Abstract: In lhis paper, we study the variable structure control of indefinite-dimensionalcontrol systems with the functional analysis method. The reaching conditions, stabilityconditions and the approximating conditions of sliding mode, as well as the generalform of the variable structure control law are given. and the elementary frame of thevariable structure control of indefinite-dimensional systems is built.

Key words: hydromagnetic, fluctuating, natural convection, magnetized plate, current density, heat transfer, semi-inner product, variable structure control, sliding mode, indefinite-dimensionalcontrol systems