System structure and singular control

The general continuous-time linear-quadratic control problem is considered. It is shown that recently developed linear system theoretic properties and algorithms play an important role in solving this singular control problem.     

Riccati equations for generalized singular estimate control systems

We consider the Bolza problem associated with boundary/point control systems governed by strongly continuous semigroups. In continuation of our work in Lasiecka and Tuffaha [I. Lasiecka and A. Tuffaha, Riccati equations for the Bolza problem arising in boundary/point control problems governed by C ₀–semigroups satisfying a singular estimate, J. Optim. Theory Appl. 136 (2008), pp. 229–246; I. Lasiecka and A. Tuffaha, A Bolza optimal synthesis problem for singular estimate control systems, Control Cybernet 38(4B) (2009), pp. 1429–1460], we yet extend the theory to a more general class of control problems that are not analytic providing sharp blow-up rates for the regularity. Solvability of the associated Riccati equations and an optimal feedback synthesis are established. The presence of unbounded control actions, such as boundary/point controls, naturally lead to a singularity at the terminal point t?=?T of the optimal control and of the corresponding feedback operator as before. The class of control systems considered in this article is a generalization to the class usually referred to in the literature as ‘Singular Estimate Control Systems’. The prototype is still that of a PDE system consisting of coupled hyperbolic parabolic dynamics interacting on an interface with point/boundary control. The distinct feature of the class considered in this article is that the degree of unboundedness in the control is stronger than that allowed in the usual singular estimate control system configuration, giving rise to less regular optimal state trajectories.     

Integral sliding mode control for T–S fuzzy descriptor systems

The present study focuses on designing the integral sliding-mode control (ISMC) for generalized Takagi–Sugeno (T–S) fuzzy singular stochastic systems by involving the Markovian jump type of system parameters. Distinct to the existing works, the present paper concerns about the derivation of sufficient conditions that ensure the global stability of considered T–S fuzzy stochastic singular Markovian jump systems with matched/mismatched uncertainties under fuzzy-based ISMC. In this regard, an improved fuzzy integral sliding manifold function with mode-dependent derivative-term coefficient is proposed where the matched uncertainties become unnecessary and the mismatched uncertainties have been disintegrated during the sliding mode phase. Based on Lyapunov stability theory, a suitable Lyapunov functional candidate by involving the information about the membership functions is constructed with singular and P-type matrices, the stochastic admissibility of corresponding sliding mode dynamics are derived. To proven the effectiveness of the proposed method, a physical experimental problem on cart and pendulum is adapted and simulated via the derived theoretical results and the corresponding results are provided.     

一类非线性离散系统基于有界反馈的奇异H_∞控制问题

主要研究基于有界控制律的一类非线性离散系统的奇异H∞控制问题.在系统不满足正则条件的情况下,分离出正则部分与非正则部分,给出基于有界反馈与二次Lyapunov函数的离散系统奇异H∞问题可解性的必要条件以及充分条件,求出的有界控制律能使得闭环系统在保证内稳定的条件下达到干扰衰减.     

Optimal control in a quantum cooling problem

The optimal control for cooling a quantum harmonic oscillator by controlling its frequency is considered. It is shown that this singular problem may be transformed with the proper choice of coordinates to an equivalent problem which is no longer singular. The coordinates used are sufficiently simple that a graphical solution is possible and eliminates the need to use a Weierstrass-like approach to show optimality. The optimal control of this problem is of significance in connection with cooling physical systems to low temperatures. It is also mathematically significant in showing the power and limitations of coordinate transformations for attacking apparently singular problems.     

Iterative solution of inconsistent systems of linear inequalities

In 1980 S.P. Han proposed a finitely terminating (in exact arithmetic) algorithm for solving an inconsistent system of linear inequalities in a least squares sense. This algorithm uses a singular value decomposition of a submatrix of the problem matrix on each iteration, making it impractical for all but only smaller problems. In this paper we show that a modification of Han's algorithm allows us to introduce an iterative approximation to the singular value decomposition solution. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Approximate Solution of Optimization Problems for Infinite-Dimensional Singular Systems

An optimal control problem is considered for a system described by a singular equation of parabolic type. The study bases on a special regularization method. We establish existence of a solution to the regularized problem, as well as the corresponding necessary optimality conditions. The results enable us to find an approximate solution to the original problem even in the absence of solvability.     

奇异脉冲系统的奇异H_∞控制

主要研究奇异脉冲系统的奇异H∞控制问题.当系统不满足正则条件时,给出奇异脉冲系统的奇异H∞控制问题可解的充分条件,控制律使得闭环系统在保证内稳定的条件下达到干扰衰减.     

增广PWA系统的奇异H_∞控制问题

讨论了PW A增广系统的奇异H∞控制问题.在一些假定和Lyapunov函数的基础上,给出了对于给定系统的奇异H∞控制可控的充要条件.     

Singular optimal controls of stochastic recursive systems and Hamilton–Jacobi–Bellman inequality

In this paper, we study the optimal singular controls for stochastic recursive systems, in which the control has two components: the regular control, and the singular control. Under certain assumptions, we establish the dynamic programming principle for this kind of optimal singular controls problem, and prove that the value function is a unique viscosity solution of the corresponding Hamilton–Jacobi–Bellman inequality, in a given class of bounded and continuous functions. At last, an example is given for illustration.     

