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Regularity Theory and Adjoint-Based Optimality Conditions for a Nonlinear Transport Equation with Nonlocal Velocity

Gröschel M, Keimer A, Leugering G, Wang Z (2014)

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Publication Type: Journal article

Publication year: 2014

Journal

Publisher: Society for Industrial and Applied Mathematics

Book Volume: 52

Pages Range: 2141--2163

Volume: 52

Issue: 4

Journal Issue: 4

DOI: 10.1137/120873832

Abstract

In this contribution the optimal boundary control problem for a first order nonlinear, nonlocal hyperbolic PDE is studied. Motivated by various applications ranging from re-entrant manufacturing systems to particle synthesis processes, we establish the regularity of solutions for W1,p-data. Based on a general L2 tracking type cost functional, the existence, uniqueness, and regularity of the adjoint system in W1,p is derived using the special structure induced from the nonlocal flux function of the state equation. The assumption of W1,p - and not Lp-regularity comes thereby due to the fact that the adjoint equation asks for more regularity to be well defined. This problem is discussed in detail, and we give a solution by defining a special type of cost functional, such that the corresponding optimality system is well defined.

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APA:

Gröschel, M., Keimer, A., Leugering, G., & Wang, Z. (2014). Regularity Theory and Adjoint-Based Optimality Conditions for a Nonlinear Transport Equation with Nonlocal Velocity. SIAM Journal on Control and Optimization, 52(4), 2141--2163. https://doi.org/10.1137/120873832

MLA:

Gröschel, Michael, et al. "Regularity Theory and Adjoint-Based Optimality Conditions for a Nonlinear Transport Equation with Nonlocal Velocity." SIAM Journal on Control and Optimization 52.4 (2014): 2141--2163.

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