We study the turnpike phenomenon for optimal control problems with mean-field dynamics that are obtained as the limit $N\rightarrow \infty$ of systems governed by a large number $N$ of ordinary differential equations. We show that the optimal control problems with large time horizons give rise to a turnpike structure of the optimal state and the optimal control. For the proof, we use the fact that the turnpike structure for the problems on the level of ordinary differential equations is preserved under the corresponding mean-field limit.

Results of stabilization for the higher order of the Kadomtsev-Petviashvili equation are presented in this manuscript. Precisely, we prove with two different approaches that under the presence of a damping mechanism and an internal delay term (anti-damping) the solutions of the Kawahara–Kadomtsev–Petviashvili equation are locally and globally exponentially stable. The main novelty of this work is that we present the optimal constant, as well as the minimal time, that ensures that the energy associated with this system goes to zero exponentially.

This work concerns Markov decision chains on a denumerable state space endowed with a bounded cost function. The performance of a control policy is assessed by a long-run average criterion as measured by a risk-seeking decision maker with constant risk-sensitivity. Besides standard continuity–compactness conditions, the framework of the paper is determined by the following conditions: (i) the state process is communicating under each stationary policy, and (ii) the simultaneous Doeblin condition holds. Within this framework it is shown that (i) the optimal superior and inferior limit average value functions coincide and are constant, and (ii) the optimal average cost is characterized via an extended version of the Collatz–Wielandt formula in the theory of positive matrices.

While the quasilinear isothermal Euler equations are an excellent model for gas pipeline flow, the operation of the pipeline flow with high pressure and small Mach numbers allows us to obtain approximate solutions by a simpler semilinear model. We provide a derivation of the semilinear model that shows that the semilinear model is valid for sufficiently low Mach numbers and sufficiently high pressures. We prove an existence result for continuous solutions of the semilinear model that takes into account lower and upper bounds for the pressure and an upper bound for the magnitude of the Mach number of the gas flow. These state constraints are important both in the operation of gas pipelines and to guarantee that the solution remains in the set where the model is physically valid. We show the constrained exact boundary controllability of the system with the same pressure and Mach number constraints.

This paper studies a mixed singular/switching stochastic control problem for a multidimensional diffusion with multiple regimes on a bounded domain. Using probabilistic partial differential equation and penalization techniques, we show that the value function associated with this problem agrees with the solution to a Hamilton–Jacobi–Bellman equation. In this way, we see that the regularity of the value function is $ \textrm{C}^{0,1}\cap \textrm{W}^{2,\infty}_{\textrm{loc}}$ .

Cooperative coordination in multi-agent systems has been a topic of interest in networked control theory in recent years. In contrast to cooperative agents, Byzantine agents in a network are capable to manipulate their data arbitrarily and send bad messages to neighbors, causing serious network security issues. This paper is concerned with resilient tracking consensus over a time-varying random directed graph, which consists of cooperative agents, Byzantine agents and a single leader. The objective of resilient tracking consensus is the convergence of cooperative agents to the leader in the presence of those deleterious Byzantine agents. We assume that the number and identity of the Byzantine agents are not known to cooperative agents, and the communication edges in the graph are dynamically randomly evolving. Based upon linear system analysis and a martingale convergence theorem, we design a linear discrete-time protocol to ensure tracking consensus almost surely in a purely distributed manner. Some numerical examples are provided to verify our theoretical results.

The population dynamics for the replicator equation has been well studied in continuous time, but there is less work that explicitly considers the evolution in discrete time. The discrete-time dynamics can often be justified indirectly by establishing the relevant evolutionary dynamics for the corresponding continuous-time system, and then appealing to an appropriate approximation property. In this paper we study the discrete-time system directly, and establish basic stability results for the evolution of a population defined by a positive definite system matrix, where the population is disrupted by random perturbations to the genotype distribution either through migration or mutation, in each successive generation.

In this paper, we study the controllability of second-order nonlinear stochastic delay systems driven by the Rosenblatt distributions in finite dimensional spaces. A set of sufficient conditions are established for controllability of nonlinear stochastic delay systems using fixed point theory, delayed sine and cosine matrices and delayed Grammian matrices. Furthermore, controllability results for second-order stochastic delay systems driven by Rosenblatt distributions via the representation of solution by delayed sine and cosine functions are presented. Finally, our theoretical results are illustrated through numerical simulation.

We establish a separation principle for a class of fractional order time-delay nonlinear differential systems. We show that a nonlinear time-delay observer is globally convergent and give sufficient conditions under which the observer-based controller stabilises the system.

