An optimal control approach to the approximation of fundamental solution groups for lossless wave equations

An approximation for the fundamental solution group for a general class of lossless wave equations is constructed using a recently developed concept of max-plus fundamental solution semigroup that arises in optimal control. In particular, by identifying all solutions of a bounded approximation of a wave equation of interest with the characteristic equations associated with a Hamiltonian, an auxiliary optimal control problem is constructed. As the dynamic programming evolution of the value function of this optimal control problem encapsulates evolution of all possible solutions of the aforementioned characteristic equations, and hence the approximating wave equation of interest, a max-plus fundamental solution semigroup for the optimal control problem can be used to construct the fundamental solution group for this wave equation.