Extremal problems for time lag parabolic systems

Extremal problems for time lag parabolic systems are presented. An optimal boundary control problem for distributed parabolic systems in which constant time lags appear in the state equations and in the boundary conditions simultaneously is solved. Such equations constitute in a linear approximation a universal mathematical model for many processes of optimal heating. The time horizon is fixed. Making use of the Dubovicki-Milutin scheme, necessary and sufficient conditions of optimality for the Neumann problem with the quadratic performance functionals and constrained control are derived.