Sufficient Conditions For Controllability Research Articles

Study approximate controllability and null controllability of neutral delay Hilfer fractional stochastic integrodifferential system with Rosenblatt process

In this paper, we investigate the sufficient conditions for approximate controllability and null controllability of neutral delay stochastic Hilfer fractional integrodifferential system with Rosenblatt process are established. The required results are obtained based on a fixed point technique, fractional calculus and stochastic analysis. Finally, an example is given to illustrate the obtained results.

Null controllability of Hilfer fractional stochastic differential equations with nonlocal conditions

<p style='text-indent:20px;'>This paper assesses a class of nonlocal Hilfer fractional stochastic differential equations (NHFSDEs) governed by fractional Brownian motion (fBm). New sufficient condition of exact null controllability for this stochastic setting has been investigated using fractional calculus, fixed point theory and a theory of resolvent operator. The derived result is new because it generalizes several of previously published results. However, the obtained results are proven by an illustration using stochastic partial differential equations to demonstrate the present application characteristic of null controllability. A filter example is demonstrated to make use of resolvent operator in a practical way.</p>

Small-time local controllability of the bilinear Schrödinger equation with a nonlinear competition

We consider the local controllability near the ground state of a 1D Schrödinger equation with bilinear control. Specifically, we investigate whether nonlinear terms can restore local controllability when the linearized system is not controllable. In such settings, it is known [K. Beauchard and M. Morancey, Math. Control Relat. Fields 4 (2014) 125-160, M. Bournissou, J. Diff. Equ. 351 (2023) 324−360] that the quadratic terms induce drifts in the dynamics which prevent small-time local controllability when the controls are small in very regular spaces. In this paper, using oscillating controls, we prove that the cubic terms can entail the small-time local controllability of the system, despite the presence of such a quadratic drift. This result, which is new for PDEs, is reminiscent of Sussmann's S (θ) sufficient condition of controllability for ODEs. Our proof however relies on a different general strategy involving a new concept of tangent vector, better suited to the infinite-dimensional setting.

Stability and controllability analysis of fractional integro-delay differential equation via delayed perturbation

This paper deals with stability and controllability analysis of the system governed by fractional integro delay differential equations. By establishing Finite time stability and Ulam–Hyers stability of fractional integro delay differential equation via delayed perturbation Mittag-Leffler type matrix function. Additionally, we give a suitable Grammian matrix by using a delayed Mittag-Leffler type matrix function. The necessary and sufficient conditions for controllability are established for linear system. Further, by employing Kransnoselskii’s fixed point theorem, the existence of controllability in non-linear case is verified. Finally, an example is provided for better understanding of theoretical aspects.

Output controllability of impulsive Boolean control networks based on hybrid‐index model

AbstractIn biological systems, impulses are often used to model mutations in gene expression. In order to describe the instantaneity of the impulse, this paper employs a hybrid‐index model of impulsive Boolean networks with state‐triggered impulses. Based on this model, the time‐domain output controllability is investigated. Using the quotient mapping method, the hybrid‐domain solutions of this model are mapped to the time‐domain solutions, mathematically. Then a necessary and sufficient condition for output controllability in the time domain is presented. Finally, an example is given to demonstrate the correctness and effectiveness of the obtained results.

Controllability under positivity constraints of a size‐structured population model with delayed birth process

In this paper, we study the approximate controllability, under positivity constraints on the control and state, of a size‐structured population system with time‐delayed behavior. In this model, the birth term is non‐local in size and time, which describes the time‐delay in the recruitment process in newborn individuals population. Our control function is localized at birth and corresponds to a supply of newborn individuals. First, we establish the well‐posedness and positivity property of the model. Second, and more importantly, we derive necessary and sufficient conditions for approximate controllability under positivity constraints. The method relies on the feedback theory of infinite‐dimensional positive systems.

Existence and controllability of non-local fractional dynamical systems with almost sectorial operators

In this article, we investigate the existence of the mild solutions and controllability of nonlocal fractional dynamical system with almost sectorial operator. The dynamical system under consideration involves Caputo fractional derivative of order α∈(0,1). The existence results are proved by using fixed point theorems with suitable assumptions. Sufficient conditions for controllability are derived using appropriate control function via Leray-Schauder fixed point theorem. An example is provided to validate the results.

Prescribed-time Tracking Control for Robotic Manipulator with Time-varying Output Constraints

This paper introduces prescribed-time tracking control of nonlinear systems for robotic manipulator under time-varying output constraints (TVOC). For the robotic manipulator driven by DC motor, we will design a controller to realize: one is to realize prescribed-time tracking control; the other is no violation of the time-varying output constraints (TVOC). First, sufficient conditions for prescribed-time tracking control for nonlinear systems with TVOC are presented. Secondly, relying on the backstepping technique, the recursive design algorithm is applied to construct the controller, which then realizes the prescribed-time tracking without the violation of TVOC. Finally, the applicability of this method in robot manipulator is verified by simulation.

