Global dynamic behavior of a predator–prey model under ratio-dependent state impulsive control

This paper studies the global dynamic behavior of a prey–predator model with square root functional response under ratio-dependent state impulsive control strategy. It is shown that the boundary equilibrium point of the controlled system is globally asymptotically stable. An order-k periodic orbit is obtained by employing the Brouwer’s fixed point theorem. Furthermore, the critical values are determined for the existence of orbitally asymptotically stable order-1 and order-2 periodic orbits in finite time. These critical values play an important role in determining different kinds of order-k periodic orbits and can also be used for designing the control parameters to obtain the desirable dynamic behavior of the controlled prey–predator system. Moreover, it is found that the local equilibrium point is also globally asymptotically stable under the control strategy. Numerical examples are provided to validate the effectiveness and feasibility of the theoretical results.     

An optimal control problem for a prey–predator system with a general functional response

An optimal control problem is studied for a prey–predator system with a general functional response. The control functions represent the rate of mixture of the populations and the cost functional is of Mayer type. The number of switching points of the optimal control is discussed in terms of the sign of a specific constant.     

Bifurcation and control of a bioeconomic model of a prey–predator system with a time delay

In this paper, we analyze the dynamical behaviour of a bioeconomic model system using differential algebraic equations. The system describes a prey–predator fishery with prey dispersal in a two-patch environment, one of which is a free fishing zone and other is a protected zone. It is observed that a singularity-induced bifurcation phenomenon appears when a variation of the economic interest of harvesting is taken into account. We have incorporated a state feedback controller to stabilize the model system in the case of positive economic interest. A discrete-type gestational delay of predators is incorporated, and its effect on the dynamical behaviour of the model is analyzed. The occurrence of Hopf bifurcation of the proposed model with positive economic profit is shown in the neighbourhood of the coexisting equilibrium point through considering the delay as a bifurcation parameter. Finally, some numerical simulations are given to verify the analytical results, and the system is analyzed through graphical illustrations.     

Non-degenerate necessary optimality conditions for the optimal control problem with equality-type state constraints

In this work, an optimal control problem with state constraints of equality type is considered. Novelty of the problem formulation is justified. Under various regularity assumptions imposed on the optimal trajectory, a non-degenerate Pontryagin Maximum Principle is proven. As a consequence of the maximum principle, the Euler–Lagrange and Legendre conditions for a variational problem with equality and inequality state constraints are obtained. As an application, the equation of the geodesic curve for a complex domain is derived. In control theory, the Maximum Principle suggests the global maximum condition, also known as the Weierstrass–Pontryagin maximum condition, due to which the optimal control function, at each instant of time, turns out to be a solution to a global finite-dimensional optimization problem.     

The existence of an extremal solution to a nonlinear system with the right-handed Riemann–Liouville fractional derivative

A monotone iterative method is applied to show the existence of an extremal solution for a nonlinear system involving the right-handed Riemann–Liouville fractional derivative with nonlocal coupled integral boundary conditions. Two comparison results are established. As an application, an example is presented to demonstrate the efficacy of the main result.     

Optimal control of harvesting coupled with boundary control in a predator—prey system

We consider boundary control and control via harvesting in a parabolic predator—prey system for a bounded region. The boundary control depicts the relationship between the boundary environment and the possibly harmful species. In addition, a proportion of the predator is harvested for profit. We choose to maximize the objective functional which incorporates the amount of the prey and the revenue of harvesting of the predator less the economic cost of sustaining a satisfactory boundary habitat and the cost due to the harvesting component. Moreover, we characterize the unique optimal control in terms of the solution to the optimality system, which is the state system coupled with the adjoint system.     

On differentiability with respect to the initial data of the solution to an SDE with a Lévy noise and discontinuous coefficients

We construct a stochastic flow generated by an stochastic differential equation with its drift being a function of bounded variation and its noise being a stable process with exponent from (1,2). It is proved that the flow is non-coalescing and Sobolev differentiable with respect to the initial data. The representation for the derivative is given.     

Stability-enhancing control of a coupled transport–diffusion system with Dirichlet actuation and Dirichlet measurement

We investigate the problem of enhancing the stability of a coupled transport–diffusion system with Dirichlet actuation and Dirichlet measurement. In the recent paper [H. Sano, Neumann boundary control of a coupled transport–diffusion system with boundary observation, J. Math. Anal. Appl. 377 (2011) 807–816], we treated the stabilization problem for the case with Neumann actuation and Dirichlet measurement, where the variable transformation of the state is performed by using the fractional power of an unbounded operator. However, we cannot use the similar transformation for the case with Dirichlet actuation and Dirichlet measurement, since it brings an ill-posed expression of the system. So, we use an algebraic approach for the formulation of the system. In this paper, it is shown that a reduced-order model with a finite-dimensional state variable is controllable and observable. The fact enables us to construct a finite-dimensional stability-enhancing controller for the original infinite-dimensional system by using a residual mode filter (RMF) approach. The novelty of this paper is the structure that the controller contains the dynamics with respect to the control variable. As a result, the state vector of the resulting closed-loop system includes the control variable as its entry.     

