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September  2014, 9(3): 501-518. doi: 10.3934/nhm.2014.9.501

On relaxation of state constrained optimal control problem for a PDE-ODE model of supply chains

Ciro D'Apice 1, , Peter I. Kogut 2, and  Rosanna Manzo 3,

1. 

Department of Information Engineering, Electrical Engineering and Applied Mathematics, University of Salerno, Via Giovanni Paolo II, 132, Fisciano (SA), Italy
2. 

Department of Differential Equations, Dnipropetrovsk National University, Gagarin av., 72, 49010 Dnipropetrovsk, Ukraine
3. 

Dept. of Information Eng., Electrical Eng. and Applied Mathematics, University of Salerno, Via Giovanni Paolo II, 132, I 84084 Fisciano (SA), Italy

Received  April 2014 Revised  July 2014 Published  October 2014

We discuss the optimal control problem (OCP) stated as the minimization of the queues and the difference between the effective outflow and a desired one for the continuous model of supply chains, consisting of a PDE for the density of processed parts and an ODE for the queue buffer occupancy. The main goal is to consider this problem with pointwise control and state constraints. Using the so-called Henig delation, we propose the relaxation approach to characterize the solvability and regularity of the original problem by analyzing the corresponding relaxed OCP.
Keywords: entropy solutions, optimal control, relaxation, supply chains, Conservation laws, Henig dilating cone..
Mathematics Subject Classification: Primary: 35L65, 90B30, 49J20; Secondary: 76N2.
Citation: Ciro D'Apice, Peter I. Kogut, Rosanna Manzo. On relaxation of state constrained optimal control problem for a PDE-ODE model of supply chains. Networks & Heterogeneous Media, 2014, 9 (3) : 501-518. doi: 10.3934/nhm.2014.9.501
References:
[1]

D. Armbruster, P. Degond and C. Ringhofer, A model for the dynamics of large queuing networks and supply chains, SIAM Journal on Applied Mathematics, 66 (2006), 896-920. doi: 10.1137/040604625.  Google Scholar

[2]

H. Attouch, G. Buttazzo and G. Michaille, Variational Analysis in Sobolev and BV Spaces: Application to PDE and Optimization, SIAM, Philadelphia, 2006.  Google Scholar

[3]

G. Bretti, C. D'Apice, R. Manzo and B. Piccoli, A continuum-discrete model for supply chains dynamics, Networks and Heterogeneous Media, 2 (2007), 661-694. doi: 10.3934/nhm.2007.2.661.  Google Scholar

[4]

G. A. Chechkin and A. Yu. Goritsky, S.N. Kruzhkov's Lectures on First-Order Quasilinear PDEs, in Analytical and Numerical Aspects of PDEs, Walter de Gruyter, 2009.  Google Scholar

[5]

C. F. Daganzo, A Theory of Supply Chains, Springer-Verlag, New York, Berlin, Heidelberg, 2003. doi: 10.1007/978-3-642-18152-8.  Google Scholar

[6]

C. D'Apice, S. Göttlich, M. Herty and B. Piccoli, Modeling, Simulation, and Optimization of Supply Chains: A Continuous Approach, SIAM, Philadelphia, 2010. doi: 10.1137/1.9780898717600.  Google Scholar

[7]

C. D'Apice, P. I. Kogut and R. Manzo, Efficient controls for one class of fluid dynamic models, JFar East J. Appl. Math., 46 (2010), 85-119.  Google Scholar

[8]

C. D'Apice and R. Manzo, A fluid-dynamic model for supply chain, Networks and Heterogeneous Media, 1 (2006), 379-398. doi: 10.3934/nhm.2006.1.379.  Google Scholar

[9]

C. D'Apice, R. Manzo and B. Piccoli, Modelling supply networks with partial differential equations, Quarterly of Applied Mathematics, 67 (2009), 419-440.  Google Scholar

[10]

C. D'Apice, R. Manzo and B. Piccoli, Existence of solutions to Cauchy problems for a mixed continuum-discrete model for supply chains and networks, Journal of Mathematical Analysis and Applications, 362 (2010), 374-386. doi: 10.1016/j.jmaa.2009.07.058.  Google Scholar

[11]

C. D'Apice, R. Manzo and B. Piccoli, Optimal input flow for a PDE-ODE model of supply chains, Commun. Math. Sci., 10 (2012), 1225-1240. doi: 10.4310/CMS.2012.v10.n4.a10.  Google Scholar

[12]

C. D'Apice, R. Manzo and B. Piccoli, Numerical schemes for the optimal input flow of a supply-chain, SIAM Journal on Numerical Analysis, 51 (2013), 2634-2650. doi: 10.1137/120889721.  Google Scholar

[13]

F. Dubois and P. L. Lefloch, Boundary conditions for nonlinear hyperbolic systems of conservation laws, Journal of Differential Equations, 71 (1988), 93-122. doi: 10.1016/0022-0396(88)90040-X.  Google Scholar

