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 L2-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.