In this paper, the exponential stability in mean square for Markovian jumping systems (MJSs) is discussed. A new dynamic model, which involves parameters uncertainties, nonlinearities, and Lévy noises, is proposed. Moreover, an adaptive sliding mode controller is built to study the stability of such a complex model. First, an integral-type sliding mode surface (SMS) is established to obtain the sliding mode motion dynamics of MJSs. By the generalized Itô formula and the Lyapunov stability theory, some sufficient conditions are obtained to make sure the exponential stability in mean square for the sliding mode motion dynamics. Second, an adaptive sliding mode control law is provided to assure the reachability of the specified SMS. Furthermore, corresponding parameters of the sliding mode controller and the SMS can be got by solving the convex optimization problem. Finally, the validity of the stability results obtained is illustrated by a numerical simulation and a practical simulation.

Prescribed-time consensus tracking for second-order nonlinear multiagent systems (MASs) with the unknown nonlinear dynamics satisfying a time-varying Lipschitz growth rate is investigated in this article. A time-varying function is introduced as a part of the controller gains, and it plays an important role in overcoming the rapid growth of nonlinear terms and in ensuring that the consensus can be achieved in a preassigned time. An integral sliding-mode control protocol, which forces the system trajectory to move on to the defined sliding manifold at the initial moment, is proposed for solving the prescribed-time consensus tracking problem of leader-following MASs with disturbances. Furthermore, we propose a slightly different control law based on terminal sliding-mode control, and under such a controller, the trajectories of each follower reach the sliding manifold in an arbitrary assigned time $T_{1}$ , and then in a specified time $T_{2}$ , the position and velocity tracking errors for all followers converge to 0 at the same time instant. Based on the graph theory, state transformations, and Lyapunov theorem, we prove that the proposed solutions are feasible and, finally, three simulation examples are provided to verify the theoretical results.

To evaluate the energy-saving effect of the controller, obtaining upper bounds on energy consumption and control time has become a worthwhile and meaningful issue to study. This article mainly discusses three contents about the Kuramoto oscillator network, including fixed-time synchronization (FxTS), prescribed-time synchronization (PTS), and energy consumption estimation. First, to reach FxTS, two sufficient conditions are proposed to guarantee that the Kuramoto oscillator network can reach fixed-time phase agreement and frequency synchronization. Unlike finite/fixed-time controllers, the prescribed-time controller in this article includes a time-varying function term, which is essential to ensure that the system achieves the prescribed-time phase agreement and frequency synchronization. At the same time, the setting-time for PTS is independent of the system initial values or controller parameters, which expands the application prospects of the system. Then, with limited setting-time as a premise, the energy consumed during the fixed/prescribed-time control process is obtained, which helps to evaluate the working time of the system. Finally, an example of a 5-node network is used to illustrate the effectiveness of FxTS and PTS in Kuramoto-oscillator networks.