On the controllability of some singularly perturbed nonlinear systems

The controllability of a large scale dynamic system which depends singularly upon a small parameter λ is considered. When λ = 0, the large scale system degenerates into a reduced order subsystem representing its slow dynamics while neglecting the fast phenomena. Another subsystem, often called a boundary layer system, represents the fast dynamics. In this paper sufficient conditions are established under which the controllability of the overall large scale system is inferred from the same property of the two subsystems.     

Invariant surfaces of standard two-dimensional systems with conservative first approximation of the third order

We simultaneously study two classes of two-dimensional time-periodic systems of differential equations with a small positive parameter, namely, systems with “slow” or “fast” time whose first-approximation systems are autonomous and conservative and do not contain terms of order higher than three. Thus, the corresponding unperturbed systems have one, two, or three rest points.For the perturbations, we indicate explicit conditions, independent of the small parameter, under which every original system of either class with coefficients three times continuously differentiable with respect to the phase variables and the parameter in a neighborhood of zero has finitely many two-dimensional invariant surfaces homeomorphic to tori for all sufficiently small parameter values. We also give formulas for these surfaces.     

On the (in)stability of quasi‐static paths of smooth systems: definitions and sufficient conditions

A concept of stability of quasi‐static paths is discussed in this paper that takes into account the existence of fast (dynamic) and slow (quasi‐static) time scales in the evolution of many mechanical systems. The proposed concept is essentially a continuity property with respect to the smallness of the initial perturbations (as in Lyapunov stability) and the smallness of the quasi‐static loading rate (that plays the role of the small parameter in singular perturbation problems). A related concept of attractiveness is also proposed. Several examples illustrate the relevance of the definitions. Sufficient conditions for attractiveness or for instability of quasi‐static paths of smooth systems are proved. Copyright © 2005 John Wiley & Sons, Ltd.     

Regular and singular periodic perturbations of an oscillator with cubic restoring force

The paper considers small periodic regular and singular perturbations of a system, whose conservative part is an oscillator with cubic restoring force. The smallness of perturbations is due to both the smallness of the neighborhood of equilibrium and the presence of a small parameter. In the absence of a small parameter, we obtain conditions for Lyapunov stability of the equilibrium position. If a small parameter is present, we derive (both for regular and singular perturbations) an equation whose positive roots are in correspondence with invariant two-dimensional tori of the perturbed system.     

On output feedback control of singularly perturbed systems

We treat the problem of robustness of output feedback controllers with respect to singular perturbations. Given a singularly perturbed control system whose boundary layer system is exponentially stable and whose reduced order system is exponentially stabilizable via a (possibly dynamical) output feedback controller, we present a sufficient condition which ensures that the system obtained by applying the same controller to the original full order singularly perturbed control system is exponentially stable for sufficiently small values of the perturbation parameter. This condition, which is less restrictive than those previously given in the literature, is shown to be always satisfied when the singular perturbation is due to the presence of fast actuators and/or sensors. Furthermore, we show explicitly that, in the linear time-invariant case, if this condition is not satisfied then there exists an output feedback controller which stabilizes the reduced order system but destabilizes the full order system.     

Controllability of nonlinear systems with restrained controls

In this paper, we examine the controllability of nonlinear differential systems which are perturbations of systems of the formx=A(t)x+g(t, u). Under appropriate assumptions, we show that integrably bounded perturbations preserve the controllability properties of the base system in the case when the control values are restrained. Our approach is based on the necessary and sufficient growth condition for controllability developed earlier by the author.     

Difference singular perturbations—I. A priori estimates

Difference approximations for differential singular perturbations with small parameter ? are considered. We point out ellipticity and coerciveness conditions which are necessary and sufficient for a two-sided a priori estimate to hold for the solutions of difference singular perturbation uniformly with respect to the ratio of both small parameters: the original one ? and the meshsize h.     

