniformly bounded variation. Introducing generalized tangent vectors to piecewise constant controls, representing shifts of discontinuities, we analyze the dependence of the solution on the control function. Then existence of an optimal control for the original problem is obtained. Finally we study the sensitivity of the cost functional J as function of controlled inflow, providing an estimate of the derivative of J with respect to switching times.


conservation laws, supply chains, optimal control

2010 Mathematics Subject Classification

35L65, 49J20, 76N25, 90B30

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