The optimal control problem in the processes described by the Goursat problem for a hyperbolic equation in variable exponent Sobolev spaces with dominating mixed derivatives

In this paper a necessary and sufficient condition, such as the Pontryagin’s maximum principle for an optimal control problem with distributed parameters, is given by a hyperbolic equation of the second order with Lp(x)Lp(x)-coefficients. The results can be used in the theory of optimal processes for distribution Pontryagin maximum principle for various controlled processes described by hyperbolic equations of second order with discontinuous coefficients in variable exponent Sobolev spaces with dominant mixed derivatives.