ABSTRACT A tracking‐type optimal control problem governed by the viscous wave equation with a distributed‐source control and ‐ control costs is investigated. For this class of PDE‐constrained linear‐convex problems, a Pontryagin maximum principle (PMP) in the PDE setting is derived, and it is shown that the pointwise maximization condition is also sufficient for optimality. Based on the PMP, a sequential quadratic Hamiltonian (SQH) method is implemented, and a sufficient decrease property is established by introducing an adaptive penalization parameter. Convergence of the SQH method is discussed for both interval‐valued and discrete‐valued control sets. For the state and adjoint equations, a second‐order finite‐difference discretization is analysed. Numerical experiments validate both the approximation properties of the discretization and the effectiveness of the PMP‐based optimization framework.
Borzì et al. (2026) studied this question.