Impulsive control of multiple Lotka–Volterra systems

In this work, we consider the control problem of multiple Lotka–Volterra system. Our means to control the population dynamics is via impulses not only in a single species, but also in multiple species, that is, some members of these populations are added to or removed from the environment impulsively at the same time. We establish the strategies for preventing all the species from going extinct by stabilizing some special positive points, which may not be the equilibrium points of the system. We give several Lotka–Volterra systems to illustrate our results by drawing their time-series graphs.     

A novel feedforward–feedback suboptimal control of linear time-delay systems

This paper presents a new approach for solving the optimal control problem of linear time-delay systems with a quadratic cost functional. In this approach, a method of successive substitution is employed to convert the original time-delay optimal control problem into a sequence of linear time-invariant ordinary differential equations (ODEs) without delay and advance terms. The obtained optimal control consists of a linear state feedback term and a forward term. The feedback term is determined by solving a matrix Riccati differential equation. The forward term is an infinite sum of adjoint vectors, which can be obtained by solving recursively the above-mentioned sequence of linear non-delay ODEs. A fast-converging iterative algorithm for this purpose is presented which provides a promising possible reduction of computational efforts. Numerical examples demonstrating the efficiency, simplicity and high accuracy of the suggested technique have been included. Simulation results reveal that just a few iterations of the proposed algorithm are required to find an accurate enough feedforward–feedback suboptimal control.     

Input–output control systems and dichotomy of variational difference equations

We propose a new and unified approach for the study of dichotomy of variational difference equations, establishing a link between control methods and basic techniques from interpolation theory. We obtain necessary and sufficient conditions for the existence of uniform dichotomy and, respectively, for uniform exponential dichotomy of variational difference equations in terms of the admissibility of general pairs of sequence spaces. We provide a classification of the main classes of sequence spaces where the input spaces and the output spaces may belong to, for each dichotomy property and prove that the hypotheses on the underlying sequence spaces cannot be removed. The obtained results extend the framework to the study of dichotomy of variational difference equations, hold without any requirement on the coefficients and are applicable to all systems of variational difference equations.     

Type-2 T–S fuzzy impulsive control of nonlinear systems

A novel impulsive control approach based on interval Type-2 T–S fuzzy model has been presented for nonlinear systems in this paper. This approach makes up for the drawback of Type-1 fuzzy impulsive control, which cannot fully handle the uncertainties in describing the complex nonlinear systems by Type-1 fuzzy membership functions and cannot give rigorous fuzzy rules. Further more, this approach uses the “broad band” effect of the Type-2 membership functions to solve the noise of training data and exterior disturbance of the Type-1 fuzzy impulsive control. By using Lyapunov theory and Lipschitz condition, which is combined with integrated approaches such as comparison methods and linear matrix inequalities, the Type-2 fuzzy impulsive controller is designed and the general asymptotical stability analysis of the systems is given. Finally, the simulation of the inverted pendulum model demonstrates the validity and superiority of the proposed method by easily determining the membership functions and choosing minimum number of fuzzy rules and the method can handle random disturbance and data uncertainties very well.     

Trend to equilibrium of renormalized solutions to reaction–cross-diffusion systems

The convergence to equilibrium of renormalized solutions to reaction–cross-diffusion systems in a bounded domain under no-flux boundary conditions is studied. The reactions model complex balanced chemical reaction networks coming from mass-action kinetics and thus do not obey any growth condition, while the diffusion matrix is of cross-diffusion type and hence nondiagonal and neither symmetric nor positive semi-definite, but the system admits a formal gradient-flow or entropy structure. The diffusion term generalizes the population model of Shigesada, Kawasaki and Teramoto to an arbitrary number of species. By showing that any renormalized solution satisfies the conservation of masses and a weak entropy–entropyproduction inequality, it can be proved under the assumption of no boundary equilibria that all renormalized solutions converge exponentially to the complex balanced equilibrium with a rate which is explicit up to a finite dimensional inequality.     

Time-discretization of nonlinear control systems with state-delay via Taylor–Lie series

The state-delay is always existent in the practical systems. Analysis of the delay phenomenon in a continuous-time domain is sophisticated. It is appropriate to obtain its corresponding discrete-time model for implementation via digital computer. In this paper, we propose a new scheme for the discretization of nonlinear systems using Taylor series expansion and the zero-order hold assumption. This scheme is applied to the sample-data representation of a nonlinear system with constant state time-delay. The mathematical expressions of the discretization scheme are presented and the effect of the time-discretization method on equilibrium properties of nonlinear control system with state time-delay is examined. The proposed scheme provides a finite-dimensional representation for nonlinear systems with state time-delay enabling existing controller design techniques to be applied to them. The performance of the proposed discretization procedure is evaluated using a nonlinear system. For this nonlinear system, various sampling rates and time-delay values are considered.     

The effects of lateral–torsional coupling on the nonlinear dynamic behavior of a rotating continuous flexible shaft–disk system with rub–impact

This study investigates the lateral–torsional coupling effects on the nonlinear dynamic behavior of a rotating flexible shaft–disk system. The system is modeled as a continuous shaft with a rigid disk in its mid span. Coriolis and centrifugal effects due to shaft flexibility are also included. The partial differential equations of motion are extracted under the Rayleigh beam theory. The assumed mode method is used to discretize partial differential equations and the resulting equations are solved via numerical methods. The analytical methods used in this work include time series, phase plane portrait, power spectrum, Poincaré map, bifurcation diagrams, and Lyapunov exponents. The main objective of the present study is to investigate the torsional coupling effects on the chaotic vibration behavior of a system. Periodic, sub-harmonic, quasi-periodic, and chaotic states can be observed for cases with and without torsional effects. As demonstrated, inclusion of the torsional–lateral coupling effects can primarily change the speed ratios at which rub–impact occurs. Also, substantial differences are shown to exist in the nonlinear dynamic behavior of the system in the two cases.     

