Nonlinear boundary control problem of hyperbolic systems

The paper is devoted to the study of optimal control of a hyperbolic system, where the control enters the system through the boundary. The hyperbolic system is the wave equation on a smooth and open domain, where the boundary condition involves the normal derivative at the boundary of z, the time derivative of z times a constant k, and a nonlinear term control. Here, z is the state and u is the control, satisfying some boundedness condition depending on k. The functional cost consists of the energy and the difference between the solution of the system at final time, and a desired state in L-2-norm. For a closed convex set, we prove the existence of an optimal control that minimises the cost functional using a priori estimates. Then, using the differentiability of the cost functional with respect to the control, we establish the characterisation by deriving necessary conditions that an optimal control must satisfy. A numerical approach is successfully illustrated by simulations.