Sufficiency theorems for target capture

A dynamical system is assumed to be governed by a set of ordinary differential equations subject to control. The set of points in state space from which there exist permissible controls that can transfer these points to a prescribed target set in a finite time interval is called a capture set. The task of determining the capture set is studied in two contexts. first, in the case of the system subject to a single control vector; and second, in the case of the system subject to two control vectors each operated independently. In the latter case, it is assumed that one controller's aim is to cause the system to attain the target, and the other's is to prevent that from occurring.Sufficient conditions are developed that, when satisfied everywhere on the interior of some subset of the state space, ensure that this subset is truly a capture set. A candidate capture set is assumed to have already been predetermined by independent methods. The sufficient conditions developed herein require the use of an auxiliary scalar function of the state, similar to a Lyapunov function.To ensure capture, five conditions must be satisfied. Four of these constrain the auxiliary state function. Basically, these four conditions require that the boundary of the controllable set be an envelope of the auxiliary state function and that that function be positive inside the capture set, approaching zero value as the target set is approached. The final condition tests the inner product of the gradient of the auxiliary state function with the system state velocity vector. If the sign of that inner product can be made negative everywhere within the test subset, then that subset is a capture set.Dedicated to Professor A. BusemannThe authors are indebted to Professors G. Leitmann and J. M. Skowronskii for their useful comments and discussion.     

Controllability Properties of Linear Mean-Field Stochastic Systems

We study some controllability properties for linear stochastic systems of mean-field type. First, we give necessary and sufficient criteria for exact terminal-controllability. Second, we characterize the approximate and approximate null-controllability via duality techniques. Using Riccati equations associated to linear quadratic problems in the control of mean-field systems, we provide a (conditional) viability criterion for approximate null-controllability. In the classical diffusion framework, approximate and approximate null-controllability are equivalent. This is no longer the case for mean-field systems. We provide sufficient (algebraic) invariance conditions implying approximate null-controllability. We also present a general class of systems for which our criterion is equivalent to approximate null-controllability property. We also introduce some rank conditions under which approximate and approximate null-controllability are equivalent. Several examples and counter-examples as well as a partial algorithm are provided.     

Playability with and without capture

A two-player qualitative differential game with a given target is considered. The first player is assumed to have two objectives: to drive the system to the target in finite time (playability) and then to keep it there indefinitely (playability with capture). Both are to be accomplished against all possible control actions of the second player. Playability without capture implies that the second player cannot avoid the target, but can prevent capture by choosing controls to drive the system out of the target and perhaps keep it out indefinitely (playability with escape) against all possible control actions of the first player. Sufficient conditions for playability with and without capture and playability with escape are derived. Means of determining states from which each case may be attained are examined. Candidates for capturing and escape subtargets, to be used in conjunction with the sufficient conditions developed here, may be obtained from the conditions shown to be necessary on boundaries of the subtargets. An example closes the paper.This research was supported by a University of Queensland Travel Grant.     

Uniform invariance principle of discontinuous systems with parameter variation

An extension of the invariance principle for a class of discontinuous righthand sides systems with parameter variation in the Filippov sense is proposed. This extension allows the derivative of an auxiliary function V, also called a Lyapunov-like function, along the solutions of the discontinuous system to be positive on some sets. The uniform estimates of attractors and basin of attractions with respect to parameters are also obtained. To this end, we use locally Lipschitz continuous and regular Lyapunov functions, as well as Filippov theory. The obtained results settled in the general context of differential inclusions, and through a uniform version of the LaSalle invariance principle. An illustrative example shows the potential of the theoretical results in providing information on the asymptotic behavior of discontinuous systems.     

Dynamical behaviors of the chaotic Brushless DC motors model

This article is concerned with the boundedness solutions of the brushless DC motor system. For this system, the global attractive set and positively invariant set are obtained based on generalized Lyapunov function stability theory and the extremum principle of function. Furthermore, the rate of the trajectories is also obtained. Numerical simulations are presented to show the effectiveness of the proposed scheme. © 2014 Wiley Periodicals, Inc. Complexity 21: 79–85, 2016     

随机泛函微分方程正整体解的存在唯一性及矩有界性

构造了Lyapunov函数V (x),然后给出一个一般条件,应用Khasminskii?Mao定理,得到非线性随机泛函微分方程(SFDEs)存在正整体解,且这个解p阶矩有界.     

Stability analysis and uniform ultimate boundedness control synthesis for a class of nonlinear switched difference systems

In this paper, we deal with stability analysis of a class of nonlinear switched discrete-time systems. Systems of the class appear in numerical simulation of continuous-time switched systems. Some linear matrix inequality type stability conditions, based on the common Lyapunov function approach, are obtained. It is shown that under these conditions the system remains stable for any switching law. The obtained results are applied to the analysis of dynamics of a discrete-time switched population model. Finally, a continuous state feedback control is proposed that guarantees the uniform ultimate boundedness of switched systems with uncertain nonlinearity and parameters.     

