The Travelling Salesman Problem (TSP) is a classical NP-hard problem that plays a crucial role in combinatorial optimization. In this paper, we are interested in the quantum search framework for the TSP because it has robust theoretical guarantees. However, we need to first search for all Hamiltonian cycles from a very large solution space, which greatly weakens the advantage of quantum search algorithms. To address this issue, one can first prepare a superposition state of all feasible solutions, and then amplify the amplitude of the optimal solution from it. We propose a quantum algorithm to generate the uniform superposition state of all N-length Hamiltonian cycles as an initial state within polynomial gate complexity based on pure quantum dynamic programming with very few ancillary qubits, which achieves exponential acceleration compared to the previous initial state preparation algorithm. As a result, we realized the theoretical minimum query complexity of quantum search algorithms for a general TSP. Compared to some algorithms that theoretically have lower query complexities but lack practical implementation solutions, our algorithm has feasible circuit implementation.

This paper is concerned with the fixed-time consensus control problem for non-strict feedback multi-agent systems with asymmetric output constraints and full state constraints. Considering the feasibility of controlling execution, a novel practical virtual control signal is developed utilizing both saturation function and hyperbolic tangent function to ensure that this signal can remain within the same restricted range as the corresponding state variable throughout entire operation process. In backstepping steps, the design of ideal virtual control signal also adopts a different form of piecewise function than before, introducing high-order polynomial functions to avoid singularity problems in the derivation process. In addition, function approximation ability of radial basis function neural networks technique is applied to estimate uncertainties derived from the system functions and controller design procedure. Moreover, universal barrier Lyapunov function approach is improved for constructing an adaptive constrained synchronization control scheme. By fixed-time stability theory, it is shown that the tracking errors of the MAS converge to an adjustable region around the origin in a fixed time and the state variables always obey their constraints. And the upper bound of the settling time is merely dependent on design parameters, which is not affected by the initial states of MAS. The effectiveness of the proposed control strategy is shown by a numerical simulation example at last. Two scenarios are provided to demonstrate the advantages of the control protocol proposed in this paper.

This article discusses the consensus tracking issue for multiagent systems, and the purpose is to develop a fixed-time consensus proposal by fuzzy adaptive method. To this end, we first set up a more general fixed-time stability criterion. By using the proposed stability criterion, a backstepping design procedure is presented to construct the fixed-time fuzzy adaptive controller. The suggested fuzzy adaptive control protocol guarantees that 1) for each agent, its closed-loop signals keep bounded; 2) the consensus tracking error tends to a small region around origin in fixed time. In addition, the virtual control signals are constructed to be the piecewise functions to prevent the singularity of their derivatives. Curve fitting method is used such that the designed virtual control signals are derivable at the point of partition. Finally, numerical simulation further checks the validity of the suggested control strategy.

This paper aims to address adaptive tracking control problem of distributed multiagent systems. Differing from some existing works, each follower under consideration is modeled by a nonlinear nonstrict feedback system, especially, the virtual and real control gains are unknown functions rather than constants. To overcome the difficulty caused by the unknown nonlinearities, radial basis function neural networks are employed to model those unknown nonlinearities. Then, adaptive neural approach and backstepping technique are combined to construct the consensus tracking control protocol. It is shown that under the action of the suggested control protocol, whole closed-loop system is stable and all the outputs of followers ultimately track the reference signal, i.e., the output of the leader, synchronously. Numerical simulation is presented to further demonstrate the efficacy of the suggested control proposal.

By studying two unsharp quantum structures, namely extended lattice ordered effect algebras and lattice ordered QMV algebras, we obtain some characteristic theorems of MV algebras. We go on to discuss automata theory based on these two unsharp quantum structures. In particular, we prove that an extended lattice ordered effect algebra (or a lattice ordered QMV algebra) is an MV algebra if and only if a certain kind of distributive law holds for the sum operation. We introduce the notions of (quantum) finite automata based on these two unsharp quantum structures, and discuss closure properties of languages and the subset construction of automata. We show that the universal validity of some important properties (such as sum, concatenation and subset constructions) depend heavily on the above distributive law. These generalise results about automata theory based on sharp quantum logic.