Energy conversion from heat to sound is desirable in some practical applications such as thermoacoustic heat engines or cooling systems. However, it is unwanted in gas turbine or aeroengine combustors. In this work, a Rijke-type thermoacoustic model with a linearly varied mean temperature configuration is developed. An acoustically compact heat source is confined and characterized by a modified form of King's law. Unlike previous models available in the literature, the mean temperature is assumed to undergo not only a sudden jump across the heat source but also linearly increasing and decreasing in the pre- and after-heating regions respectively. Such mean temperature configuration is consistent with the experimental measurement. Coupling the heat source model with a Galerkin series expansion of the acoustic fluctuations provides a platform to gain insight on (1) the nonlinearity of the thermoacoustic system, (2) onset of limit cycle oscillations, (3) predicting its non-normality behaviors, (4) energy distribution and transfer between neighboring eigenmodes, and (5) evaluating the performance of feedback controllers. Pseudospectra and transient energy growth analyses are then performed. It reveals that the system is non-normal. And it is associated with transient growth of acoustical energy. The non-normality is found to be less intensified in comparison with that in a system with an invariant mean temperature from pre- and after-heating regions. To mitigate these limit cycle oscillations, the heat-to-sound coupling is interrupted by implementing multiple monopole-like actuators driven by a LQG (Linear Quadratic Gaussian) controller. For comparison, a pole-placement controller is also implemented. Approximately 76 dB sound pressure level reduction is achieved. However, implementing the LQG controller is shown to be associated with transient growth of acoustical energy, which has potential to trigger thermoacoustic instability. The present work opens up new applicable way to model thermoacoustic systems in the presence of a mean temperature gradient. Furthermore, it reveals new potential risk of applying active controllers to stabilize thermoacoustic systems. (C) 2016 Elsevier Ltd. All rights reserved.