not sure understand your question.

The barrier option's price is determined by current time to maturity and current stock price. Nothing to do with the past path.

i want to use the expectation of the barrier option in the control variate method,Y(b)=Y-b(X-E(X)).but i don't know how to calculate E(X).I have the formula of the barrier option,but it is the function of the underlying price and time,I don't know waht two values should be if i want to calculate the expectation.

For control variant method, E(X) should be some thing we know explicitly. It has many choices

Your task now is to simulate Y, which is the option's price under stochastic interest rate framework.

What E(X) you can choose is the Barrier option price under constant interest rate model.

Of course, in theory, you can also choose the standard European Call for X, but that may not make much improvement on the simulation accuracy

Barrier option price under constant interest rate model, everything is the same as standard BS model, current time to maturity and current stock price.
I you feel confused of the terminal or starting point. You can just simulate the discounted payoff. The average discounted payoff is the option's price. E(X) is the expectation of discounted payoff, Y is the discounted payoff in the stochastic interest rate framwork and so does X

i still don't know what is the meaning of  ''E(X) is in the close form''. the formula of the barrier option price with deterministic interest rate is the function of S and t. if S is the average value of underlying price at  time T I simulated, what will t be if I want to calculate E(X)?can you help me explain it more explicitly?
thanks very much!

Ok, I give a simple example

suppose you want to get the price of an exotic option such as barrier or look back.

The naive MC is just simulate, say 10000 stock paths and calculate the corresponding payoff, then average all the paths and get the price.

The control variant method is you choose a control variant X. The  control variate's expectation should be known. In this example, I choose european call option which has a well-known BS formula to calcuate E(X) which is the close form solution of the european call.

What you now need to do is that simulate the payoff of the barrier option Y and the european call option X simutanously. Then you construct a new variable Y(b)=Y+b(X-E(X)), E(X) is known, you have the current stock price, the time to maturity and all the other necessary parameters. b is also available via some algorithm. So now you average Y(b) and get the barrier option price

Y(b) has a less variance than Y if you choose the control variant properly
In your case, you want to get the barrier option' price under stochastic interest rate, so what I recommand, is you choose the barrier option under constant interest rate framework as the control variant. It may or may not work. Anyway, you can make a try. You can also use interest rate as an control variant as you proposed, but it may not work.

It's too late, I should go to bed. Hope its clear now. If it is still not clear, you can check some related books or just google or baidu it out.
Best,