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## Controlling of clock synchronization in WSNs: structure of optimal solutions

We considered the control problem for wireless sensor networks with a single time server node and a large number of client nodes. The cost functional of this control problem accumulates clock synchronization errors in the clients nodes and the energy consumption of the server over some time interval $[0,T]$. For all possible parameter values we found the structure of optimal control function. It was proved that for any optimal solution $\widehat{R}\left(t\right)$ there exist a time moment $\tau,$ $0\leq\tau<T$, such that $\hat{u}(t)=0,$ $t\in[\tau,T]$, i.e., the sending messages at times close to $T$ is not optimal. We showed that for sufficiently large $u_{1}$ the optimal solutions contain singular arcs. We found conditions on the model parameters under which different types of the optimal control are realized.