We focus on the convergence rate of the alternating direction method of multipliers (ADMM) in a complex domain. First, the complex form of variational inequality (VI) is established by using the Wirtinger calculus technique. Second, the $O(1/K)$ convergence rate of the ADMM in a complex domain is provided. Third, the ADMM in a complex domain is applied to the least absolute shrinkage and selectionator operator (LASSO). Finally, numerical simulations are provided to show that ADMM in a complex domain has the $O(1/K)$ convergence rate and that it has certain advantages compared with the ADMM in a real domain.

In this work, we examine the mathematical analysis and numerical simulation of pattern formation in a subdiffusive multicomponents fractional-reaction-diffusion system that models the spatial interrelationship between two preys and predator species. The major result is centered on the analysis of the system for linear stability. Analysis of the main model reflects that the dynamical system is locally and globally asymptotically stable. We propose some useful theorems based on the existence and permanence of the species to validate our theoretical findings. Reliable and efficient methods in space and time are formulated to handle any space fractional reaction-diffusion system. We numerically present the complexity of the dynamics that are theoretically discussed. The simulation results in one, two and three dimensions show some amazing scenarios.

We investigate the state feedback pinning synchronization of fractional-order complex networks. Based on the stability theory of fractional-order differential systems and state feedback control by a single controller, synchronization conditions for fractional-order complex networks are given. We assume that the coupling matrix is irreducible, and provide a numerical example to illustrate the validity of the proposed conclusions.

The aim of this work is to prove the existence of a positive almost periodic solution to a multifinite time delayed nonlinear differential equation that describes the so-called hematopoiesis model. The approach uses the Hilbert projective metric in a cone. With some additional assumptions, we construct a fixed point theorem to prove the desired existence and uniqueness of the solution.

We study the numerical performance of a continuous data assimilation (downscaling) algorithm, based on ideas from feedback control theory, in the context of the two-dimensional incompressible Navier-Stokes equations. Our model problem is to recover an unknown reference solution, asymptotically in time, by using continuous-in-time coarse-mesh nodal-point observational measurements of the velocity field of this reference solution (subsampling), as might be measured by an array of weather vane anemometers. Our calculations show that the required nodal observation density is remarkably less than what is suggested by the analytical study; and is in fact comparable to the number of numerically determining Fourier modes, which was reported in an earlier computational study by the authors. Thus, this method is computationally efficient and performs far better than the analytical estimates suggest.

This paper revisits the concept of rough paths of inhomogeneous degree of smoothness (geometric Π-rough paths in our terminology) sketched by Lyons in 1998. Although geometric Π-rough paths can be treated as p-rough paths for a sufficiently large p, and the theory of integration of Lipγ one-forms (γ > p–1) along geometric p-rough paths applies, we prove the existence of integrals of one-forms under weaker conditions. Moreover, we consider differential equations driven by geometric Π-rough paths and give sufficient conditions for existence and uniqueness of solution.

We introduce a class of discrete-time stochastic processes, called disjunctive processes, which are important for reliable simulations in random iteration algorithms. Their definition requires that all possible patterns of states appear with probability 1. Sufficient conditions for nonhomogeneous chains to be disjunctive are provided. Suitable examples show that strongly mixing Markov chains and pairwise independent sequences, often employed in applications, may not be disjunctive. As a particular step towards a general theory we shall examine the problem arising when disjunctiveness is inherited under passing to a subsequence. An application to the verification problem for switched control systems is also included.

This paper presents an integrated guidance and control (IGC) design method for an unmanned aerial vehicle with static stability which is described by a nonlinear six-degree-of-freedom (6-DOF) model. The model is linearized by using small disturbance linearization. The dynamic characteristics of pitching mode, rolling mode and Dutch rolling mode are obtained by analysing the linearized model. Furthermore, an IGC design procedure is also proposed in conjunction with a proportional–integral–derivative (PID) control method and fuzzy control method. A PID controller is applied in the control loop of the elevator and aileron, and the attitude angle and attitude angular velocity are used as compensation feedback, giving a simple and low-order control law. A fuzzy control method is applied to perform the cross-coupling control of rolling and yawing. Finally, the 6-DOF simulation shows the effectiveness of the developed method.

We present an application of optimal control theory to a simple SIR disease model of avian influenza transmission dynamics in birds. Basic properties of the model, including the epidemic threshold, are obtained. Optimal control theory is adopted to minimize the density of infected birds subject to an appropriate system of ordinary differential equations. We conclude that an optimally controlled seasonal vaccination strategy saves more birds than when there is a low uniform vaccination rate as in resource-limited places.