Abstract: We consider a thermoelastic system described by partial differential equations with initial and boundary conditions. We defined an appropriate Hilbert space and a partial differential operator based on the considered thermoelastic system, and transferred the system to a first-order evolution equation in the Hilbert space. Since the characteristic of the variable structure is a sliding mode motion, that is, a lower dimensional motion, the state of the system can be restricted under some manifold in the state space of the system. We discussed variable structure control and designed a sliding mode of the th ermoelastic system so as to make the system realize sliding mode motion during a finite time interval no matter which position the motion started from, and the whole system to be stable. By means of the semigroup of linear operators we proved the feasibility of the equivalent control equation to be appl ied to the thermoelastic sysytem. Eventually, we have designed a sliding mode control of the system and have shown that the thermoelastic system is exponentially asymptotical stable under the control designed.

Key words: thermoelastic problem, sliding mode control, distributed parameter system, semigroup of linear operators