On Necessary Optimality Conditions for Singular Controls in Dynamical Systems with a Delay in Control

Abstract

In this article, an optimal control problem with a delay in control is considered. The second-order necessary condition is obtained for the optimality of singular (in the sense of the maximum principle) control. Also, the notion of degenerate singular control of order k (k?≥?1) is introduced and for optimality of this, the high-order necessary condition is obtained. Moreover, while studying the problem, one of the strengthened version of an analog of the maximum principle is shown. Finally, the rich content of the obtained results is illustrated by specific examples.     

Lyapunov functions in justification theorems for asymptotics

A formal asymptotic solution is considered for a nonlinear system of ordinary differential equations in a neighborhood of a singular point. The problem of existence of an exact solution with such an asymptotics and the problem of stability of this solution are solved. The main tool in these studies is the Lyapunov function for a system linearized on a formal solution.     

TRAVELING WAVES CONNECTING EQUILIBRIUM AND PERIODIC ORBIT FOR DELAYED LATTICE DIFFERENTIAL EQUATION

A class of lattice with time delay and nonlocal response is considered.By transforming the lattice delay differential system into an integral equations in a Banach space,we reduces a singular perturbation problem to a regular perturbation problem.Traveling wave solution therefore is obtained by applying Liapunov-Schmidt method and the implicit function theorem.     

基于扰动补偿趋近律的非匹配离散广义准滑模控制

研究了非匹配不确定离散广义系统准滑模控制策略的综合问题.给出了具有前级状态向量的动态切换函数,使得系统能够在切换带内稳定.设计了两种带有扰动补偿的离散广义趋近律,消除了常规滑模控制策略中不确定项必须有界的限制,且不必满足匹配条件.给出了系统准滑动模态保持逐步穿越切换面的必要条件,减小了准滑动模态切换带的带宽.所设计的滑模控制器在有限时间内可达切换面,削弱了系统抖振,有效地改善了系统动态品质.最后,数值算例验证了该控制策略的可行性.     

On a class of singular stochastic control problems driven by Lévy noise

A class of singular stochastic control problems whose value functions satisfy an invariance property was studied by Lasry and Lions (2000). They have shown that, within this class, any singular control problem is equivalent to the corresponding standard stochastic control problem. The equivalence is in the sense that their value functions are equal. In this work, we clarify their idea and extend their work to allow Lévy type noise. In addition, for the purpose of application, we apply our result to an optimal trade execution problem studied by Lasry and Lions (2007).     

Analysis of an Optimal Control Problem Related to the Anaerobic Digestion Process

Our aim in this work is to synthesize optimal feeding strategies that maximize, over a time period, the biogas production in a continuously filled bioreactor controlled by its dilution rate. Such an anaerobic process is described by a four-dimensional dynamical system. Instead of modeling the optimization of the biogas production as a Lagrange-type optimal control problem, we propose a slightly different optimal control approach in this paper: We study the minimal time control problem to reach a target point, which is chosen in such a way that it maximizes the biogas production at steady state. Thanks to the Pontryagin maximum principle and the geometric control theory, we provide an optimal feedback control for the minimal time control problem, when the initial conditions are taken within the invariant and attractive manifold of the system. The optimal synthesis exhibits turnpike and anti-turnpike singular arcs and a cut locus.     

Blockwise estimate of the fundamental matrix of linear singularly perturbed differential systems with small delay and its application to uniform asymptotic solution

A singularly perturbed system of linear differential equations with a small delay is considered. Estimates of blocks of the fundamental matrix solution to this system uniformly valid for all sufficiently small values of the parameter of singular perturbations are obtained in the cases of time-independent and time-dependent coefficients of the system. In the first case the system is considered on an infinite time-interval, while in the second case it is considered on a finite one. These estimates are applied to justify a uniform asymptotic solution of an initial-value problem for this system in both cases.     

STABILITY OF TIME VARYING SINGULAR DIFFERENTIAL SYSTEMS WITH DELAY

In this paper, stability of time varying singular differential systems with delay is considered. Based on variation formula and Gronwall-Bellman integral inequality, we obtain the exponential estimation of the solution and the sufficient conditions under which the considered system is stable and exponentially asymptotically stable. These results will be very useful to further research on Roust stability and control design of uncertain singular control systems with delay.     

On the well-posedness of a weighted boundary-value problem for systems of ODEs with singularity

We consider a system of linear second-order ODEs with an irregular singular point. The matrix that relates to the equation of this system is nonnormal. That implies some complications with respect to the statement of the weighted boundary-value condition at the singular point. We propose here a well-posed weighted boundary-value problem to the considered system of ODEs and prove the existence and uniqueness of its solution.     

Center conditions and bifurcation of limit cycles at degenerate singular points in a quintic polynomial differential system

The center problem and bifurcation of limit cycles for degenerate singular points are far to be solved in general. In this paper, we study center conditions and bifurcation of limit cycles at the degenerate singular point in a class of quintic polynomial vector field with a small parameter and eight normal parameters. We deduce a recursion formula for singular point quantities at the degenerate singular points in this system and reach with relative ease an expression of the first five quantities at the degenerate singular point. The center conditions for the degenerate singular point of this system are derived. Consequently, we construct a quintic system, which can bifurcates 5 limit cycles in the neighborhood of the degenerate singular point. The positions of these limit cycles can be pointed out exactly without constructing Poincaré cycle fields. The technique employed in this work is essentially different from more usual ones. The recursion formula we present in this paper for the calculation of singular point quantities at degenerate singular point is linear and then avoids complex integrating operations.