Optimal control for infinite dimensional stochastic differential equations with infinite Markov jumps and multiplicative noise

In this paper we solve an infinite-horizon linear quadratic control problem for a class of differential equations with countably infinite Markov jumps and multiplicative noise. The global solvability of the associated differential Riccati-type equations is studied under detectability hypotheses. A nonstochastic, operatorial approach is used. Some properties of the linear stochastic systems, such as stability, stabilizability and detectability, are also discussed on the basis of a new solution representation result. A generalized Ito's formula which applies to infinite dimensional stochastic differential equations with countably infinite Markov jumps is also provided.     

Stability of discrete-time positive evolution operators on ordered Banach spaces and applications

In this paper we discuss stability problems for a class of discrete-time evolution operators generated by linear positive operators acting on certain ordered Banach spaces. Our approach is based upon a new representation result that links a positive operator with the adjoint operator of its restriction to a Hilbert subspace formed by sequences of Hilbert–Schmidt operators. This class includes the evolution operators involved in stability and optimal control problems for linear discrete-time stochastic systems. The inclusion is strict because, following the results of Choi, we have proved that there are positive operators on spaces of linear, bounded and self-adjoint operators which have not the representation that characterize the completely positive operators. As applications, we introduce a new concept of weak-detectability for pairs of positive operators, which we use to derive sufficient conditions for the existence of global and stabilizing solutions for a class of generalized discrete-time Riccati equations. Finally, assuming weak-detectability conditions and using the method of Lyapunov equations we derive a new stability criterion for positive evolution operators.     

Spectral properties of non-local differential operators

In this article we consider the spectral properties of a class of non-local operators that arise from the study of non-local reaction-diffusion equations. Such equations are used to model a variety of physical and biological systems with examples ranging from Ohmic heating to population dynamics. The operators studied here are bounded perturbations of linear (local) differential operators. The non-local perturbation is in the form of an integral term. It is shown here that the spectral properties of these non-local operators can differ considerably from those of their local counterpart. Multiplicities of eigenvalues are studied and new oscillation results for the associated eigenfunctions are presented. These results highlight problems with certain similar results and provide an alternative formulation. Finally, the stability of steady states of associated non-local reaction-diffusion equations is discussed.     

A Criterion for Stationary Solutions of Retarded Linear Equations with Additive Noise

In this work, we shall consider stationary (mild) solutions for a class of retarded functional linear differential equations with additive noise in Hilbert spaces. We first introduce a family of Green operators for the stochastic systems and establish stability results which will play an important role in the investigation of stationary solutions. A criterion imposed on the Green operators is presented to identify a unique stationary solution for the systems considered. Under strong quasi-Feller property, it is shown that this criterion is a sufficient and necessary condition to guarantee a unique stationary solution, based on a method having its origins in optimal control theory.     

Numerical Solution for Linear-Quadratic Control Problems of Markov Jump Linear Systems and Weak Detectability Concept

A method for solving the linear-quadratic problem of Markov jump linear systems is developed in this paper, relying on the assumption of weak detectability. The concept of weak detectability generalizes previous concepts relevant to this class of systems, and most importantly, it allows us to revisit the quadratic control problem. In the main result of the paper, we show that, for weakly detectable systems, the solution obtained with the new method converges to the solution of the coupled algebraic Riccati equation that arises in the control problem if and only if the system is mean-square stabilizable. The paper shows how the concepts and the method involved are applied by means of numerical examples and comparisons.     

Optimal control of linear stochastic evolution equations in Hilbert spaces and uniform observability

In this paper we study the existence of the optimal (minimizing) control for a tracking problem, as well as a quadratic cost problem subject to linear stochastic evolution equations with unbounded coefficients in the drift. The backward differential Riccati equation (BDRE) associated with these problems (see [2], for finite dimensional stochastic equations or [21], for infinite dimensional equations with bounded coefficients) is in general different from the conventional BDRE (see [10], [18]). Under stabilizability and uniform observability conditions and assuming that the control weight-costs are uniformly positive, we establish that BDRE has a unique, uniformly positive, bounded on ℝ ₊ and stabilizing solution. Using this result we find the optimal control and the optimal cost. It is known [18] that uniform observability does not imply detectability and consequently our results are different from those obtained under detectability conditions (see [10]).      