Random perturbations of dynamical systems and diffusion processes with conservation laws

In this paper we consider random perturbations of dynamical systems and diffusion processes with a first integral. We calculate, under some assumptions, the limiting behavior of the slow component of the perturbed system in an appropriate time scale for a general class of perturbations. The phase space of the slow motion is a graph defined by the first integral. This is a natural generalization of the results concerning random perturbations of Hamiltonian systems. Considering diffusion processes as the unperturbed system allows to study the multidimensional case and leads to a new effect: the limiting slow motion can spend non-zero time at some points of the graph. In particular, such delay at the vertices leads to more general gluing conditions. Our approach allows one to obtain new results on singular perturbations of PDEs. Mathematics Subject Classification (2001): 60H10; 34C29; 35B20     

Nonstandard singular perturbation systems and higher index differential-algebraic systems

In order to obtain trajectory approximation results for a given singular perturbation system (SPS), two systems are derived from it: the slow and the fast one. Tikhonov's theorem gives sufficient conditions on them to ensure a good approximation for a standard SPS, i.e., its corresponding slow system is a differential-algebraic system (DAS) of index 1. In this paper it is shown that a nonstandard SPS with the parameter set to zero can be seen as a DAS of higher index. This connection allows us to obtain a Tikhonov's theorem when this DAS is of index 2.     

Robustness of the controllability for the heat equation under the influence of multiple impulses and delays

For many control systems in real life, impulses and delays are intrinsic properties that do not modify their behavior. Thus, we conjecture that under certain conditions the abrupt changes and delays as perturbations of a system, that could model a real situation, do not modify properties such as controllability. In this regard, we prove the approximate controllability of the semilinear heat equation under the influence of multiple impulses and delays, this is done by using new techniques, avoiding fixed point theorems, employed by A.E. Bashirov et al.     

Oscillatory and transient dynamics of a slow–fast predator–prey system with fear and its carry-over effect

In almost every ecological system, growth of various interacting species evolve in different time scales and the implementation of this time scale difference in the corresponding mathematical model exhibits some rich and complex oscillatory dynamics. In this article, we consider a predator–prey model with Beddington–DeAngelis functional response in which the prey reproduction is affected by the predation induced fear and its carry-over effect. Considering the growth of prey species occurs on a faster time scale than that of predator, the proposed system reduces to a ‘slow–fast predator–prey’ system. Using the geometric singular perturbation theory and asymptotic expansion technique, we investigate the system both analytically and numerically, and observe a wide range of rich and complex dynamics such as canard cycles (with or without head) near the singular Hopf-bifurcation threshold and relaxation oscillation cycles. The system experiences a canard explosion through which a rapid transition from small amplitude limit cycle to large amplitude limit cycle occurs in a tiny parametric interval. These types of complex oscillatory dynamics are absent in non slow–fast systems. Furthermore, it is shown that the interplay between fear and its carry-over effect, and the variation of time scale parameter may lead to a regime shift of the oscillatory dynamics. We also study the impact of fear and its carry-over effect on the properties of long transient dynamics. Thus our study provides some valuable biological insights of a slow–fast predator–prey system which will aid in understanding the interplay between fear and its carry-over effect.     

Asymptotic Analysis and Solution of a Finite-Horizon H ∞ Control Problem for Singularly-Perturbed Linear Systems with Small State Delay

A finite-horizon H state-feedback control problem for singularly-perturbed linear time-dependent systems with a small state delay is considered. Two approaches to the asymptotic analysis and solution of this problem are proposed. In the first approach, an asymptotic solution of the singularly-perturbed system of functional-differential equations of Riccati type, associated with the original H problem by the sufficient conditions of the existence of its solution, is constructed. Based on this asymptotic solution, conditions for the existence of a solution of the original H problem, independent of the small parameter of singular perturbations, are derived. A simplified controller with parameter-independent gain matrices, solving the original H problem for all sufficiently small values of this parameter, is obtained. In the second approach, the original H problem is decomposed into two lower-dimensional parameter-independent H subproblems, the reduced-order (slow) and the boundary-layer (fast) subproblems; controllers solving these subproblems are constructed. Based on these controllers, a composite controller is derived, which solves the original H problem for all sufficiently small values of the singular perturbation parameter. An illustrative example is presented.     

广义分布参数系统的状态反馈稳定性问题(1)

关于系统的状态反馈稳定性问题的研究一直是现代控制理论研究的重要问题之一.广义分布参数系统是比分布参数系统更广的一类系统,在研究复合材料热导体中的温度分布等问题时会出现这样的系统.本文讨论了H ilbert空间中一阶广义分布参数系统的状态反馈稳定性问题.应用泛函分析及线性算子半群理论的方法给出了使闭环广义分布参数系统渐进稳定的充要条件,充分条件及状态反馈的构造性表达式.这对研究广义分布参数系统的状态反馈稳定性问题具有重要的理论价值.     

