## 1 Introduction  
Sensor arrays have been widely used in a variety of applications including estimation of the direction of arrival (DOA) of signals [1, 2], tracking systems, location of sources [3], and suppression of interference signals [4], amongst others. Adaptive array systems are able to locate and track signals (of users and interferences), dynamically adjusting the sensor alignment to maximize reception, and minimizing interference using signal-processing algorithms [5]. In applications where the aim is to suppress interference signals associated with various different sources, adaptive sensor arrays can be used together with power control techniques. The development of adaptive sensor arrays using beamforming together with power control can be used to achieve better system performance, with lower consumption of energy for transmission [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. Methods used to resolve the problem of combining beamforming with power control have been described [6, 7], where an algorithm has been proposed that is capable of beamforming in the uplink channel, followed by the adjustment of power in this channel and in the downlink channel. In this case, the weights for the downlink channel are considered the same as those for the uplink channel. This algorithm improves the performance of the system using a Signal to Interference plus Noise Ratio (SINR). In [8], a duality constrained least-mean-square (DCLMS) algorithm was proposed that utilizes LMS to find the optimum beamforming weights, while at same time controlling the power in both the uplink and downlink channels. A reference signal is used for beamforming, avoiding the use of additional algorithms for computation of the arrival directions of the signals. Since it is based on LMS, the algorithm presents low computational complexity, and the adaptation step for control of convergence can be selected empirically in order to better address the objectives of application of the algorithm. Updating of the transmission power only occurs after convergence of the beamforming process, taking account of successive tests of convergence and loss of performance in non-stationary environments. A similar algorithm is presented in [9], but with proportionality between the uplink and downlink weights. In other work [10], minimization of the transmission power in the channels is performed for each antenna individually. All these approaches assume a priori knowledge of the channel, so that it is possible to calculate the reception SINR, which then enables calculations for updating of the transmission powers in the uplink and downlink channels. The work described in [11] presents a joint optimization of beamforming and power control in a coordinated multicell downlink system which attends multiple users per cell to maximize the minimum weighted signal-to-interference-plus-noise ratio. The optimal solution and distributed algorithm with a fast convergence rate are obtained using the nonlinear Perron-Frobenius theory and the multicell network duality. Despite operating in a distributed manner, the iterative algorithm requires instantaneous power update in a coordinated cluster by means of backhaul. In [12], an algorithm is proposed that explores techniques of beamforming at the source and destination nodes, together with control of transmission power, in order to minimize the total transmission power of the source so that a minimum SINR threshold can be maintained in each receiver. This objective is achieved using an iterative algorithm that combines these techniques. The work described in [13] proposes an algorithm for power control together with beamforming in the receiver, with multiple adaptive base stations, for communication in the uplink channel. An iterative optimization algorithm is proposed, and the results show that the transmission power can besignificantly reduced, even with a smaller number of multiple base stations, which is of considerable interest for uplink channel communications.