Energy-aware unmanned aerial vehicle (UAV) trajectory design has been a crucial issue to address in UAV-assisted wireless networks. However, unlike traditional offline designs, online UAV trajectory designs in target tracking scenarios are non-trivial in concurrently satisfying the total propulsion energy consumption constraint and the initial-final location constraint due to the undetermined information. To address this issue, we propose a novel online UAV trajectory optimization approach for the weighted sum-predicted posterior Cramér-Rao bound (PCRB) minimization, which guarantees the feasibility of satisfying the two mentioned constraints. Specifically, our proposed approach designs the UAV trajectory by solving two subproblems: the candidate trajectory optimization problem and the energy-aware backup trajectory optimization problem. Then, we propose an algorithm converging to the global optimal solution to the candidate trajectory optimization problem based on Dinkelbach's transform and the Lasserre hierarchy. The energy-aware backup trajectory optimization problem is efficiently solved by the successive convex approximation method. Numerical results illustrate the effectiveness of our proposed approach and its significant superiority to the existing approach and benchmark regarding sensing performance and energy utilization flexibility.