Reachable Set Control for Discrete-Time Takagi–Sugeno Fuzzy Singular Markov Jump System

This article investigates the problem of reachable set control for a class of discrete-time Takagi-Sugeno fuzzy singular Markov jump systems (T-S FSMJSs) with time-varying delay and disturbance for the first time. By designing the state feedback and output feedback controllers, and based on the Lyapunov functional, two new sufficient conditions for the reachable set control are established. The state feedback and output feedback controller gains are obtained by optimizing the iterative algorithm. Then, all the state trajectories of the closed-loop T-S FSMJSs are contained by the intersection of ellipsoids. In the end, two examples with practical application background are presented to show the validity and usability of the proposed methods.

A Study on Existence and Controllability of Conformable Impulsive Equations

We study the existence/uniqueness of conformable fractional type impulsive nonlinear systems as well as the controllability of linear/semilinear conformable fractional type impulsive controlled systems. Using the conformable fractional derivative approach, we introduce the conformable controllability operator and the conformable controllability Gramian matrix in order to obtain the necessary and sufficient conditions for the complete controllability of linear impulsive conformable systems. We present a set of sufficient conditions for the controllability of the conformable semilinear impulsive systems.

On the small-time local controllability

A class of polynomial control systems is considered. The local properties of the corresponding reachable sets are studied by using a general differential–geometrical approach based on the Campbell–Baker–Hausdorff (C–B–H) formula. The main result is a new sufficient condition of small-time local controllability. Two four-dimensional examples illustrate this result.

Controllability and observability of linear time-varying fractional systems

We consider linear time-varying control problems involving the Caputo fractional derivative. By introducing the controllability Gramian matrix, we prove that the invertibility of the Gramian is a necessary and sufficient condition for controllability. Furthermore, we prove equivalence between controllability and observability of the associated adjoint problem. Using methods from linear control theory for systems with integer-order derivatives, and adapting them to the fractional setting, we derive the form of the optimal control in weighted $$L^2$$ space as well as of the optimal control in $$L^{\infty }$$ . We provide an example to illustrate the obtained results.

Some sufficient conditions of existence and trajectory controllability for impulsive and initial value fractional order functional differential equations

For a class of impulsive fractional differential equation with fractional order r∈(1,2) by means of solution operator Sr(t), we develop the mild solution and deal with the existence results of mild solution for the considered equation with state-dependent delay through three fixed point theorems, namely Banach, Nonlinear Leray–Schauder Alternative and Krasnoselskii’s fixed point theorem. Next, we define the mild solution via an operator function Wr,β(τ) for abstract version of the considered equation. Further, we study the trajectory controllability (which is a stronger notion than other controllability) of the equation. In the last, we present some examples to illustrate the established results through partial fractional differential equations.

Controllability and observability for linear quaternion-valued impulsive differential equations

In this paper, the controllability and observability for linear quaternion-valued impulsive differential equations (QIDEs) are investigated. We mainly establish some sufficient and necessary conditions for state controllability and state observability of linear QIDEs. By virtue of the isomorphism between quaternion vector space and complex variables space as well as the adjoint matrix of quaternion matrix, all the theoretical results in the sense of complex-valued and quaternion-valued are equivalent to each other. Finally, the validity of theoretical results acquired is indicated by two instances shown.

Finite-time controllability and stabilization of probabilistic logical systems with state-dependent constraint via subset transition method

Constraints are very common for practical control systems. For logical systems, the existing technique of pre-feedback is an effective way of treating state-dependent constraints in control when the state is measurable. However, it is inapplicable for the case when measurement information is not available. In this situation, in order for the control input to not violate the state-dependent constraint, the control at each step must be selected from the common admissible controls of all possible states. Motivated by this observation, in this study, we propose a novel technique, termed the subset transition method, for finite-time controllability and stabilization of probabilistic logical dynamic control system (PLDCS) with a state-dependent control constraint. The main idea of this method is to construct an unconstrained deterministic logical control system over the power set of the state space, called the subset transition system (SubSTS), characterizing the transitional dynamics between subsets under common admissible controls. We prove that a control sequence is admissible with respect to all states in an initial subset if and only if it does not steer the SubSTS from the initial subset to the empty set. Based on this, necessary and sufficient conditions for set controllability and set stabilizability are obtained. Examples are presented to demonstrate the application of the obtained results.

