On the class of proper linear differential systems with unbounded coefficients

Proper linear differential systems (whose coefficients are not necessarily bounded on the half-line) are defined as systems for which there exists a generalized Lyapunov transformation reducing them to a diagonal system with constant coefficients (Basov). We prove that Lyapunov’s original definition of a proper system and the Perron and Vinograd criteria hold for the class of proper systems as well as for the class of proper systems with uniformly bounded coefficients. We show that the Lyapunov properness criterion for a triangular system fails for systems with unbounded coefficients; namely, we construct an improper system with the following properties: the Lyapunov exponents of all nonzero solutions of that system are finite and exact, and for an arbitrary reduction of this system by a generalized Lyapunov transformation to triangular form, its diagonal coefficients have finite exact mean values, whose set with regard of multiplicities is independent of the choice of the transformation. In addition, we show that the main property of proper systems with uniformly bounded coefficients (preservation of conditional exponential stability as well as the dimension of the exponentially stable manifold and the exponent of the asymptotic behavior of solutions under perturbations of higher-order smallness) holds for proper systems with unbounded coefficients as well.     

On the boundedness of outer polyhedral estimates for reachable sets of linear systems

New properties of outer polyhedral (parallelepipedal) estimates for reachable sets of linear differential systems are studied. For systems with a stable matrix, it is determined what the orientation matrices are for which the estimates possessing the generalized semigroup property are bounded/unbounded on an infinite time interval. In particular, criteria are found (formulated in terms of the eigenvalues of the system’s matrix and the properties of bounding sets) that guarantee for previously mentioned tangent estimates and estimates with a constant orientation matrix that either there are initial orientation matrices for which the corresponding estimate tubes are bounded or all these tubes are unbounded. For linear stationary systems, a system of ordinary differential equations and algebraic relations is derived that determines estimates with constant orientation matrices for reachable sets that have no generalized semigroup property but are tangent and also bounded if the matrix of the system is stable.     

A computationally efficient robust tube based MPC for linear switched systems

This article considers the robust regulation problem for a class of constrained linear switched systems with bounded additive disturbances. The proposed solution extends the existing robust tube based model predictive control (RTBMPC) strategy for non-switched linear systems to switched systems. RTBMPC utilizes nominal model predictions, together with tightened sets constraints, to obtain a control policy that guarantees robust stabilization of the dynamic systems in presence of bounded uncertainties. In this work, similar to RTBMPC for non-switched systems, a disturbance rejection proportional controller is used to ensure that the closed loop trajectories of the switched linear system are bounded in a tube centered on the nominal system trajectories. To account for the uncertainty related to all sub-systems, the gain of this controller is chosen to simultaneously stabilize all switching dynamics. The switched system RTBMPC requires an on-line solution of a Mixed Integer Program (MIP), which is computationally expensive. To reduce the complexity of the MIP, a sub-optimal design with respect to the previous formulation is also proposed that uses the notion of a pre-terminal set in addition to the usual terminal set to ensure stability. The RTBMPC design with the pre-terminal set aids in determining the trade-off between the complexity of the control algorithm with the performance of the closed-loop system while ensuring robust stability. Simulation examples, including a Three-tank benchmark case study, are presented to illustrate features of the proposed MPC.     

Stochastic Stability of Some Mechanical Systems

We discuss the behavior, for large values of time, of two linear stochastic mechanical systems. The systems are similar mathematically in that they contain a white noise in their parameters. The initial data may be random as well but are independent of white noise. The expected energy is calculated in both cases. It is well known that for free nonstochastic mechanical systems with viscous damping, the energy approaches zero as time increases. We check that this behavior takes place for the stochastic systems under consideration in the case when the initial data are random but the parameters are not. When the parameters contain a random noise the expected energy may be infinite, approach zero, remain bounded, or increase with no bound. This regime is similar to but more interesting than the known regime for the solutions of differential equations with time dependent periodic coefficients that describes the behavior of a mechanical system with characteristics that are periodic functions of time. We give necessary and sufficient conditions for stability of both systems in terms of the structure of the set of roots of an auxiliary equation.     