The study of a predator–prey system with group defense and impulsive control strategy

A predator–prey system with group defense and impulsive control strategy is established. By using Floquet theorem and small amplitude perturbation skills, a locally asymptotically stable prey-eradication periodic solution is obtained when the impulsive period is less than some critical value. Otherwise, if the impulsive period is larger than the critical value, the system is permanent. By using bifurcation theory, we show the existence and stability of positive periodic solution when the pest-eradication lost its stability. Further, numerical examples show that the system considered has more complicated dynamics, such as: (1) quasi-periodic oscillating, (2) period-doubling bifurcation, (3) period-halving bifurcation, (4) non-unique dynamics (meaning that several attractors coexist), (5) attractor crisis, etc. Finally, the biological implications of the results and the impulsive control strategy are discussed.     

Existence and positivity of a system k-ε with a production term of the Rayleigh-Taylor type

In this paper, we study a modelization of the turbulence which allows us to have a control on the positivity of the kinetic turbulence energy k and the dissipation growth rate of the energy ε. We use for that purpose the maximum principle on a new system modelling the turbulence, written on the variables θ and φ first introduced by Lewandowski (θ = k/ε, φ = ε²/k³).Estimates on θ and φ are given for a turbulent system with a Rayleigh-Taylor type term under a hypothesis of low compressibility of the mean flow, which is more general than the hypothesis of Lewandowski.In a second part, we study a simpler convection diffusion system (the diffusion is a constant) in which there is still the Rayleigh-Taylor term. We show that the presence of this term gives greater solutions of the k, ε system, hence proving that these terms are turbulent kinetic energy production terms.     

Global existence and blow-up of solutions to a parabolic system with nonlocal sources and boundaries

This paper deals with a semi-linear parabolic system with nonlinear nonlocal sources and nonlocal boundaries.By using super-and sub-solution techniques,we first give the sufficient conditions that the classical solution exists globally and blows up in a finite time respectively,and then give the necessary and sufficient conditions that two components u and v blow up simultaneously.Finally,the uniform blow-up profiles in the interior are presented.     

Discretization of interior point methods for state constrained elliptic optimal control problems: optimal error estimates and parameter adjustment

An adjustment scheme for the relaxation parameter of interior point approaches to the numerical solution of pointwise state constrained elliptic optimal control problems is introduced. The method is based on error estimates of an associated finite element discretization of the relaxed problems and optimally selects the relaxation parameter in dependence on the mesh size of discretization. The finite element analysis for the relaxed problems is carried out and a numerical example is presented which confirms our analytical findings.     

Stochastic stability of Duffing–Mathieu system with delayed feedback control under white noise excitation

The asymptotic Lyapunov stability with probability one of Duffing–Mathieu system with time-delayed feedback control under white-noise parametric excitation is studied. First, the time-delayed feedback control force is expressed approximately in terms of the system state variables without time delay. Then, the averaged Itô stochastic differential equations for the system are derived by using the stochastic averaging method and the expression for the Lyapunov exponent of the linearized averaged Itô equations is derived. Finally, the effects of time delay in feedback control on the Lyapunov exponent and the stability of the system are analyzed. Meanwhile, the stability conditions for the system with different time delays are also obtained. The theoretical results are well verified through digital simulation.     

Uniqueness and stability of positive steady state solutions for a ratio-dependent predator–prey system with a crowding term in the prey equation

This paper deals with a ratio-dependent predator–prey system with a crowding term in the prey equation, where it is assumed that the coefficient of the functional response is less than the coefficient of the intrinsic growth rates of the prey species. We demonstrate some special behaviors of solutions to the system which the coexistence states of two species can be obtained when the crowding region in the prey equation only is designed suitably. Furthermore, we demonstrate that under some conditions, the positive steady state solution of the predator–prey system with a crowding term in the prey equation is unique and stable. Our result is different from those ones of the predator–prey systems without the crowding terms.     

Mean–variance optimal portfolios in the presence of a benchmark with applications to fraud detection

We first study mean–variance efficient portfolios when there are no trading constraints and show that optimal strategies perform poorly in bear markets. We then assume that investors use a stochastic benchmark (linked to the market) as a reference portfolio. We derive mean–variance efficient portfolios when investors aim to achieve a given correlation (or a given dependence structure) with this benchmark. We also provide upper bounds on Sharpe ratios and show how these bounds can be useful for fraud detection. For example, it is shown that under some conditions it is not possible for investment funds to display a negative correlation with the financial market and to have a positive Sharpe ratio. All the results are illustrated in a Black–Scholes market.     

Output tracking for an Euler–Bernoulli beam equation with moment boundary control and shear boundary disturbance

This paper is concerned with performance output tracking for an Euler–Bernoulli beam equation with moment boundary control and shear boundary disturbance. An infinite-dimensional disturbance estimator is designed to estimate the total disturbance. By compensating the total disturbance, a servomechanism corresponding to the reference signal and servomechanism-based output feedback control law are designed. It is proved that under such control law, the performance output tracks exponentially the reference signal and the involved states of closed-loop system are bounded. The most important contribution is to deal with the shear boundary term stemmed from the error system between the disturbance estimator and the original system. The admissibility does not hold for such shear boundary term, while the corresponding boundary terms in the existing literature was proved to be admissible. Two key steps are presented to cope with such problem: First, the semigroup generation and exponential stability for a coupled beam system are verified by Riesz basis approach; second, the admissibility of a control operator for semigroup governed by such coupled beam system is proved. Moreover, Sobolev embedding theorem is introduced to simplify the proof of the boundedness of the closed-loop systems with respect to the available literature. Some numerical simulations are presented to illustrate the effectiveness.