Stability and stabilizability of linear infinite-dimensional discrete-time systems

This paper contains three results on stability and stabilizabilityof linear time invariant infinite-dimensional discrete-timesystems. (1) Power stability is characterized in transfer-functionterms using the concepts of stabilizability and detectability.(2) Under the assumption that the input operator is campact,we present a necessary and sufficient condition for stabilizabilityinvolving spectral properties of the system operator and a projectionof the infinite-dimensional system onto a certain finite-dimensionalsubspace of the state space. (3) It is shown that, if the inputand output spaces are finite-dimensional, then stabilizationby finite-dimensional dynamic output feedback is possible ifand only if the systems is detectable and stabilizable.     

具有热储备的可修复平行系统在由常规错误引起失效下的渐进稳定性

讨论了具有热储备和两个独立相同部件的平行系统在由常规错误引起失效下的渐进稳定性.首先,利用Banach空间的Volttera算子方程得到了非负动态解的存在唯一性;然后,利用强连续线性算子半群理论证明了系统正的动态解的存在唯一性,而由于初始值不在定义域内,故得到的是mild解.但在t＞0时系统古典解存在唯一,所以此时mild解即为古典解.最后,利用线性算子半群稳定性的结果,证明了该动态解在范数意义下收敛到稳态解,进而得到了系统的渐进稳定性.     

Approximate controllability of a fractional stochastic partial integro-differential systems via noncompact operators

Objective: in this article, we discuss the approximate controllability problems of a new class of fractional impulsive stochastic partial integro-differential systems in separable Hilbert spaces. Methods: by applying the fractional calculus, the measure of noncompactness, properties of fractional resolvent operators and fixed point theorems. Results: we prove our main results without the hypotheses of compactness on the operator generated by the linear part of systems. Instead we suppose that the nonlinear term only satisfies a weakly compactness condition. Conclusion: the approximate controllability for the control systems with noncompact operators is established. Finally, an example is given for the illustration of the obtained theoretical results.     

Boundedness and closedness of linear relations

This paper focuses on boundedness and closedness of linear relations, which include both single-valued and multi-valued linear operators. A new (single-valued) linear operator induced by a linear relation is introduced, and its relationships with other two important induced linear operators are established. Several characterizations for closedness, closability, bundedness, relative boundedness and boundedness from below (above) of linear relations are given in terms of their induced linear operators. In particular, the closed graph theorem for linear relations in Banach spaces is completed, and stability of closedness of linear relations under bounded and relatively bounded perturbations is studied. The results obtained in the present paper generalize the corresponding results for single-valued linear operators to multi-valued linear operators, and some improve or relax certain assumptions of the related existing results.     

Quadratic control for linear periodic systems

We propose a new quadratic control problem for linear periodic systems which can be finite or infinite dimensional. We consider both deterministic and stochastic cases. It is a generalization of average cost criterion, which is usually considered for time-invariant systems. We give sufficient conditions for the existence of periodic solutions.Under stabilizability and detectability conditions we show that the optimal control is given by a periodic feedback which involves the periodic solution of a Riccati equation. The optimal closed-loop system has a unique periodic solution which is globally exponentially asymptotically stable. In the stochastic case we also consider the quadratic problem under partial observation. Under two sets of stabilizability and detectability conditions we obtain the separation principle. The filter equation is not periodic, but we show that it can be effectively replaced by a periodic filter. The theory is illustrated by simple examples.This work was done while this author was a visiting professor at the Scuola Normale Superiore, Pisa.     

Controllability of Fractional Functional Evolution Equations of Sobolev Type via Characteristic Solution Operators

The paper is concerned with the controllability of fractional functional evolution equations of Sobolev type in Banach spaces. With the help of two new characteristic solution operators and their properties, such as boundedness and compactness, we present the controllability results corresponding to two admissible control sets via the well-known Schauder fixed point theorem. Finally, an example is given to illustrate our theoretical results.     

Minimization of a functional over the set of causal operators of causal Hilbert space

The minimization problem for a quadratic functional defined on the set of nonwarning (causal) operators acting in a causal Hilbert space can be regarded as an abstrat analog of the Wiener problem on constructing the optimal nonwarning filter. A similar problem also arises in the linear control problem with the quadratic performance criterion (in this case the transfer operators of a closed control system serve as causal ones). The introduction of causal operators in filtering theory and control theory is a mathematical expression of the causality principle, which must be taken into account for a number of problems. In the present paper we attempt to systematize the mathematical foundations of the abstract linear filtering theory, for which its basic results are expressed in terms of operators describing the filtering problem. We introduce and study a class of finite operators, a natural generalization of the class of causal operators, and give a solution of the minimization problem for a quadratic positive functional defined on the set of causal operators acting in a “discrete” causal space. Bibliography: 54 titles. Translated fromProblemy Matematicheskogo Analiza, No. 14, 1995. pp. 143–187.     

