## 1 Introduction
### Contributions and Structure of the Paper  
This paper also deals with the problem of maximizing the lifetime of a mobile wireless sensor network and focuses on the energy spent during the task of communication among sensor nodes or between sensor nodes and sink nodes. It presents a novel convex optimization-based procedure (or method) that controls what we define as the communication activity levels of the sensor nodes, by dynamically assigning a number of time or time-frequency slots for communication with the sink nodes within a frame. The optimization concentrates on the communication task, since this is the most energy-consuming process of a sensor node [18]. Autonomous sensor node mobility is not considered here, since in this specific case large energy consumption would result from the mechanisms that produce mobility, as for instance in robots. Node movement can happen in a non-autonomous fashion, as for example when sensor nodes are attached to vehicles, animals, soldiers in the battle field, deployed in the sea, and so on. In all of these cases, the communication task will indeed consume most of the energy. In order to confer even more challenge to the problem solution besides mobility,it is assumed that the sensor network may adopt a wireless power transmission technique for battery recharging. The main attributes of the proposed method are:  
* To the best of our knowledge, it differs from any previous one in the literature. The main novelty resides in the form of saving energy and in the formulation and solution of the corresponding optimization instance.
* It is flexible, allowing for the control of the periodicity in which the optimization tasks are performed, according to the sensor nodes' mobility speeds: lower speeds require less frequent optimizations.
* It also allows for a flexible control over the trade-off between equalizing the sensor nodes' energy consumptions to make them die altogether and minimize the wasted (not used) energy, or relaxing on the energy consumptions in favor of longer lifetimes by means of bursty transmissions followed by silent periods. When the equalization of energy consumptions is privileged, the communication with the sink node is frequent and avoids the waste of bandwidth caused by the bursty transmission mode when there is not enough data to send to the sink node. The frequent communication also avoids the storage of a high amount of sensory data until the next transmission is allowed in the bursty communication mode.
* It is simple, though fully consistent with well-known guidelines for the design of lifetime optimization algorithms reported in the literature.
* It performs closely to or better than a greedy algorithm recently proposed in the literature and strictly developed under the above-mentioned guidelines, with the advantage of not requiring the real-time channel state and residual energy information needed by the greedy solution.
* The reported results show that the optimized network can achieve substantially longer lifetimes when compared with the non-optimized one.  
The remainder of the paper is organized as follows: Section 2 presents the problem statement and formulation. In Section 3, the corresponding solution via convex optimization is given. Numerical results and discussions concerning the performance of the proposed solution are reported in Section 4. Section 5 concludes the paper and gives some directions for future research.