Controllability of a Class of Heterogeneous Networked Systems

This paper examines the controllability of a class of heterogeneous networked systems where the nodes are linear time-invariant systems (LTI), and the network topology is triangularizable. The literature contains necessary and sufficient conditions for the controllability of such systems where the control input matrices are identical in each node. Here, we extend this result to a class of heterogeneous systems where the control input matrices are distinct in each node. Additionally, we discuss the controllability of a more general system with triangular network topology and obtain necessary and sufficient conditions for controllability. Theoretical results are supplemented with numerical examples.

Structural controllability in timed continuous Petri nets

This work deals with the controllability analysis in timed continuous Petri nets (TCPNs) under infinite server semantics, a class of linear hybrid systems. In the literature, this problem has been addressed by analyzing each configuration (that defines a linear mode) of the system. However, the number of configurations may grow exponentially. Here, by using a global structural approach, we study the net rank-controllability (NRC), a structural property of the TCPN. Under the assumption of liveness, it is shown that NRC is a sufficient condition for controllability; nevertheless, if liveness is not fulfilled then controllability is not guaranteed by NRC The advantage of this approach is that NRC is characterized in terms of global structural objects of the net, thus, avoiding the analysis by configurations. In this sense, some new structural sufficient conditions for NRC are introduced for general TCPNs. Finally, polynomial-time algorithms for the verification of NRC are provided.

Controllability Criteria on Discrete-Time Impulsive Hybrid Systems With Input Delay

Due to the importance of the hybrid systems, the controllability for linear discrete-time impulsive hybrid systems with input delay (DIHSID) is investigated by resorting to the geometric and algebraic analytical methods in this article. First, the null reachability and controllability geometric conditions are studied. Specifically, the null reachable set and controllable set for every impulsive and switching sequence are obtained by the properties of the invariant subspace. Besides, a new subspace sequence is constructed to analyze the null reachability and controllability of DIHSID. Second, the complete controllability of DIHSID is investigated by algebraic methods. In the form of Gramian matrices, several sufficient complete controllability conditions for DIHSID are established without assuming the nonsingularity of all impulsive matrices. Furthermore, a less conservative complete controllability criterion that is necessary and sufficient is developed by introducing a row matrix of some Gramian matrices. Finally, two illustrating examples show the effectiveness of the developed controllability theory.

Improved multi-agent controllability processing technique based on equitable partition

In this paper, we study the controllability of multi-agent systems by equitable partition and automorphism. For the case that cells are incompletely connected outside but completely connected inside, a necessary condition for controllability is given from the perspective of the rank of connection matrix. For the case of multiple cells being completely connected outside and incompletely connected inside, in terms of the eigenvalues and eigenvectors of L and Lπ, several sufficient and necessary conditions for controllability are presented. Once the quotient graph is controllable under single input or all nodes in nontrivial cells are leaders, the lower bound of controllable subspace is determined. Finally, we give the gap between the necessary condition and the sufficient condition for controllability from the aspect of equitable partition. One highlight of the results in this paper is that we show sufficient conditions to judge controllability by equitable partition and automorphism, which, for specific cases, provides one method that how to break through the defect that equitable partition can only obtain necessary conditions.

Relative Controllability for a Class of Linear Impulsive Systems

Hybrid systems are systems that involve continuous and discrete event dynamical behaviors. A impulsive system is a special hybrid system. The continuous dynamics of impulsive systems are usually described by ordinary differential equations and the discrete event dynamics with instantaneously rapid jumps are described by switching laws. Various complex dynamical phenomena that can be modeled by impulsive systems arise in many areas of modern science and technology such as economics, physics, chemistry, biology, information science, radiotherapy, acupuncture, robotics, neural networks, automatic control, artificial intelligence, space technology, and telecommunications, etc. In modern control theory, controllability is one of the most important dynamical properties of considered impulsive systems, therefore, the controllability problem is regarded as one of the fundamental issues of impulsive systems. The basic questions for controllability of impulsive systems as well as for the ordinary systems without impulses and with control function are to obtain useful criteria that allow us to identify whether given dynamic systems are controllable or not. Up to now there have been being many investigation results for controllability of different kinds of impulsive systems with respect to the terminal state constraints of a point type. The purpose of this paper is to study relative controllability with respect to the terminal state constraint of a general type for a class of linear time-varying impulsive systems. In this paper, several types of criteria for relative controllability of such systems are established by a algebraic method, that is, specially speaking, by the matrix rank method. Some corresponding necessary and sufficient conditions for controllability of linear time-invariant impulsive systems are also obtained more compactly. Meanwhile, for given impulsive systems some equivalent relationships between different kinds of controllability are investigated and our criteria for relative controllability are compared with the existing results. A simple example is given to illustrate the utility of our criteria.