On the boundedness and unboundedness of external polyhedral estimates for reachable sets of linear differential systems

The boundedness and unboundedness properties of external polyhedral (paralle-lepiped-valued) estimates are investigated for reachable sets of linear differential systems with a stable matrix. Boundedness and unboundedness criteria on an infinite time interval are presented for two types of estimates (“touching” estimates, which were introduced earlier, and estimates with constant orientation matrix). Conditions for the system matrix and bounding sets are given under which there are bounded estimates among the estimates of the mentioned types, under which there are unbounded estimates, and under which all the estimates are bounded or all the estimates are unbounded. In terms of the exponents of the estimates, the possible rate of their growth is described. For two-dimensional systems, the classification and comparison of possible situations of the boundedness or unboundedness for estimates of both types are given and boundedness criteria for estimates with special (orthogonal and “quasi-orthogonal”) constant orientation matrices are found. Results of numerical modeling are presented.     

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.     

Estimating reachable sets for two-dimensional linear discrete systems

We are concerned with the problem of estimating the reachable set for a two-dimensional linear discrete-time system with bounded controls. Different approaches are adopted depending on whether the system matrix has real or complex eigenvalues. For the complex eigenvalue case, the quasiperiodic nature of minimum time trajectories is exploited in developing a simple, but often accurate, procedure. For the real eigenvalue case, over estimates of reachable sets can be trivially obtained using a decomposition method.The second author was supported by funds supplied by the John M. Bennett Faculty Fellowship, Trinity University, San Antonio, Texas.     

Differential equations for ellipsoidal estimates for reachable sets of a nonlinear dynamical control system

The problem of estimating trajectory tubes of a nonlinear control dynamical system with uncertainty in initial data is considered. It is assumed that the dynamical system has a special structure, in which nonlinear terms are quadratic in phase coordinates and the values of the uncertain initial states and admissible controls are subject to ellipsoidal constraints. Differential equations are found that describe the dynamics of the ellipsoidal estimates of reachable sets of the nonlinear dynamical system under consideration. To estimate reachable sets of the nonlinear differential inclusion corresponding to the control system, we use results from the theory of ellipsoidal estimation and the theory of evolution equations for set-valued states of dynamical systems under uncertainty.     

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 THE REACHABLE SEMIGROUP OF BILINEAR CONTROL SYSTEMS ON LIE GROUP

This paper studies the reachability and the structure of reachable semigroup of bilinear control systems on Lie group. In the second section some equivalency lemmas are given, which not only simplify the proofs of the main results, but discover some properties of systems also. In the third section some conditions are advanced that the reachable semigroup of system is weakly symmetric by means of the study of one prameter subgroups. This study is discussed by manifold theory and matrix theory, respectively. In the last seetion, some topological properties of the reachable semigroup are advanced.     

具有边界控制的线性Timoshenko型系统的指数稳定性

研究多孔弹性材料在实际应用中的稳定性问题.多孔物体的动力学行为由线性Timoshenko型方程描述,这样的系统一般只是渐近稳定但不指数稳定,假定系统在一端简单支撑,另一端自由,在自由端对系统施加边界反馈控制,讨论闭环系统的适定性和指数稳定性.首先,证明了由闭环系统决定的算子A是预解紧的耗散算子、生成C0压缩半群,从而得到了系统的适定性.进一步通过对系统算子A的本征值的渐近值估计,得到算子谱分布在一个带域,相互分离的,模充分大的本征值都是A的简单本征值.通过引入一个辅助算子A0,利用算子A0的谱性质以及算子A与A0之间的关系,得到了A的广义本征向量的完整性以及Riesz基性质.最后利用Riesz基性质和谱分布得到闭环系统的指数稳定性.     

Approximating Reachable Sets for n-dimensional Linear Discrete Systems

We present a technique for approximating the reachable set fromthe origin for n-dimensional discrete-time linear systems subjectto bounded control, where the state matrix is stable and diagonalizable.The procedure is based on decomposing the system into one- andtwo-dimensional subsystems for which the reachable set of eachof the subsystems can be over-estimated by a polygon. Thesepolygons are then used to create an n-dimensional polyhedronwhich contains the reachable set of the original system     

Ellipsoidal Bounds on Reachable Sets of Dynamical Systems with Matrices Subjected to Uncertain Perturbations1

Linear dynamical systems described by finite-difference or ordinary differential equations are considered. The matrix of the system is uncertain or subject to disturbances, and only the bounds on admissible perturbations of the matrix are known. Outer ellipsoidal estimates of reachable sets of the system are obtained and equations describing the evolution of the approximating ellipsoids are derived. An example is presented.     

广义算子半群与广义分布参数系统的适定性

首先,针对广义分布参数系统的求解问题,提出了由Hilbert空间中有界线性算子所引导的广义算子半群和广义积分半群;其次,讨论了广义预解算子的性质、广义算子半群与广义积分半群的性质;最后,研究了广义分布参数系统的适定性问题.     