Elliptic difference singular perturbations

Summary Difference approximations for differential singular perturbations with small parameter ɛ are considered. We point out ellipticity and coerciveness conditions which arenecessary andsufficient for a two-sided a priori estimate to hold for the solution of difference singular perturbation uniformly with respect to the ratio of both small parameters: the original one ɛ and the meshsize h. Entrata in Redazione il 27 maggio 1978.     

分数阶时变广义系统稳定性分析

基于Caputo分数阶导数,研究了分数阶时变广义线性系统和分数阶时变广义非线性系统的稳定性问题.首先利用相关不等式,给出了一个时变广义线性系统无脉冲且稳定的充分条件.然后,通过慢子系统来判断快子系统的变化,并利用Riccati方程,建立了分数阶时变广义非线性系统是渐近稳定的判定准则.最后,给出了算例和Simulink仿真结果,以说明结论的正确性.     

Singular Hopf Bifurcation in Systems with Fast and Slow Variables

Summary. We study a general nonlinear ODE system with fast and slow variables, i.e., some of the derivatives are multiplied by a small parameter. The system depends on an additional bifurcation parameter. We derive a normal form for this system, valid close to equilibria where certain conditions on the derivatives hold. The most important condition concerns the presence of eigenvalues with singular imaginary parts, by which we mean that their imaginary part grows without bound as the small parameter tends to zero. We give a simple criterion to test for the possible presence of equilibria satisfying this condition. Using a center manifold reduction, we show the existence of Hopf bifurcation points, originating from the interaction of fast and slow variables, and we determine their nature. We apply the theory, developed here, to two examples: an extended Bonhoeffer—van der Pol system and a predator-prey model. Our theory is in good agreement with the numerical continuation experiments we carried out for the examples. Received October 24, 1996; revised October 31, 1997; accepted November 3, 1997     

Coercive singular perturbations: Eigenvalue problems and bifurcation phenomena

Summary The method based upon a constructive reduction of coercive singular perturbations to regular ones, introduced in 1977 (see [4]) and developed later on (see [9–11]) is applied for computing the asymptotic expansions for eigenvalues of coercive singular perturbations, when the small parameter goes to zero. The same method turns out to be useful for investigating the asymptotic behaviour of solutions to quasi-linear coercive singular perturbations in the neighbourhood of the bifurcation points. It can be applied to classes of quasi-linear singular perturbations whose principal linear part in local representation is coercive and the nonlinear part is analytic in some ball in the solution space with values in the data space. The results are summarized in [7, 8].     

Distance to Ill-Posedness in Linear Optimization via the Fenchel-Legendre Conjugate

We consider the parameter space of all the linear inequality systems, in the n-dimensional Euclidean space and with a fixed index set, endowed with the topology of the uniform convergence of the coefficient vectors. A system is ill-posed with respect to the consistency when arbitrarily small perturbations yield both consistent and inconsistent systems. In this paper, we establish a formula for measuring the distance from the nominal system to the set of ill-posed systems. To this aim, we use the Fenchel-Legendre conjugation theory and prove a refinement of the formula in Ref. 1 for the distance from any point to the boundary of a convex set.This research has been partially supported by grants BFM2002–04114-C02 (01–02) from MEC (Spain) and FEDER (EU) and by grants GV04B-648 and GRUPOS04/79 from Generalitat Valenciana (Spain).     

Infinite horizon quadratic control of linear singularly perturbed systems with small state delays: an asymptotic solution of Riccati-type equations

Email: valery{at}techunix.technion.ac.il Received on January 31, 2006; Accepted on October 5, 2006 An infinite horizon linear-quadratic optimal control problemfor a singularly perturbed system with multiple point-wise anddistributed small delays in the state variable is considered.The set of Riccati-type equations, associated with this problemby the control optimality conditions, is studied. Since thesystem in the control problem is singularly perturbed, the equationsof this set are also perturbed by a small parameter of the singularperturbations. The zero-order asymptotic solution to this setof equations is constructed and justified. Based on this asymptoticsolution, parameter-free sufficient conditions for the existenceand uniqueness of solution to the original optimal control problemare established.     

Approximate boundary controllability of Sobolev-type stochastic differential systems

The objective of this paper is to investigate the approximate boundary controllability of Sobolev-type stochastic differential systems in Hilbert spaces. The control function for this system is suitably constructed by using the infinite dimensional controllability operator. Sufficient conditions for approximate boundary controllability of the proposed problem in Hilbert space is established by using contraction mapping principle and stochastic analysis techniques. The obtained results are extended to stochastic differential systems with Poisson jumps. Finally, an example is provided which illustrates the main results.