[14]

E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, Boston, 1984. doi: 10.1007/978-1-4684-9486-0.  Google Scholar

[15]

S. Göttlich, M. Herty and A. Klar, Network models for supply chains, Comm. Math. Sci., 3 (2005), 545-559. doi: 10.4310/CMS.2005.v3.n4.a5.  Google Scholar

[16]

S. Göttlich, M. Herty and A. Klar, Modelling and optimization of supply chains on complex networks, Comm. Math. Sci., 4 (2006), 315-330. doi: 10.4310/CMS.2006.v4.n2.a3.  Google Scholar

[17]

K. Han, T. L. Friesz and T. Yao, A variational approach for continuous supply chain networks, SIAM J. Control Optim., 52 (2014), 663-686. doi: 10.1137/120868943.  Google Scholar

[18]

D. Helbing, S. Lämmer, T. Seidel, P. Seba and T. Platkowsk, Physics, stability and dynamics of supply networks, Phys. Rev., E, 70 (2004), 66-116. doi: 10.1103/PhysRevE.70.066116.  Google Scholar

[19]

M. Herty, A. Klar and B. Piccoli, Existence of solutions for supply chain models based on partial differential equations, SIAM J. Math. Anal., 39 (2007), 160-173. doi: 10.1137/060659478.  Google Scholar

[20]

C. Kirchner, M. Herty, S. Göttlich and A. Klar, Optimal Control for Continuous Supply Network Models, Netw. Heterog. Media, 1 (2006), 675-688. doi: 10.3934/nhm.2006.1.675.  Google Scholar

[21]

P. I. Kogut and G. Leugering, Optimal Control Problems for Partial Differential Equations on Reticulated Domains. Approximation and Asymptotic Analysis, Birkhäuser Verlag, Boston, 2011. doi: 10.1007/978-0-8176-8149-4.  Google Scholar

[22]

P. I. Kogut and R. Manzo, On Vector-valued approximation of state constrained optimal control problems for nonlinear hyperbolic conservation laws, Journal of Dynamical and Control Systems, 19 (2013), 381-404. doi: 10.1007/s10883-013-9184-5.  Google Scholar

[23]

M. La Marca, D. Armbruster, M. Herty and C. Ringhofer, Control of continuum models of production systems, IEEE Trans. Automatic Control, 55 (2010), 2511-2526. doi: 10.1109/TAC.2010.2046925.  Google Scholar

[24]

P. D. Lax, Hyperbolic System of Conservation Laws and the Mathematical Theory of Shock Waves, Society of Industrial and Applied Mathematics, Philadelfia, Pa., 1973.  Google Scholar

[25]

M. Miranda, Comportamento delle successioni convergenti di frontiere minimali, Rend. Sem. Mat. Univ. Padova, 38 (1967), 238-257.  Google Scholar

[26]

R. Schiel, Vector Optimization ans Control with PDEs and Pointwise State Constraints, PhD thesis, Friedrich-Alexander-Universität Erlangen-Nürnberg, 2014. Google Scholar

[27]

D. M. Zhuang, Density result for proper efficiencies, SIAM J. on Control and Optimiz., 32 (1994), 51-58. doi: 10.1137/S0363012989171518.  Google Scholar

show all references

References:
[1]

D. Armbruster, P. Degond and C. Ringhofer, A model for the dynamics of large queuing networks and supply chains, SIAM Journal on Applied Mathematics, 66 (2006), 896-920. doi: 10.1137/040604625.  Google Scholar

[2]

H. Attouch, G. Buttazzo and G. Michaille, Variational Analysis in Sobolev and BV Spaces: Application to PDE and Optimization, SIAM, Philadelphia, 2006.  Google Scholar

[3]

G. Bretti, C. D'Apice, R. Manzo and B. Piccoli, A continuum-discrete model for supply chains dynamics, Networks and Heterogeneous Media, 2 (2007), 661-694. doi: 10.3934/nhm.2007.2.661.  Google Scholar

[4]

G. A. Chechkin and A. Yu. Goritsky, S.N. Kruzhkov's Lectures on First-Order Quasilinear PDEs, in Analytical and Numerical Aspects of PDEs, Walter de Gruyter, 2009.  Google Scholar

[5]

C. F. Daganzo, A Theory of Supply Chains, Springer-Verlag, New York, Berlin, Heidelberg, 2003. doi: 10.1007/978-3-642-18152-8.  Google Scholar

[6]

C. D'Apice, S. Göttlich, M. Herty and B. Piccoli, Modeling, Simulation, and Optimization of Supply Chains: A Continuous Approach, SIAM, Philadelphia, 2010. doi: 10.1137/1.9780898717600.  Google Scholar

[7]

