Energy Conserving Norms for the Solution of Hyperbolic Systems of Partial Differential Equations

The problem of finding an energy conserving norm for the solution of the hyperbolic system of partial differential equations du/dt = Adu/dx, subject to boundary conditions, is reduced to the problem of characterizing those matrices appearing in the boundary conditions which satisfy two specific matrix equations. Necessary and sufficient conditions on the coefficient matrix A and the matrices appearing in boundary conditions are derived for an energy conserving norm to exist. Thus, these conditions serve as tests on a given system which determine whether or not the solution will have its energy conserved in some norm. In addition, some examples of specific systems and boundary conditions are provided. I. Motivation. Consider the hyperbolic system of partial differential equations (1.1) ^= A^f0rr>OandOOandO • • • > X„. Suppose that p of these eigenvalues are positive and q are negative and that p + q = n (we, therefore, exclude the possibility of vanishing eigenvalues); then, we may perform the partitions