The variational optimality condition in the problem of minimizing the finite state norm by a composite system of hyperbolic and ordinary differential equations

An optimal control problem for a system of linear first-order hyperbolic equations is studied. The boundary conditions are determined from controlled systems of ordinary differential equations. A nonclassical exact formulae for the increment of a linear performance index (a finite state norm) is suggested. Based on this result, a variational optimality condition is proved. The original optimal control problems for a hyperbolic system is reduced to the problem for systems of ordinary differential equations.