Null controllability of systems with control constraints and state saturation

Sufficient conditions for the null controllability of discrete-time systems with control constraints and state saturation are presented. This is accomplished in two stages: First, we establish a result for the asymptotic stability of such systems under zero input conditions. Next, we utilize this stability result to obtain a condition for the null controllability of such systems. We include specific examples to demonstrate the applicability of the present results.     

Null controllability with vanishing energy for discrete-time systems

In this paper null controllability with vanishing energy for discrete-time systems is considered. As in the case of continuous time systems necessary and sufficient conditions in terms of an algebraic Riccati equation are given. Then necessary and sufficient conditions involving the eigenvalues of the state matrix are given. Reachability and controllability with vanishing energy are also considered and necessary and sufficient conditions for them are given. Finally applications to sampled-data systems, systems with impulse control and periodic systems are discussed.     

Burgers方程的精确解

引入一个变换,将二阶非线性偏微分方程—Burgers方程降阶为一阶的非线性方程,再直接求解该方程,得出了Burgers方程精确解的新形式,并与已有结果完全吻合.这种方法也适合于求解其他非线性偏微分方程.     

A lumped Galerkin method for solving the Burgers equation

A numerical solution of the one-dimensional Burgers equation is obtained using a lumped Galerkin method with quadratic B-spline finite elements. The scheme is implemented to solve a set of test problems with known exact solutions. Results are compared with published numerical and exact solutions. The proposed scheme performs well. A linear stability analysis shows the scheme to be unconditionally stable.     

求解 Burgers 方程的决定方程的两种方法研究

论文研究了两种求解偏微分方程的决定方程的方法,一种是运用向量场及其延拓方法,另一种是通过符号计算软件 maple 自动求解软件包。论文以 Burgers 方程为例,证明两种方法得到的结果相同。但运用 maple 自动求解软件包能够避免复杂的代数计算,提高计算的速度与准确率,适用于求解复杂的高阶非线性偏微分方程。     

Uniform boundary controllability of a discrete 1-D wave equation

A numerical scheme for the controlled discrete 1-D wave equation is considered. We prove the convergence of the boundary controls of the discrete equations to a control of the continuous wave equation when the mesh size tends to zero when time and space steps coincide. This positive result is in contrast with previous negative ones for space semi-discretizations.     

Burgers方程的精确孤立波解的符号计算

借助于符号计算Maple,给出了一种构造非线性波动方程行波解的直接代数方法,该方法的主要特点是充分利用Riccati方程.使用此方法得到Burgers方程的多组精确行波解,其中包括一些新的孤立波解,这种方法也适用于求解其它的非线性波动方程（组）.     

On the exact and numerical solution of the time-delayed Burgers equation

The time-delayed Burgers equation is introduced and the improved tanh-function method is used to construct exact multiple soliton and triangular periodic solutions. For an understanding of the nature of the exact solutions that contained the time-delay parameter, we calculated the numerical solutions of this equation by using the Adomian decomposition method to the boundary value problem.     

Null controllability of an infinite dimensional SDE with state- and control-dependent noise

We consider a controlled stochastic linear differential equation with state- and control-dependent noise in a Hilbert space H. We investigate the relation between the null controllability of the equation and the existence of the solution of “singular” Riccati operator equations. Moreover, for a fixed interval of time, the null controllability is characterized in terms of the dual state. Examples of stochastic PDEs are also considered.     

The integration of the output controllability index within the dynamic operability framework in process system design

This paper presents the integration of the output controllability index within the dynamic operability framework to facilitate controllability and economic assessment of process system design for regulatory cases. The framework utilises the geometric representation of feasible operating region as well as input, output and disturbance spaces, in terms of their convex hulls. The geometric operations amongst these convex hulls replace most of the inequalities on the original framework. In addition to the best feasible operating conditions, which are the original outcome, the framework also provides a controllability index involving all of the output variables (generalised integral absolute error, GIAE) and the expected variation of profit on disturbance rejection dynamics. The applicability of the proposed framework is demonstrated on a nonlinear chemical process system with a fixed control structure.     

Sufficient conditions for controllability of a class of distributed parameter systems

In the paper we present certain results on the question of controllability for (i) linear evolution equations (parabolic and hyperbolic) with distributed controls and (ii) a class of linear parabolic equations with boundary controls.     

A notion of possible controllability for uncertain linear systems with structured uncertainty

This paper introduces a notion of possible controllability for a class of uncertain linear systems with structured uncertainty described by averaged integral quadratic constraints. This notion relates to the question of when a state is controllable for some possible value of the uncertainty. The notion of possible controllability is motivated by a desire to extend the theory of minimal realization for linear time invariant systems to the case of uncertain systems with structured uncertainty.     

Null controllability of Kolmogorov-type equations

We study the null controllability of Kolmogorov-type equations $\partial _t f + v^\gamma \partial _x f - \partial _v^2 f = u(t,x,v) 1_{\omega }(x,v)$ in a rectangle $\Omega $ , under an additive control supported in an open subset $\omega $ of $\Omega $ . For $\gamma =1$ , with periodic-type boundary conditions, we prove that null controllability holds in any positive time, with any control support $\omega $ . This improves the previous result by Beauchard and Zuazua (Ann Ins H Poincaré Anal Non Linéaire 26:1793–1815, 2009), in which the control support was a horizontal strip. With Dirichlet boundary conditions and a horizontal strip as control support, we prove that null controllability holds in any positive time if $\gamma =1$ or if $\gamma =2$ and $\omega $ contains the segment $\{v=0\}$ , and only in large time if $\gamma =2$ and $\omega $ does not contain the segment $\{v=0\}$ . Our approach, inspired from Benabdallah et al. (C R Math Acad Sci Paris 344(6):357–362, 2007), Lebeau and Robbiano (Commun Partial Differ Equ 20:335–356, 1995), is based on two key ingredients: the observability of the Fourier components of the solution of the adjoint system, uniformly with respect to the frequency, and the explicit exponential decay rate of these Fourier components.     

Uniform controllability of semidiscrete approximations of parabolic control systems

Controlling an approximation model of a controllable infinite dimensional linear control system does not necessarily yield a good approximation of the control needed for the continuous model. In the present paper, under the main assumptions that the discretized semigroup is uniformly analytic, and that the control operator is mildly unbounded, we prove that the semidiscrete approximation models are uniformly controllable. Moreover, we provide a computationally efficient way to compute the approximation controls. An example of application is implemented for the one- and two-dimensional heat equation with Neumann boundary control.     

Notes on the definition of behavioural controllability

We take another look at the behavioural definition of controllability for discrete one-dimensional (1D) systems, and its extension to multidimensional (nD) systems defined on or . We suggest that the current definition for nD systems is inappropriate for systems defined on the domain , and that care has to be taken even with the 1D definition. We propose a new definition which applies to all standard classes of discrete nD systems.     

On the controllability of piecewise-linear hypersurface systems

This paper investigates the global controllability of piecewise-linear (hypersurface) systems, which are defined as control systems that are subject to affine dynamics on each of the components of a finite polyhedral partition. Various new tools are developed for the study of the problem including a classification of the facets of the polyhedra in the partition. Necessary and sufficient conditions for complete controllability are obtained via the study of a suitably defined controllability connection matrix of polyhedra.     

Stochastic controllability of linear systems with Markovian jumps

The notion of stochastic controllability for linear systems subject to Markovian jumps in parameter values is studied. An algebraic necessary and sufficient condition is obtained in terms of an easily computable rank test.