scholarly journals Review of the Latest Progress in Controllability of Stochastic Linear Systems and Stochastic GE-Evolution Operator

Mathematics ◽  
2021 ◽  
Vol 9 (24) ◽  
pp. 3240
Author(s):  
Zhaoqiang Ge
Keyword(s):  
Linear Systems ◽  
Approximate Controllability ◽  
Evolution Operator ◽  
Sufficient Conditions ◽  
Exact Controllability ◽  
Spatial Dimension ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Stochastic Linear Systems ◽  
Singular Linear Systems

According to the spatial dimension, equation type, and time sequence, the latest progress in controllability of stochastic linear systems and some unsolved problems are introduced. Firstly, the exact controllability of stochastic linear systems in finite dimensional spaces is discussed. Secondly, the exact, exact null, approximate, approximate null, and partial approximate controllability of stochastic linear systems in infinite dimensional spaces are considered. Thirdly, the exact, exact null and impulse controllability of stochastic singular linear systems in finite dimensional spaces are investigated. Fourthly, the exact and approximate controllability of stochastic singular linear systems in infinite dimensional spaces are studied. At last, the controllability and observability for a type of time-varying stochastic singular linear systems are studied by using stochastic GE-evolution operator in the sense of mild solution in Banach spaces, some necessary and sufficient conditions are obtained, the dual principle is proved to be true, an example is given to illustrate the validity of the theoretical results obtained in this part, and a problem to be solved is introduced. The main purpose of this paper is to facilitate readers to fully understand the latest research results concerning the controllability of stochastic linear systems and the problems that need to be further studied, and attract more scholars to engage in this research.

Finite-approximate controllability of fractional evolution equations: variational approach

2018 ◽  
Vol 21 (4) ◽  
pp. 919-936 ◽  
Author(s):  
Nazim I. Mahmudov
Keyword(s):  
Hilbert Spaces ◽  
Approximate Controllability ◽  
Evolution Equations ◽  
Nonlocal Conditions ◽  
Sufficient Conditions ◽  
Exact Controllability ◽  
Heat Equations ◽  
Natural Conditions ◽  
Finite Dimensional ◽  
Semilinear Evolution Equations

Abstract In this work we extend a variational method to study the approximate controllability and finite dimensional exact controllability (finite-approximate controllability) for the fractional semilinear evolution equations with nonlocal conditions in Hilbert spaces. Assuming the approximate controllability of the corresponding linear equation we obtain sufficient conditions for the finite-approximate controllability of the fractional semilinear evolution equation under natural conditions. The obtained results are generalization and continuation of the recent results on this issue. Applications to heat equations are treated.

On exact controllability of linear perturbed boundary systems: a semigroup approach

2020 ◽  
Vol 37 (4) ◽  
pp. 1548-1573
Author(s):  
Marieme Lasri ◽  
Hamid Bounit ◽  
Said Hadd
Keyword(s):  
Control System ◽  
Control Systems ◽  
Linear Systems ◽  
Banach Spaces ◽  
Exact Controllability ◽  
Dimensional Control ◽  
Neutral Equations ◽  
Infinite Dimensional ◽  
Semigroup Approach ◽  
Perturbed Linear Systems

Abstract The purpose of this paper is to investigate the robustness of exact controllability of perturbed linear systems in Banach spaces. Under some conditions, we prove that the exact controllability is preserved if we perturb the generator of an infinite-dimensional control system by appropriate Miyadera–Voigt perturbations. Furthermore, we study the robustness of exact controllability for perturbed boundary control systems. As application, we study the robustness of exact controllability of neutral equations. We mention that our approach is mainly based on the concept of feedback theory of infinite-dimensional linear systems.

APPROXIMATE CONTROLLABILITY OF SECOND-ORDER STOCHASTIC DIFFERENTIAL EQUATIONS WITH IMPULSIVE EFFECTS

2010 ◽  
Vol 24 (14) ◽  
pp. 1559-1572 ◽  
Author(s):  
RATHINASAMY SAKTHIVEL ◽  
YONG REN ◽  
N. I. MAHMUDOV
Keyword(s):  
Biological Sciences ◽  
Differential Equations ◽  
Approximate Controllability ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Second Order ◽  
Impulsive Effects ◽  
Infinite Dimensional ◽  
Infinite Dimensional Dynamical Systems ◽  
Necessary And Sufficient

Many practical systems in physical and biological sciences have impulsive dynamical behaviors during the evolution process which can be modeled by impulsive differential equations. In this paper, the approximate controllability of nonlinear second-order stochastic infinite-dimensional dynamical systems with impulsive effects is considered. By using the Holder's inequality, stochastic analysis and fixed point strategy, a new set of necessary and sufficient conditions are formulated which guarantees the approximate controllability of the nonlinear second-order stochastic system. The results are obtained under the assumption that the associated linear system is approximately controllable.

