A theoretical analysis of formation flight as a nonlinear self-organizing phenomenon

This study analyses the existence, stability and self-organizationof formation flight utilized by migrant birds. Air is approximatedas an incompressible inviscid flow, while birds are modelledas elliptically loaded lifting-lines. Application of conventionalwing theory leads to newly derived, basic equations that describethe problem as a dynamical system of multiple wings interactingwith each other through induced flow field. Formation flightis defined as the steady-state solution of the basic equations,in particular the solution that all the birds fly at the samespeed. In the case of a prescribed thrust, constant transverseinterval between adjacent birds, and a flock of physically identicalbirds, analytical study of the basic equations reveals the factsthat (1) formation flight is self-organized and (2) this formationflight is stable. The new implication is that a configurationof formation emerges as a result of nonlinear dynamical interactionbetween many birds and that this nonlinear dynamical systemdoes not exhibit chaotic behaviour. Numerical calculation hasalso been done for cormorant-type birds with the same transverseinterval between flock members. The proposed numerical schemequickly converges to very accurate results owing to the recentlyderived, closed-form expression of induced velocity distributionaround an elliptically loaded lifting-line. Transverse intervalsbetween birds are found to be a more important factor than thenumber of birds. Configurations of formations are found to beinverted U rather than inverted V. In these formations everybird enjoys the same amount of drag reduction.     

Multivariable identification and controller design of an integrated flight control system

This paper investigates the multivariable identification and controller design for the longitudinal channel of a Boeing 747 transport. The transfer function matrix of the system is identified using the prediction error (PE) identification method with multivariable ARX model. An ellipsoidal parametric uncertainty set is constructed from the covariance matrix of the identified parameters. It contains the parameters of actual system at a certain probability level. The identified models and the associated uncertainty sets are validated by measuring the worst-case ν-gap and then compared with the maximum value of the generalized stability margin. In automatic flight control system or autopilots, multiple specifications criteria are needed to be satisfied concurrently, such as good holding (small static altitude holding error), fast response, smooth transition (less oscillation, overshoot). The design of a Multiple Simultaneous Specifications (MSS) controller effectively and practically is a very significant and challenging job. Liu and Mills [H.H.T. Liu, J.K. Mills, Multiple specification design in flight control system, in: Proceedings of the American Control Conference, Chicago, Illinois, 2000, pp. 1365–1369] proposed a MSS controller design method using a convex combination approach. In this paper, we apply the method [H.H.T. Liu, J.K. Mills, Multiple specification design in flight control system, in: Proceedings of the American Control Conference, Chicago, Illinois, 2000, pp. 1365–1369; H.H.T. Liu, Design combination in integrated flight control, in: Proceedings of the American Control Conference, Arlington, Virginia, 2001, pp. 494–499; H.H.T. Liu, Multi-objective design for an integrated flight control system: a combination with model reduction approach, in: Proceedings of IEEE International Symposium on Computer Aided Control System Design, Glasgow, 2002, pp. 21–26] to design a MSS controller based on the identified models of the Boeing 747 transport aircraft longitudinal channel. The controllers are also validated by simulation using the true plant transfer functions.     

A coupled nonlinear hyperbolic-parabolic system

A boundary initial value problem for a quasi-linear hyperbolic system in one space variable is coupled to a boundary initial value problem for a quasi-linear parabolic equation in two space variables. The coupling occurs through one of the boundary conditions for the hyperbolic system and the source term in the parabolic equation. Such a coupling can arise in the consideration of gas flowing in a porous medium and out of that medium via a pipe. A local existence and uniqueness theorem is demonstrated. The proof involves the method of characteristics, Bernstein's estimates for parabolic partial differential equations, and the contracting mapping theorem.     

On a control problem for a nonlinear second-order system

A control problem for a nonlinear second-order system of differential equations in the presence of uncontrollable effects is investigated. A solution algorithm is proposed in the case when one phase coordinate of the system is measured at discrete moments. The algorithm is stable with respect to information noises and computational errors. Results of a computer experiment are presented.     

A nonlinear diffusion system coupled via nonlinear boundary flux

In this paper, we deal with the global existence and nonexistence of solutions to a nonlinear diffusion system coupled via nonlinear boundary flux. By constructing various kinds of sub- and super-solutions and using the basic properties of M-matrix, we give the necessary and sufficient conditions for global existence of nonnegative solutions. The critical curve of Fujita type is conjectured with the aid of some new results, which extend the recent results of Wang et al. [Nonlinear Anal. 71 (2009) 2134-2140] and Li et al. [J. Math. Anal. Appl. 340 (2008) 876-883] to more general equations.     