Optimal distributed control of nonlinear Cahn–Hilliard systems with computational realization

The goal of this work is to present some optimal control aspects of distributed systems described by nonlinear Cahn–Hilliard equations (CH). Theoretical conclusions on distributed control of CH system associated with quadratic criteria are obtained by the variational theory (see [17]). A computational result is stated by a new semi-discrete algorithm, constructed on the basis of the finite-element method with the updated (nonlinear) conjugate gradient method for minimizing the performance index efficiently. Finally, the implementation of a laboratory demonstration is included to show the efficiency of the proposed nonlinear scheme.     

Prediction of Energy Barriers due to Nucleation in Solid-to-Solid Phase–Transitions

This contribution generally aims at the application of the multiscale-FEM to problems arising from inhomogeneities of arbitrary shape within a solid matrix material. In the context of our work, the inhomogeneity represents a nucleus of a new solid phase. According to the theory of homogeneous nucleation, the model predicts whether it is energetically preferable to initiate the growth of the nucleus or not. The implementation of this procedure into a micromechanical model for shape-memory alloys is assumed to be suitable to reconstruct the so-called stress-drop after the initiation of phase-transformation as observed in experiments. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Projective synchronization of chaotic fractional-order energy resources demand–supply systems via linear control

A fractional-order energy resources demand–supply system is proposed. A projective synchronization scheme is proposed as an extension on the synchronization scheme of Odibat et al. (2010). The scheme is applied to achieve projective synchronization of the chaotic fractional-order energy resource demand–supply systems. Numerical simulations are performed to verify the effectiveness of the proposed synchronization scheme.     

Global existence of weak solutions to quasilinear degenerate Keller–Segel systems of parabolic–parabolic type

This paper deals with the quasilinear degenerate Keller–Segel system (KS) of parabolic–parabolic type. The global existence of weak solutions to (KS) is established when $q<m+\frac{2}{N}$ (m denotes the intensity of diffusion and q denotes the nonlinearity) without restriction on the size of initial data; note that $q=m+\frac{2}{N}$ corresponds to generalized Fujita?s exponent. The result improves both Sugiyama (2007) [14, Theorem 1] and Sugiyama and Kunii (2006) [15, Theorem 1] in which it is assumed that $q?m$.     

Duality of Gauß–Manin systems associated to linear free divisors

We investigate differential systems occurring in the study of particular non-isolated singularities, the so-called linear free divisors. We obtain a duality theorem for these ${\mathcal{D}}$ -modules taking into account filtrations, and deduce degeneration properties of certain Frobenius manifolds associated to linear sections of the Milnor fibres of the divisor.     

Global existence and blowing-up of solutions to a class of coupled reaction–convection–diffusion systems

This paper concerns the global existence and blowing-up of solutions to the homogeneous Neumann problem of a coupled reaction–convection–diffusion system. The critical Fujita curve is determined and blowing-up theorem of Fujita type is established. An interesting phenomenon is that the critical Fujita curve even could be the infinite due to the convection.     

Synchronization and anti-synchronization of Lu and Bhalekar–Gejji chaotic systems using nonlinear active control

This paper aims at synchronization and anti-synchronization between Lu chaotic system, a member of unified chaotic system, and recently developed Bhalekar–Gejji chaotic system, a system which cannot be derived from the member of unified chaotic system. These synchronization and anti-synchronization have been achieved by using nonlinear active control since the parameters of both the systems are known. Lyapunov stability theory is used and required condition is derived to ensure the stability of error dynamics. Controller is designed by using the sum of relevant variables in chaotic systems. Simulation results suggest that proposed scheme is working satisfactorily.     

T–S fuzzy-model-based robust stabilization for a class of nonlinear discrete-time networked control systems

In this paper, the robust stabilization problem is investigated for a class of nonlinear discrete-time networked control systems (NCSs). To study the system stability and facilitate the design of fuzzy controller, Takagi–Sugeno (T–S) fuzzy models are employed to represent the system dynamics of the nonlinear discrete-time NCSs with effects of the approximation errors taken into account, and a unified model of NCSs in the T–S fuzzy model is proposed by modeling the approximation errors as norm-bounded uncertainties in system metrics, where non-ideal network Quality of Services (QoS), such as data dropout and network-induced delay, are coupled in a unified framework. Then, based on the Lyapunov–Krasovskii functional, sufficient conditions are derived for the existence of a fuzzy controller. By these criteria, two approaches to design a fuzzy controller are developed in terms of linear matrix inequalities (LMIs). Finally, illustrative examples are provided to show the effectiveness of the proposed methods.     

A dynamical approach to the large-time behavior of solutions to weakly coupled systems of Hamilton–Jacobi equations

We investigate the large-time behavior of the value functions of the optimal control problems on the n-dimensional torus which appear in the dynamic programming for the system whose states are governed by random changes. From the point of view of the study on partial differential equations, it is equivalent to consider viscosity solutions of quasi-monotone weakly coupled systems of Hamilton–Jacobi equations. The large-time behavior of viscosity solutions of this problem has been recently studied by the authors and Camilli, Ley, Loreti, and Nguyen for some special cases, independently, but the general cases remain widely open. We establish a convergence result to asymptotic solutions as time goes to infinity under rather general assumptions by using dynamical properties of value functions.