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Niveau: Supérieur, Doctorat, Bac+8

Control of the Continuity Equation with a Non Local Flow Rinaldo M. Colombo? Michael Herty† Magali Mercier‡. September 29, 2009 Abstract This paper focuses on the analytical properties of the solutions to the continuity equa- tion with non local flow. Our driving examples are a supply chain model and an equation for the description of pedestrian flows. To this aim, we prove the well posed- ness of weak entropy solutions in a class of equations comprising these models. Then, under further regularity conditions, we prove the differentiability of solutions with re- spect to the initial datum and characterize this derivative. A necessary condition for the optimality of suitable integral functionals then follows. 2000 Mathematics Subject Classification: 35L65, 49K20, 93C20 Keywords: Optimal Control of the Continuity Equation; Non-Local Flows. 1 Introduction We consider the scalar continuity equation in N space dimensions { ∂t?+ div ( ? V (?) ) = 0 (t, x)?R+ ? RN ?(0, x) = ?0(x) x?RN (1.1) with a non local speed function V . This kind of equations appears in numerous examples, a first one being the supply chain model introduced in [3, 4], where V (?) = v ( ∫ 1 0 ?(x) dx ) , see Section 3.

Control of the Continuity Equation with a Non Local Flow Rinaldo M. Colombo? Michael Herty† Magali Mercier‡. September 29, 2009 Abstract This paper focuses on the analytical properties of the solutions to the continuity equa- tion with non local flow. Our driving examples are a supply chain model and an equation for the description of pedestrian flows. To this aim, we prove the well posed- ness of weak entropy solutions in a class of equations comprising these models. Then, under further regularity conditions, we prove the differentiability of solutions with re- spect to the initial datum and characterize this derivative. A necessary condition for the optimality of suitable integral functionals then follows. 2000 Mathematics Subject Classification: 35L65, 49K20, 93C20 Keywords: Optimal Control of the Continuity Equation; Non-Local Flows. 1 Introduction We consider the scalar continuity equation in N space dimensions { ∂t?+ div ( ? V (?) ) = 0 (t, x)?R+ ? RN ?(0, x) = ?0(x) x?RN (1.1) with a non local speed function V . This kind of equations appears in numerous examples, a first one being the supply chain model introduced in [3, 4], where V (?) = v ( ∫ 1 0 ?(x) dx ) , see Section 3.

- differentiability properties
- local operator
- r0 ?
- weak entropy
- well blow
- differentiability
- nonlinear local
- ?0 ?

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Published by | mijec |

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Language | English |

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