C. D'Apice, P. I. Kogut and R. Manzo, Efficient controls for one class of fluid dynamic models, JFar East J. Appl. Math., 46 (2010), 85-119.  Google Scholar

[8]

C. D'Apice and R. Manzo, A fluid-dynamic model for supply chain, Networks and Heterogeneous Media, 1 (2006), 379-398. doi: 10.3934/nhm.2006.1.379.  Google Scholar

[9]

C. D'Apice, R. Manzo and B. Piccoli, Modelling supply networks with partial differential equations, Quarterly of Applied Mathematics, 67 (2009), 419-440.  Google Scholar

[10]

C. D'Apice, R. Manzo and B. Piccoli, Existence of solutions to Cauchy problems for a mixed continuum-discrete model for supply chains and networks, Journal of Mathematical Analysis and Applications, 362 (2010), 374-386. doi: 10.1016/j.jmaa.2009.07.058.  Google Scholar

[11]

C. D'Apice, R. Manzo and B. Piccoli, Optimal input flow for a PDE-ODE model of supply chains, Commun. Math. Sci., 10 (2012), 1225-1240. doi: 10.4310/CMS.2012.v10.n4.a10.  Google Scholar

[12]

C. D'Apice, R. Manzo and B. Piccoli, Numerical schemes for the optimal input flow of a supply-chain, SIAM Journal on Numerical Analysis, 51 (2013), 2634-2650. doi: 10.1137/120889721.  Google Scholar

[13]

F. Dubois and P. L. Lefloch, Boundary conditions for nonlinear hyperbolic systems of conservation laws, Journal of Differential Equations, 71 (1988), 93-122. doi: 10.1016/0022-0396(88)90040-X.  Google Scholar

[14]

E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, Boston, 1984. doi: 10.1007/978-1-4684-9486-0.  Google Scholar

[15]

S. Göttlich, M. Herty and A. Klar, Network models for supply chains, Comm. Math. Sci., 3 (2005), 545-559. doi: 10.4310/CMS.2005.v3.n4.a5.  Google Scholar

[16]

S. Göttlich, M. Herty and A. Klar, Modelling and optimization of supply chains on complex networks, Comm. Math. Sci., 4 (2006), 315-330. doi: 10.4310/CMS.2006.v4.n2.a3.  Google Scholar

[17]

K. Han, T. L. Friesz and T. Yao, A variational approach for continuous supply chain networks, SIAM J. Control Optim., 52 (2014), 663-686. doi: 10.1137/120868943.  Google Scholar

[18]

D. Helbing, S. Lämmer, T. Seidel, P. Seba and T. Platkowsk, Physics, stability and dynamics of supply networks, Phys. Rev., E, 70 (2004), 66-116. doi: 10.1103/PhysRevE.70.066116.  Google Scholar

[19]

M. Herty, A. Klar and B. Piccoli, Existence of solutions for supply chain models based on partial differential equations, SIAM J. Math. Anal., 39 (2007), 160-173. doi: 10.1137/060659478.  Google Scholar

[20]

C. Kirchner, M. Herty, S. Göttlich and A. Klar, Optimal Control for Continuous Supply Network Models, Netw. Heterog. Media, 1 (2006), 675-688. doi: 10.3934/nhm.2006.1.675.  Google Scholar

[21]

P. I. Kogut and G. Leugering, Optimal Control Problems for Partial Differential Equations on Reticulated Domains. Approximation and Asymptotic Analysis, Birkhäuser Verlag, Boston, 2011. doi: 10.1007/978-0-8176-8149-4.  Google Scholar

[22]

P. I. Kogut and R. Manzo, On Vector-valued approximation of state constrained optimal control problems for nonlinear hyperbolic conservation laws, Journal of Dynamical and Control Systems, 19 (2013), 381-404. doi: 10.1007/s10883-013-9184-5.  Google Scholar

[23]

M. La Marca, D. Armbruster, M. Herty and C. Ringhofer, Control of continuum models of production systems, IEEE Trans. Automatic Control, 55 (2010), 2511-2526. doi: 10.1109/TAC.2010.2046925.  Google Scholar

[24]

P. D. Lax, Hyperbolic System of Conservation Laws and the Mathematical Theory of Shock Waves, Society of Industrial and Applied Mathematics, Philadelfia, Pa., 1973.  Google Scholar

[25]

M. Miranda, Comportamento delle successioni convergenti di frontiere minimali, Rend. Sem. Mat. Univ. Padova, 38 (1967), 238-257.  Google Scholar

[26]

R. Schiel, Vector Optimization ans Control with PDEs and Pointwise State Constraints, PhD thesis, Friedrich-Alexander-Universität Erlangen-Nürnberg, 2014. Google Scholar

[27]

D. M. Zhuang, Density result for proper efficiencies, SIAM J. on Control and Optimiz., 32 (1994), 51-58. doi: 10.1137/S0363012989171518.  Google Scholar

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