A Sensitivity and Adjoint Calculus for Discontinuous Solutions of Hyperbolic Conservation Laws with Source Terms

We present a sensitivity and adjoint calculus for the control of entropy solutions of scalar conservation laws with controlled initial data and source term. The sensitivity analysis is based on shift-variations which are the sum of a standard variation and suitable corrections by weighted indicator functions approximating the movement of the shock locations. Based on a first order approximation by shift-variations in L1 we introduce the concept of shift-differentiability, which is applicable to operators having functions with moving discontinuities as images and implies differentiability for a large class of tracking-type functionals. In the main part of the paper we show that entropy solutions are generically shift-differentiable at almost all times t>0 with respect to the control. Hereby we admit shift-variations for the initial data which allows us to use the shift-differentiability result repeatedly over time slabs. This is useful for the design of optimization methods with time domain decomposition. Our analysis, especially of the shock sensitivity, combines structural results by using generalized characteristics and an adjoint argument. Our adjoint-based shock sensitivity analysis does not require us to restrict the richness of the shock structure a priori and admits shock generation points. The analysis is based on stability results for the adjoint transport equation with discontinuous coefficients satisfying a one-sided Lipschitz condition. As a further main result we derive and justify an adjoint representation for the derivative of a large class of tracking-type functionals.

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