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.     

Null controllability for a parabolic equation with nonlocal nonlinearities

In this paper, we establish a local null controllability result for a nonlinear parabolic PDE with nonlocal nonlinearities. The result relies on the (global) null controllability of similar linear equations and a fixed point argument. We also analyze other similar controllability problems and we present several open questions.     

On null controllability of linear systems in Banach spaces

Our aim here is to give characterizations for the null controllability of linear systems in general Banach spaces. The starting point of this paper is the work of Pengnian Chen and Huashu Qin (Systems Control Lett. 45 (2002) 155), which solves the problem of exact controllability. In fact, the results of (Systems Control Lett. 45 (2002) 155) say that, in case of setting in general Banach spaces, there are the same characterizations of exact controllability as in the case of reflexive Banach spaces. An open problem raised in (Systems Control Lett. 45 (2002) 155) is the validity of a similar result for null controllability. We give here a positive answer to this problem. However, it seems to us that the technique of (Systems Control Lett. 45 (2002) 155) does not work for null controllability, and, consequently, our approach is completely different.     

Stochastic maximum principle for SPDEs with noise and control on the boundary

In this paper we prove necessary conditions for optimality of a stochastic control problem for a class of stochastic partial differential equations that is controlled through the boundary. This kind of problem can be interpreted as a stochastic control problem for an evolution system in a Hilbert space. The regularity of the solution of the adjoint equation, that is a backward stochastic equation in infinite dimension, plays a crucial role in the formulation of the maximum principle.     

Stability radii of infinite dimensional systems with stochastic uncertainty and their optimization

In this paper we consider infinite dimensional systems which are subjected to stochastic structured multiperturbations. We first characterize the stability radii of these systems in terms of a Lyapunov equation and the corresponding Lyapunov inequalities. Then we investigate the problem of maximizing the stability radius by linear state feedback. We show that the supremal achievable stability radius can be determined via the resolution of a parametrized Riccati equation. Illustrative examples are included. Copyright © 2006 John Wiley & Sons, Ltd.     

Approximate controllability of infinite dimensional linear systems in nonreflexive state spaces

This paper deals with the problem of approximate controllability of infinite dimensional linear systems in nonreflexive state spaces. A necessary and sufficient condition for approximate controllability via L^p（[0, T], U）, 1≤p〈∞ is obtained,where L^p（ [0, T], U） is the control function space.     

Approximate controllability of infinite dimensional linear systems in nonreflexive state spaces

This paper deals with th e problem of approximate controllability of infinite di mensional linear systems in nonreflexive state spaces. A necessary and sufficient condition for approximate controllability via L~p[KG*1/3]p([0,T],U),1≤p<∞ is obtain ed,where L~p[KG*1/3]p([0,T],U) is the control function space.     

Solvability of indefinite stochastic Riccati equations and linear quadratic optimal control problems

A new approach to study the indefinite stochastic linear quadratic (LQ) optimal control problems, which we called the “equivalent cost functional method”, is introduced by Yu (2013) in the setup of Hamiltonian system. On the other hand, another important issue along this research direction, is the possible state feedback representation of optimal control and the solvability of associated indefinite stochastic Riccati equations. As the response, this paper continues to develop the equivalent cost functional method by extending it to the Riccati equation setup. Our analysis is featured by its introduction of some equivalent cost functionals which enable us to have the bridge between the indefinite and positive-definite stochastic LQ problems. With such bridge, some solvability relation between the indefinite and positive-definite Riccati equations is further characterized. It is remarkable the solvability of the former is rather complicated than the latter, hence our relation provides some alternative but useful viewpoint. Consequently, the corresponding indefinite linear quadratic problem is discussed for which the unique optimal control is derived in terms of state feedback via the solution of the Riccati equation. In addition, some example is studied using our theoretical results.     

Approximate controllability results of semilinear integrodifferential equations with infinite delays

We study the approximate controllability of control systems governed by a class of semilinear integrodifferential equations with infinite delays. Sufficient conditions for approximate controllability of semilinear control systems are established provided the approximate controllability of the corresponding linear control systems. The results are obtained by using fixed-point theorems and semigroup theory. As an illustration of the application of the approximate controllability results, a simple example is provided.     