Optimal control for nonlinear dynamical system of microbial fed-batch culture

In fed-batch culture of glycerol bio-dissimilation to 1, 3-propanediol (1, 3-PD), the aim of adding glycerol is to obtain as much 1, 3-PD as possible. So a proper feeding rate is required during the process. Taking the concentration of 1, 3-PD at the terminal time as the performance index and the feeding rate of glycerol as the control function, we propose an optimal control model subject to a nonlinear dynamical system and constraints of continuous state and non-stationary control. A computational approach is constructed to seek the solution of the above model in two aspects. On the one hand we transcribe the optimal control model into an unconstrained one based on the penalty functions and an extension of the state space; on the other hand, by approximating the control function with simple functions, we transform the unconstrained optimal control problem into a sequence of nonlinear programming problems, which can be solved using gradient-based optimization techniques. The convergence analysis of this approximation is also investigated. Numerical results show that, by employing the optimal control policy, the concentration of 1, 3-PD at the terminal time can be increased considerably.     

Optimal control of a nonlinear singular system with state constraints

A control system described by a nonlinear equation of parabolic type is considered in the situation where there may be no global solution. A particular optimal control problem subject to state constraints is studied. A proof of the existence of an optimal control is presented. The penalty method is used to obtain necessary conditions for optimal control. A proof of the convergence of this method is given. The successive approximation method is used to obtain an approximate solution for the conditions derived. Translated fromMatematicheskie Zametki, Vol. 60, No. 4, pp. 511–518, October, 1996.     

Approximation theorem for a nonlinear control system with sliding modes

We consider the question of validity of the extension of a nonlinear control system by introducing the so-called sliding modes (i.e., by convexifying the set of admissible velocities) in the presence of constraints imposed on the endpoints of trajectories. We prove that a trajectory of the extended system can be approximated by trajectories of the original system if the equality constraints of the extended system are nondegenerate in the first order. The proof is based on a nonlocal estimate for the distance to the zero set of the nonlinear operator corresponding to the extended system, and involves a specific iteration process of corrections. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2007, Vol. 256, pp. 102–114.     

Optimal control of a nonlinear mechanical system by instantaneous impulses

A method is proposed for the numerical construction of an optimal control for a nonlinear system moving under the action of instantaneous impulses. The magnitudes of impulses and the moments of their action are variable. The algorithm is based on the sequential linearization of the system of equations of motion and the projection of the antigradient of the objective function on the set of constraints.Translated from Dinamicheskie Sistemy, No. 4, pp. 3–10, 1985.     

Bifurcation continuation,chaos and chaos control in nonlinear Bloch system

A detailed analysis is undertaken to explore the stability and bifurcation pattern of the nonlinear Bloch equation known to govern the dynamics of an ensemble of spins, controlling the basic process of nuclear magnetic resonance. After the initial analysis of the parameter space and stability region identification, we utilize the MATCONT package to analyze the detailed bifurcation scenario as the two important physical parameters γ (the normalized gain) and c (the phase of the feedback field) are varied. A variety of patterns are revealed not studied ever before. Next we explore the structure of the chaotic attractor and how the identification of unstable periodic orbit (UPO) can be utilized to control the onset of chaos.     

Asymptotic analysis of a nonlinear stochastic eco-epidemiological system with feedback control

This paper proposes a new stochastic eco-epidemiological model with nonlinear incidence rate and feedback controls. First, we prove that the stochastic system has a unique global positive solution. Second, by constructing a series of appropriate stochastic Lyapunov functions, the asymptotic behaviors around the equilibria of deterministic model are obtained,and we demonstrate that the stochastic system exists a stationary Markov process. Third, the conditions for persistence in the mean and exti...     

Optimal control problem for cancer invasion parabolic system with nonlinear diffusion

ABSTRACT

In this paper, we study a distributed optimal control problem of a coupled nonlinear system of reaction–diffusion equations. The system consists of three partial differential equations to represent cancer cell density, matrix-degrading enzymes concentration and oxygen concentration, and an ordinary differential equation to describe the extracellular matrix concentration. Our aim is to minimize the growth of cancer cells by controlling the production of matrix-degrading enzymes. First, we prove the existence and uniqueness of solutions of the direct problem. Then, we prove the existence of an optimal control. Finally, we derive the first-order optimality conditions and prove the existence of weak solutions of the adjoint problem.