Robust H∞ control of stochastic linear systems with input delay by predictor feedback

2017 ◽  
Vol 40 (7) ◽  
pp. 2396-2407
Author(s):  
Ali Javadi ◽  
Mohammad Reza Jahed-Motlagh ◽  
Ali Akbar Jalali
Keyword(s):  
Linear Systems ◽  
Multiplicative Noise ◽  
Sufficient Conditions ◽  
Input Delay ◽  
Linear Matrix ◽  
Matrix Inequalities ◽  
External Disturbances ◽  
Delayed Systems ◽  
Simulation Results ◽  
Stochastic Linear Systems

This study investigates the prediction-based (dynamic) stabilization of linear systems with input delay in the presence of external disturbances and multiplicative noise modelled as Itô type stochastic differential equations. Conventional memory-less (static) controllers are widely used for the stabilization of both deterministic and stochastic delayed systems. However, using these methods the upper bound for delay is strongly restricted. Motivated by acceptable performances of dynamic controllers for deterministic delayed systems, the extension of these methods for stochastic delayed systems is considered in this paper. The structure of the dynamic controller for stabilization of stochastic delayed systems is firstly derived utilizing the prediction vector. Then two sufficient conditions are given in the form of linear matrix inequalities that in the case of feasibility provide the stabilizing gain of the controller. Finally, simulation results are given to illustrate the effectiveness of the proposed method.

scholarly journals Relative controllability of a stochastic system using fractional delayed sine and cosine matrices

2021 ◽  
Vol 26 (6) ◽  
pp. 1031-1051
Author(s):  
JinRong Wang ◽  
T. Sathiyaraj ◽  
Donal O’Regan
Keyword(s):  
Fixed Point ◽  
Stochastic System ◽  
Fixed Point Theory ◽  
Sufficient Conditions ◽  
Exact Controllability ◽  
Point Theory ◽  
Linear Operators ◽  
Relative Controllability ◽  
Finite Dimensional ◽  
Pure Delay

In this paper, we study the relative controllability of a fractional stochastic system with pure delay in finite  dimensional stochastic spaces. A set of sufficient conditions is obtained for relative exact controllability using fixed point theory, fractional calculus (including fractional delayed linear operators and Grammian matrices) and local assumptions on nonlinear terms. Finally, an example is given to illustrate our theory.

scholarly journals POINTED HOPF ALGEBRAS WITH CLASSICAL WEYL GROUPS

2012 ◽  
Vol 23 (06) ◽  
pp. 1250066
Author(s):  
SHOUCHUAN ZHANG ◽  
YAO-ZHONG ZHANG
Keyword(s):  
Hopf Algebras ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weyl Groups ◽  
Drinfeld Modules ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Pointed Hopf Algebras ◽  
Nichols Algebras ◽  
Necessary And Sufficient

We prove that Nichols algebras of irreducible Yetter–Drinfeld modules over classical Weyl groups A ⋊ 𝕊nsupported by 𝕊nare infinite dimensional, except in three cases. We give necessary and sufficient conditions for Nichols algebras of Yetter–Drinfeld modules over classical Weyl groups A ⋊ 𝕊nsupported by A to be finite dimensional.

scholarly journals Minimization of nonsmooth integral functionals

1992 ◽  
Vol 15 (4) ◽  
pp. 673-679
Author(s):  
Nikolaos S. Papageorgiou ◽  
Apostolos S. Papageorgiou
Keyword(s):  
Sobolev Spaces ◽  
Convex Analysis ◽  
Optimization Problems ◽  
Sufficient Conditions ◽  
Integral Functionals ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Dimensional State Space ◽  
Sufficient Conditions For Optimality ◽  
Necessary And Sufficient

In this paper we examine optimization problems involving multidimensional nonsmooth integral functionals defined on Sobolev spaces. We obtain necessary and sufficient conditions for optimality in convex, finite dimensional problems using techniques from convex analysis and in nonconvex, finite dimensional problems, using the subdifferential of Clarke. We also consider problems with infinite dimensional state space and we finally present two examples.

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