Using lifting technology,the linear periodic time-varying system is converted to a linear periodic time-invariant system,and the linear quadratic performance index of the linear periodic time-varying system is converted to that of a linear periodic time-invariant system.

We study one kind of linear forward backward stochastic control system and find a explicit solution of control under linear quadratic index by stochastic maximum principle, finally we prove that it is the unique optimal control.

By minimizing the quadratic performance index, a simple adaptive control algorithm is obtained.

In this paper, based on the optimal control thought of the quadratic performance index, system performance index incremental-contribution method is put forward to make it apply the elastic-plastic state for viscous dampers.

The quadratic index is introduced in order to realize the optimal control of the single-neuron adaptive PSD controller.

From the start of analysing the stability of the airconditioning control system according to linear system the calculating formulas of the optimal PID parameters is inferred by the quadratic form performance index to solve the parameters optimal question of the controller and the weightel optimal method.