Reachable Set Bounding for Linear Discrete-Time Systems with Delays and Bounded Disturbances

This paper addresses the problem of reachable set bounding for linear discrete-time systems that are subject to state delay and bounded disturbances. Based on the Lyapunov method, a sufficient condition for the existence of ellipsoid-based bounds of reachable sets of a linear uncertain discrete system is derived in terms of matrix inequalities. Here, a new idea is to minimize the projection distances of the ellipsoids on each axis with different exponential convergence rates, instead of minimization of their radius with a single exponential rate. A smaller bound can thus be obtained from the intersection of these ellipsoids. A numerical example is given to illustrate the effectiveness of the proposed approach.     

Estimates of reachable sets of multidimensional control systems with nonlinear interconnections

The paper is devoted to the problem of constructing external estimates for the reachable set of a multidimensional control system by means of vector estimators. A system is considered that permits a decomposition into several independent subsystems with simple structure (for example, linear subsystems), which are connected to each other by means of nonlinear interconnections. For each of the subsystems, an external estimate of the reachable set is assumed to be known; this estimate is representable in the form of a level set of some function satisfying a differential inequality. An estimate for the reachable set of the combined system is constructed with the use of estimates for subsystems. The method of deriving the estimates is based on constructing comparison systems for analogs of vector Lyapunov functions (value functions).     

APPROXIMATE CONTROLLABILITY RESULTS FOR INTEGRO-QUASILINEAR EVOLUTION EQUATIONS VIA TRAJECTORY REACHABLE SETS

In this article, we study the approximate controllability results for an integroquasilinear evolution equation with random impulsive moments under sufficient conditions.The results are obtained by the theory of C_0 semigroup of bounded linear operators on evolution equations and using trajectory reachable sets. Finally, we generalize the results too with and without fixed type impulsive moments.     

Hyperbolic spline interpolation algorithms

Isogeometric interpolation by hyperbolic splines is formulated as a differential multipoint boundary value problem. A discretization of this problem results in the necessity of solving a linear system with a five-diagonal matrix. This system can be ill-conditioned if the data are nonuniformly distributed. It is shown that this system can be split into tridiagonal systems with the property of diagonal dominance. The latter do not require that hyperbolic functions be evaluated. Their solution is numerically stable and can be efficiently parallelized on the basis of the superposition principle. For quasiuniform grids, these systems have positive definite matrices. Algorithms for parallelizing calculations in the case of tri- and five-diagonal systems are given.     

Gaussian upper bounds for heat kernels of second order complex elliptic operators with unbounded diffusion coefficients on arbitrary domains

In this paper we obtain Gaussian upper bounds for the integral kernel of the semigroup associated with second order elliptic differential operators with complex unbounded measurable coefficients defined in a domain Ω of ? ^N and subject to various boundary conditions. In contrast to the previous literature the diffusions coefficients are not required to be bounded or regular. A new approach based on Davies-Gaffney estimates is used. It is applied to a number of examples, including degenerate elliptic operators arising in Financial Mathematics and generalized Ornstein-Uhlenbeck operators with potentials.     

On external estimates for reachable sets of nonlinear control systems

The paper is devoted to the problem of constructing external estimates for reachable sets of a nonlinear control system. The estimates are constructed in the form of level sets of smooth functions in the space of states satisfying differential inequalities. In the system under consideration, the linear part is found, for which the corresponding functions are assumed to be known. The method proposed for estimating trajectories of a nonlinear system is based on modifying estimates for the linear part and on applying the comparison principle.     

Some properties of majorants of Lyapunov exponents for systems with unbounded coefficients

A number of well-known results on majorants of Lyapunov exponents of linear differential systems with bounded coefficients on the half-line (such as their simultaneous attainability in the class of infinitesimal perturbations, belonging to the second Baire class in the compact-open topology, etc.) are generalized to systems with unbounded coefficients.     

A geometric approach for reachability and observability of linear switched impulsive systems

This paper is concerned with the reachability and observability of linear switched impulsive systems with singular impulse matrices. First some new concepts with respect to the reachability and unobservability are introduced. Especially, span reachability is proposed because the reachable sets of switched impulsive systems do not always constitute subspaces. Then the geometric characterization of the span reachable and unobservable sets is presented. Moreover, the relations between the span reachable set, unobservable set and the invariant subspaces of such systems are discussed. Finally, corresponding criteria applied to linear impulsive systems and linear switched systems are also discussed.     

Recessively Compact Sets: Properties and Uses

We develop the concept of recessive compactness recently introduced by Luc and Penot. Then we employ this idea to extend some important results of functional analysis such as closed image criteria, a theorem on a family of unbounded sets having a finite intersection property, an existence condition for a variational inequality problem on a noncompact set, a fixed point theorem for nonexpansive maps on unbounded sets, and an existence result for periodic solutions of a nonlinear differential equation in a Hilbert space without a-priori estimates for the solutions of the equation to stay in a bounded region.