Invariant convex bodies for strongly elliptic systems

We consider uniformly strongly elliptic systems of the second order with bounded coefficients. First, sufficient conditions for the invariance of convex bodies are obtained for linear systems without zero order term on bounded domains and quasilinear systems of special form on bounded domains and on a class of unbounded domains. These conditions are formulated in algebraic form. They describe relation between the geometry of the invariant convex body and the coefficients of the system. Next, necessary conditions, which are also sufficient, for the invariance of some convex bodies are found for elliptic homogeneous systems with constant coefficients in a half-space. The necessary conditions are derived by using a criterion on the invariance of convex bodies for normalized matrix-valued integral transforms also obtained in the paper. In contrast with the previous studies of invariant sets for elliptic systems, no a priori restrictions on the coefficient matrices are imposed.     

On Russell's method of controllability via stabilizability

Russell has observed that a linear system is controllable provided it is stabilizable in both positive and negative time. We give a version of this result valid for nonlinear systems, and illustrate its use by giving new proofs of two classical results from control theory, the first involving bounded perturbations of controllable linear systems, and the second involving controllability of linear systems by bounded controls.This research was supported by the Natural Sciences and Engineering Research Council of Canada.     

Distribution semigroups and control systems

We introduce the new concept of a distributional control system. This class of systems is the natural generalization of distribution semigroups to input/state/output systems. We showthat, under the Laplace transform, this new class of systems is equivalent to the class of distributional resolvent linear systems which we introduced in an earlier article. There we showed that this latter class of systems is the correct abstract setting in which to study many non-well-posed control systems such as the heat equation with Dirichlet control and Neumann observation. In this article we further show that any holomorphic function defined and polynomially bounded on some right half-plane can be realized as the transfer function of some exponentially bounded distributional resolvent linear system.     

Estimates of solutions of some systems of differential equations with large retardation

The behavior of the solutions of differential equations with asymptotically large retardation is studied. In the case of linear systems with bounded coefficients and a class of nonlinear differential equations, estimates are derived which depend explicitly on the retardation.Translated from Matematicheskie Zametki, Vol. 6, No. 4, pp. 483–488, October, 1969.     

Discrete Dichotomies and Asymptotic Behavior for Abstract Retarded Functional Difference Equations in Phase Space

We study the problem of asymptotic behavior between weighted bounded solutions of a system of homogeneous linear functional difference equations and its perturbation under non-classical dichotomic properties and also we obtain some results about approximation. We apply our results to Volterra difference systems with infinite delay.     

NORMAL切换系统在干扰下的性质

考虑了具有有界干扰NORMAL切换系统的稳定性分析,运用Lyapunov函数给出了干扰的界限.主要解决了切换系统在有界干扰下的稳定性分析,给出了当系统干扰满足一定的条件时,切换系统在任意的切换条件下轨迹渐近稳定.同时讨论了带干扰的线性切换系统的镇定问题.问题的讨论中对系统的要求少,因此更具一般性.     

Eigenstructure assignment for linear parameter-varying systems with applications

The aim of this paper is to show that a recently proposed technique for eigenstructure assignment of linear time-invariant systems can be extended to solve the corresponding eigenstructure assignment problem for linear parameter-varying systems, whose state-space matrices depend on a set of time-varying parameters that are bounded and available online. In particular, the design of eigenstructure assignment is performed without requiring any conditions on the closed-loop eigenvalues, and provides a simple, complete and analytical parametric approach as well as the most degrees of design freedom for the eigenstructure assignment problem of linear parameter-varying systems. A parameter-varying attitude control system of refueling spacecraft in-orbit is used to demonstrate the usefulness and practicality of the proposed approach.     