Controllability of series connections of arbitrarily many linear systems

In this paper we study the controllability of series connections of arbitrarily many linear systems. As the main result, we completely determine the controllability and the possible controllability indices of a system obtained by a special series connection of arbitrarily many linear systems.     

Controllability of the wave equation with moving point control

This paper deals with approximate and exact controllability of the wave equation in finite time with interior point control acting along a curve specified in advance in the system's spatial domain. The structure of the control input is dual to the structure of the observations which describe the measurements of velocity and gradient of the solution of the dual system, obtained from the moving point sensor. A relevant formalization of such a control problem is discussed, based on transposition. For any given timeinterval [0,T] the existence of the curves providing approximate controllability inH _D ^–[n/2]–1 ()×H _D ^–[n/2]–1 () (wheren stands for the space dimension) is established with controls fromL ²(0,T; R ⁿ +1). The same curves ensure exact controllability inL ²() × H^–1() if controls are allowed to be selected in [L (0,T; R ⁿ+1)]. Required curves can be constructed to be continuous on [0,T).This work was supported in part by NSF Grant ECS 89-13773 and NASA Grant NAG-1-1081.     

关于有界性的勒茹米辛型定理

本文对于泛函微分方程建立了关于有界性的新的勒茹米辛型定理,其中不仅可避免采用不易寻找的函数Ｐ,而且还可以采用若干个含有状态变量ｘ的部分变元的李雅普诺夫函数,这样,本文中所得的结果大大改进了已知结果。一方面,它更易于应用;另一方面,保证有界性的条件较少限制,并且,还给出了一些例子来说明所得结果的优越性。     

非局部半线性积微分系统的可控制性

本文运用半群理论和Schauder不动点定理,在Banach空间研究了一类非局部半线性积微分系统的可控制性.最后,举例说明了所得结果.     

Controllability of Functional Integro-Differential Inclusions with an Unbounded Delay

In this paper, a sufficient condition is established for the controllability of neutral functional integro-differential inclusions with an unbounded delay in Banach spaces. The approach used is a fixed-point theorem for condensing maps due to Martelli and the theory of analytic semigroup of linear operators. Communicated by F. Zirilli Research supported by NNSF of China, by the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of the Ministry of Education of China, and by the Qing Lan Talent Engineering Fund QL-05-164 of Lanzhou Jiaotong University. The authors are grateful to Professor F. Zirilli and two anonymous referees for valuable suggestions improving this paper.     

Controllability of Semilinear Evolution Equations with Nonlocal Conditions

In this paper we examine the controllability problems of certain evolution equations with nonlocal conditions. Using the Schaefer fixed-point theorem, we obtain sufficient conditions for controllability and we give an application.     

On guaranteed stability of uncertain linear systems via linear control

This paper addresses the problem of stabilizing an uncertain linear system. The uncertaintyq(·) which enters the dynamics is nonstistical in nature. That is, noa priori statistics forq(·) are assumed; only boundsQ on the admissible variations ofq(·) are taken as given. The results given here applied to so-called matched systems differ from previous results in two ways. Firstly, the stabilizing control in this paper is linear; for this same class of problems, many of the existing results would require a nonlinear control. Furthermore, those results which do in fact yield linear controls are only valid when a certain matrix (q) (formed using the given data) is negative definite for allq Q. In contrast, the theory given here only requires compactness of the bounding setQ. Secondly, we show that the so-called matching conditions (used in earlier work) can be generalized so as to encompass a larger class of dynamical systems.This research was supported by the US Department of Energy under Contract No. ET-78-S-01-3390.     

Controllability of nonlinear delay systems

A necessary and sufficient condition for complete controllability of differential systems with delay is proved, which extends analogous results for usual differential systems.     

Controllability of semilinear systems

The most general results known about controllability of semilinear systems are found in Refs. 1 and 2. The authors of these papers used different approaches, and no connection between them has ever been pointed out. The purpose of this paper is to present a general theorem under which these results easily follow.This paper is taken from a chapter of the author's doctoral thesis written at University of California, Los Angeles, California.     

Existence and Controllability Results for First-and Second-Order Functional Semilinear Differential Inclusions with Nonlocal Conditions

In this paper, we prove existence and controllability results for first- and second-order semilinear differential inclusions in Banach spaces with nonlocal conditions.     

On Stability and Boundedness for Lipschitzian Differential Inclusions: The Converse of Lyapunov's Theorems

For Lipschitzian differential inclusions, we prove that the existence of suitable Lyapunov functions is necessary for uniform stability and uniform boundedness of solutions.