This paper studies the problem of target control and how a virtual ellipsoid can avoid the static obstacle. During the motion to the target set, the virtual ellipsoid can achieve a motion under collision avoidance by keeping the distance between the ellipsoid and obstacle. We present solutions to this problem in the class of closed-loop (feedback) controls based on Hamilton–Jacobi–Bellman (HJB) equation. Simulation results verify the validity and effectiveness of our algorithm.

There is tremendous interest in designing feedback strategy control for clusters in modern control theory. We propose a novel numerical solution to target team control problems by using the Hamilton formalism methods. In order to ensure the smooth wireless information exchange, all members of the team are located in a virtual ellipsoidal container during the whole movement process. An ellipsoidal container tube is constructed as the external state constraint of the team. The corresponding value function is then formulated based on collision avoidance conditions and energy constraints in the process of the team motion. Time-dependent partial differential equations are formulated based on Hamilton formalism, which have been solved numerically by using the traditional finite difference method (FDM). The objective of the presented method is to obtain optimal control and motion trajectory of the cluster at each moment. Lastly, we conduct a simulation study of unmanned aerial vehicles (UAVs) to demonstrate the performance of the proposed method.

This paper investigates the problem of coordinated control for multiple surface unmanned vehicles in a cluster under the influence of time-varying disturbances such as wind field and tides, based on state feedback controllers. With the rapid development of offshore cluster formation control technology, studying the cluster formation control problem of surface unmanned vehicles in multiple clusters under the influence of wind field and tides in complex and changing sea environments has important practical significance. This paper proposes a complete solution based on the Hamiltonian method. Firstly, each cluster is set to be located within a ellipsoidal virtual container throughout the entire motion process, and the trajectory of the virtual ellipsoid is used as an external state constraint for the cluster. Based on this, the dynamic equation of the system and the corresponding value function under the influence of wind field and tides on the sea surface are established, and the Hamilton–Jacobi–Bellman equation is used to solve the energy constraint and mutual avoidance between clusters and their members to obtain the optimal control and trajectory of each cluster. Finally, numerical simulation results demonstrate the effectiveness of the proposed method.

The explicit formulas of operations, in particular addition and multiplication, of $p $-adic integers are presented. As applications of the results, at first the explicit formulas of operations of Witt vectors with coefficients in $\mathbb{F}_{2}$ are given; then, through solving a problem of Browkin about the transformation between the coefficients of a $p$-adic integer expressed in the ordinary least residue system and the numerically least residue system, similar formulas for Witt vectors with coefficients in $\mathbb{F}_{3}$ are obtained.