Is it possible to design such a microwave oven working on the same frequency
but the intensity of microwaves could be variable by a controller .If yes then how?
Control of the magnitude of power delivered to the microwave oven chamber can be achieved by control of the magnitude of the voltage applied to the microwave magnetron.
This can be achieved by connecting a Variac in the input ac line leading to the oven.
The output voltage from the Variac can be manually adjusted from zero to full voltage as supplied by the electric power company. Ideally the magnetron will not oscillate and not draw current and not provide power until the transformer within the oven power supply develops a certain voltage level. Depending on the magnetron this voltage may be in the range of 3 to 4 kilovolts. This voltage is called the Hartree voltage. A small increase in the Variac setting will cause the magnetron to draw a small current and generate a small amount of power. Further (slow) increase in the Variac setting will increase the voltage to the magnetron and the current/power will increase. Thus ideally one can adjust the oven power level. Note as the current is increased the frequency of the magnetron will vary somewhat. This is called frequency pushing.
Now it is not clear to me how the level of the input ac voltage affects the general electronics of the oven. This can be tested to see if the number of half-wave voltage pulses applied to the magnetron within a given time interval is affected (recall the number of pulses at a given voltage establishes the total energy delivered to the oven within that time period). If the circuit behaves badly one can try setting for maximum energy implying the pulses are continuously generated. If this works you will have a continuous stream of pulses with about a 4 millisecond pulse width separated by a time interval of t=1/60 seconds or some 16.6 milliseconds.
Be sure to put a liter or so of water in the oven chamber to act as a lossy load. At times the water may become superheated and exhaust from the container; steam burns can be very painful.
If this does not work then disconnect the ac line, remove the oven cover, very carefully using a metal rod with well insulated handle (long screw driver with plastic handle) to short the terminals of the capacitor, and then short circuit the terminals with an insulated clip lead. I suggest you read up on microwave oven repair prior to doing this unless you are familiar with high voltage and charged capacitors. Check the circuit diagram provided with the oven and disconnect leads leading to the control panel.
Now a typical kilowatt oven will draw some 12 ampere. One might start with a low power oven (few hundred watts or less) and thus use perhaps a 5 ampere rated Variac. The energy and power per pulse will be reduced. The fan for cooling the magnetron may run slow; a temperature sensor will kill the input voltage if the magnetron overheats.