带有随机跳跃干扰的线性二次随机最优控制问题

给出一类布朗运动和泊松过程混合驱动的正倒向随机微分方程解的存在唯一性结果,应用这一结果研究带有随机跳跃干扰的线性二次随机最优控制问题,并得到最优控制的显式形式,可以证明最优控制是唯一的.然后,引入和研究一类推广的黎卡提方程系统,讨论该方程系统的可解性并由该方程的解得到带有随机跳跃干扰的线性二次随机最优控制问题最优的线性反馈.     

Suboptimal stochastic linear feedback control of linear systems with state- and control-dependent noise: The incomplete information case

The stochastic regulation problem for linear systems with state- and control-dependent noise and a noisy linear output equation is considered. The optimal quadratic cost output-feedback control law in a class of linear controllers is found. This problem was first addressed in the early 1970s and solved, in the complete information case, by Wonham. In this paper we give the solution of the problem in the incomplete information case, that is, for a linear output equation corrupted by Gaussian noise. Moreover, a different method is used here, giving the solution in a more direct way even in the complete information case.     

Approximate controllability of stochastic nonlinear third‐order dispersion equation

In this work, we consider a class of control systems governed by the stochastic nonlinear third‐order dispersion equation in Hilbert spaces. We first prove the existence of mild solutions of stochastic nonlinear third‐order dispersion equation by using fixed point theory, infinite dimensional semigroup properties, stochastic analysis techniques, and then a new set of sufficient conditions are formulated which guarantees the approximate controllability of the main problem. Finally, an example is provided to illustrate the application of the obtained results.Copyright © 2012 John Wiley & Sons, Ltd.     

正倒向随机微分方程与一类线性二次随机最优控制问题

讨论一类正倒向随机微分方程解的存在唯一性及其对应的一类线性二次随机最优控制问题,利用单调性方法证明了一类特殊的正倒向随机微分方程解的存在唯一性定理,利用该结果研究一类耦合了一个倒向随机微分方程的线性随机控制系统广义最优指标随机控制问题,得到由正倒向随机微分方程的解所表示的唯一最优控制的显式表达式,并得到精确的线性反馈及其对应的Riccati方程.     

一类带随机跳跃的完全耦合的线性二次随机控制问题

研究一类带随机跳跃的完全耦合的线性二次随机控制问题. 得到了最优控制的显式解, 并可以证明最优控制是唯一的. 引入了一类推广的黎卡提方程并讨论了其可解性. 利用这一类推广的黎卡提方程的解, 得到了上述带随机跳跃的最优控制问题的线性状态反馈调节器.     

Approximate controllability results for impulsive neutral differential inclusions of Sobolev-type with infinite delay

The paper is mainly concerned with the approximate controllability results for a class of impulsive neutral differential control systems. First, we establish a set of sufficient conditions for the approximate controllability for a class of impulsive differential inclusions of Sobolev-type with infinite delay. With the help of semigroup theory and Bohnenblust-Karlin's fixed point theorem, we prove our main results. Then, we extend the result to study the approximate controllability for a class of impulsive neutral differential inclusions of Sobolev-type with infinite delay. Finally, an example is also presented to illustrate our theoretical results.     

Approximate controllability of the n -th order infinite dimensional systems with controls delayed by the control devices

The main aim of this article is to derive general conditions for a few types of controllability at once for an arbitrary order abstract differential equation and arbitrary eigenvalues multiplicities, instead of conditions for fixed order equation and single eigenvalues. Another innovation of this article is taking into account delays caused by electronic control microcontrollers. This was possible thanks to analysis of the n-th order linear system in the Frobenius form, generating Jordan transition matrix of the confluent Vandermonde form. Using the explicit analytical form of the inverse confluent Vandermonde matrix enabled us to receive general conditions of different types of controllability for the infinite dimensional systems. We derived this analytical form of the inverse confluent Vandermonde matrix using new results from the linear algebra, presented in the paper by S. Hou and W. Pang, “Inversion of confluent Vandermonde matrices”, Int. J. Comput. Math. Appl., 43, pp. 1539–1547, 2002.