We establish necessary conditions for the optimality of smooth boundary and initial controls in a semilinear hyperbolic system of the first order. The problem adjoint to the original problem is a semilinear hyperbolic system without initial conditions. The suggested approach is based on the use of special variations of continuously differentiable controls. The existence of global generalized solutions for a semilinear first-order hyperbolic system in a domain unbounded in time is proved. The proof is based on the use of the Banach fixed-point theorem and a space metric with weight functions.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 67, No. 2, pp. 185–201, February, 2014.
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Derev’yanko, T.O., Kyrylych, V.M. Problem of Optimal Control for a Semilinear Hyperbolic System of Equations of the First Order with Infinite Horizon Planning. Ukr Math J 67, 211–229 (2015). https://doi.org/10.1007/s11253-015-1075-3
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DOI: https://doi.org/10.1007/s11253-015-1075-3