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

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 W 1,p–data. Based on a general L2 tracking type cost functional, the existence, uniqueness, and regularity of the adjoint system in W 1,p is derived using the special structure induced from the nonlocal flux function of the state equation. The assumption of W 1,pand 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.