Pathological asymptotic behavior of control systems in Banach space

New phenomena arising when a linear dynamical system is defined on an infinite dimensional Banach space, although negligible from an engineering standpoint when only a finite time-interval is considered, become crucial when the asymptotic (feedback) behavior of the system is of interest. Pathologies with respect to the correspondent finite dimensional case are displayed even when the operator acting on the state is bounded.In particular, although in such case, the classical controllability and observability theory admits a natural generalization to infinite dimensions, the finite dimensional relationships between controllability and stabilizability fails. A few examples are given of systems that are approximately controllable and yet are not stabilizable: Moreover, such examples are drawn from a class of systems that can never be exactly controllable. The analysis is carried out using the perturbation theory of the spectrum. Another new feature of the infinite dimensionality of the state space is that even if the spectrum of an operator has the max of its real part equal to 0, yet the associated homogeneous differential equation may be globally asymptotically stable: Its consequence on stabilizability is also examined.     

On linear almost periodic pulse systems

For a linear almost periodic system under pulse influence, the conditions are established under which this system is reducible (by a linear change of variables with a discontinuous almost periodic matrix) to a system without pulses but with a Bohr almost periodic right-hand side. The set of linear almost periodic pulse systems possessing only bounded solutions is studied.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 1, pp. 105–113, January, 1993.     

A Synthesis of the Control of Dynamical Systems on the Basis of the Method of Lyapunov Functions

We consider the problem of the synthesis of a bounded control reducing a dynamical system to the given terminal state in a finite time. Two approaches to solve the problem, based on methods of the theory of stability of motion, are provided. One of them is applicable to nonlinear Lagrange mechanical systems with undetermined parameters, while another is applicable to linear systems. The characteristic property is that the Lyapunov functions are defined implicitly in both cases. We make a comparison between these approaches.     

Controlling the Lorenz system: combining global and local schemes

The Lorenz equations are one of the best-known and analyzed systems exhibiting chaotic behavior. In this paper, a new control scheme for the Lorenz system combining local and global techniques is introduced. This scheme is based on a feedback law which is only applied in a bounded state space region of control (SSRC). The SSRC is determined by the enclosure of the Lorenz attractor.     

具有线性扰动的线性脉冲微分系统的有界增长

本文介绍了脉冲微分系统的有界增长的概念，并进一步讨论了具有线性扰动的线性脉冲微分系统的有界增长问题，得到了反映无扰和扰动两种情形下脉冲微分系统有界增长关系的定理。     

Invariance of deficiency indices under perturbation for discrete Hamiltonian systems

This paper deals with discrete Hamiltonian systems with one singular endpoint. Using Hermitian linear relation generalized by linear Hamiltonian system, the invariance of the minimal and maximal deficiency indices under bounded perturbation for discrete Hamiltonian systems is built. This parallels the well-known results for linear Hamiltonian differential systems obtained by F.V. Atkinson.     

An Improved Delay-Dependent Criterion for Asymptotic Stability of Uncertain Dynamic Systems with Time-Varying Delays

In this paper, the problem of stability analysis for uncertain dynamic systems with time-varying delays is considered. The parametric uncertainties are assumed to be bounded in magnitude. Based on the Lyapunov stability theory, a new delay-dependent stability criterion for the system is established in terms of linear matrix inequalities, which can be solved easily by various efficient convex optimization algorithms. Two numerical examples are illustrated to show the effectiveness of proposed method.     

Randomized system trajectories

The notion of system trajectory of a time-varying input-output, dynamical system is reviewed. By introducing a probability measure on a class of such systems a stochastic system, the randomized system, is defined. The randomized system has a trajectory induced by the trajectories of the original systems. A theorem is proved giving fairly general conditions under which the randomized system trajectory is generated by a strongly continuous semigroup of bounded linear operators in a Banach space. An example is presented for a system represented by a quadratic integral operator.Research supported in part by National Science Foundation under Grant No. ECS-8005960.     

Stabilization of stationary affine control systems with discrete time

We obtain new sufficient conditions for the local and global asymptotic stabilization of the zero solution of a nonlinear affine control system with discrete time and with constant coefficients by a continuous state feedback. We assume that the zero solution of the free system is Lyapunov stable. For systems with linear drift, we construct a bounded control in the problem of global asymptotic state and output stabilization. Corollaries for bilinear systems are obtained.