Quantitative stability of full random two-stage stochastic programs with recourse

In this paper, we consider quantitative stability analysis for two-stage stochastic linear programs when recourse costs, the technology matrix, the recourse matrix and the right-hand side vector are all random. For this purpose, we first investigate continuity properties of parametric linear programs. After deriving an explicit expression for the upper bound of its feasible solutions, we establish locally Lipschitz continuity of the feasible solution sets of parametric linear programs. These results are then applied to prove continuity of the generalized objective function derived from the full random second-stage recourse problem, from which we derive new forms of quantitative stability results of the optimal value function and the optimal solution set with respect to the Fortet–Mourier probability metric. The obtained results are finally applied to establish asymptotic behavior of an empirical approximation algorithm for full random two-stage stochastic programs.     

Stochastic uniform observability of linear differential equations with multiplicative noise

The aim of this paper is to give a deterministic characterization of the uniform observability property of linear differential equations with multiplicative white noise in infinite dimensions. We also investigate the properties of a class of perturbed evolution operators and we used these properties to give a new representation of the covariance operators associated to the mild solutions of the investigated stochastic differential equations. The obtained results play an important role in obtaining necessary and sufficient conditions for the stochastic uniform observability property.     

On the existence and stability of approximate solutions of perturbed vector equilibrium problems

In this paper we consider several concepts of approximate minima of a set in normed vector spaces and we provide some results concerning the stability of these minima under perturbation of the underlying set with a sequence of sets converging in the sense of Painlevé-Kuratowski to the initial set. Then, we introduce the concept of approximate solution for equilibrium problem governed by set-valued maps and we study the stability of these solutions. The particular case of linear continuous operators is considered as well.     

Approximation properties of a generalization of positive linear operators

In the present paper we introduce a generalization of positive linear operators and obtain its Korovkin type approximation properties. The rates of convergence of this generalization is also obtained by means of modulus of continuity and Lipschitz type maximal functions. The second purpose of this paper is to obtain weighted approximation properties for the generalization of positive linear operators defined in this paper. Also we obtain a differential equation so that the second moment of our operators is a particular solution of it. Lastly, some Voronovskaja type asymptotic formulas are obtained for Meyer-König and Zeller type and Bleimann, Butzer and Hahn type operators.     

Banach空间中关于(H,η)-增生算子的一类新变分包含组问题

In this paper,we first introduce a new class of generalized accretive operators named(H,η)-accretive in Banach space.By studying the properties of(H,η)-accretive,we extend the concept of resolvent operators associated with m-accretive operators to the new(H,η)-accretive operators.In terms of the new resolvent operator technique,we prove the existence and uniqueness of solutions for this new system of variational inclusions.We also construct a new algorithm for approximating the solution of this system and discuss the convergence of the sequence of iterates generated by the algorithm.     

Evolution equations of higher order in banach spaces

The present paper deals with a minimal extension of the classical semigroup theory for equations of any order in Banach spaces with closed densely defined linear operators as coefficients. We do not ask anymore from our operators than in the case of first-order equations, i.e., Semigroups. We present here a generalization of Myadera-Phillips--Feller theorem, of Hille theorem and some other results. The method is quite general. We focus our attention on a particular operator solution (main propagator or abstract initial value Green function) and we assume some properties about it. From this we can obtain all needed information about complementary operator solutions, among others.     

The complete solution to the Sylvester-polynomial-conjugate matrix equations

In this paper we propose two new operators for complex polynomial matrices. One is the conjugate product and the other is the Sylvester-conjugate sum. Then some important properties for these operators are proved. Based on these derived results, we propose a unified approach to solving a general class of Sylvester-polynomial-conjugate matrix equations, which include the Yakubovich-conjugate matrix equation as a special case. The complete solution of the Sylvester-polynomial-conjugate matrix equation is obtained in terms of the Sylvester-conjugate sum, and such a proposed solution can provide all the degrees of freedom with an arbitrarily chosen parameter matrix.