Sufficient Conditions For Controllability Research Articles

A study on approximate controllability of linear impulsive equations in Hilbert spaces

In this paper, we study an approximate controllability for the impulsive linear evolution equations in Hilbert spaces. We give a representation of solution in terms of semigroup and impulsive operators. We present the necessary and sufficient conditions for approximate controllability of linear impulsive evolution equation in terms of impulsive resolvent operator. An example is provided to illustrate the application of the obtained results.

Fast finite-time time-varying formation tracking control of multi-agent systems with external disturbance and region constraints

In this paper, fast finite-time time-varying formation tracking control for second-order multi-agent systems with external disturbance and region constraints is investigated. The ubiquitous control input response delay is considered. A predictor-based state transformation is employed to obtain the ideal equivalent systems without control input response delay. Considering the difficulty of measuring external disturbance, a fast finite-time observer with the sliding mode technique is proposed to estimate the disturbance, which positively compensates the system. By adopting the sliding mode control strategy, a novel fast finite-time formation tracking controller is proposed. Sufficient conditions for the formation tracking control with region constraints are obtained. Numerical simulations are employed to validate the feasibility and effectiveness of the theoretical results.

Control theory for fractional differential Sylvester matrix equations with Caputo fractional derivative

In this manuscript, we introduce the concept of first-time controllability and observability concerning fractional differential Sylvester matrix equations employing the Caputo fractional derivative. Our work establishes necessary and sufficient conditions for controllability and observability, wherein controllability equates to having a controllability matrix with full rank and observability aligns with an observability Gramian matrix that is nonsingular. Furthermore, we provide several theorems addressing the observability of fractional differential Sylvester matrix equations. Finally, we offer two illustrative examples to demonstrate the efficacy and application of our established results.

Controllability and observability of conformable fractional finite dimensional linear systems

In this research paper, we investigate the controllability and observability of continuous-time fractional dynamical systems. The characterisation of all fractional continuous one-parameter semi groups on a finite dimensional vector space as fractional matrix-valued exponential functions is given. This characterisation is used to express the solution of continuous-time fractional dynamical system. Furthermore, we analyse controllability and observability of linear conformable fractional systems in the mathematical context of linear operators on a finite dimensional vector space. By defining the controllability and observability maps, we establish necessary and sufficient conditions for controllability and observability of this type of systems. We also present the Hautus test for controllability and observability, and introduce controllability and observability Gramians, both for finite and infinite time interval. Finally, we present several illustrative applications and examples to validate all the obtained results.

An investigation on the partial approximate controllability results for nonlocal neutral fractional differential systems via approximation method

This article primarily investigates the existence and partial approximate controllability of Caputo fractional neutral differential systems with nonlocal conditions in Hilbert spaces. It employs various techniques, including fractional calculus, semigroup theory, variational approach, approximation methods, and the Schauder's fixed-point theorem. New sufficient conditions for partial approximate controllability of the control system are derived, eliminating the need for compactness or Lipschitz conditions on the nonlocal function. Additionally, an example is provided to demonstrate the obtained results.

Model‐free distributed optimal control for general discrete‐time linear systems using reinforcement learning

AbstractThis article proposes a novel data‐driven framework of distributed optimal consensus for discrete‐time linear multi‐agent systems under general digraphs. A fully distributed control protocol is proposed by using linear quadratic regulator approach, which is proved to be a sufficient and necessary condition for optimal control of multi‐agent systems through dynamic programming and minimum principle. Moreover, the control protocol can be constructed by using local information with the aid of the solution of the algebraic Riccati equation (ARE). Based on the Q‐learning method, a reinforcement learning framework is presented to find the solution of the ARE in a data‐driven way, in which we only need to collect information from an arbitrary follower to learn the feedback gain matrix. Thus, the multi‐agent system can achieve distributed optimal consensus when system dynamics and global information are completely unavailable. For output feedback cases, accurate state information estimation is established such that optimal consensus control is realized. Moreover, the data‐driven optimal consensus method designed in this article is applicable to general digraph that contains a directed spanning tree. Finally, numerical simulations verify the validity of the proposed optimal control protocols and data‐driven framework.

Approximate controllability of impulsive evolution stochastic functional differential equations driven by a fractional Brownian motion

In this paper, we study the approximate controllability of certain class of impulsive evolution stochastic functional differential equations, with variable delays, driven by a fractional Brownian motion in a separable real Hilbert space. We derive a new set of sufficient conditions for approximate controllability using a stochastic analysis of fractional Brownian motion with Hurst parameter H ∈ (1/2,1) and a Schaefer's fixed point theorem. An example is considered at the end of the paper to illustrate the obtained abstract results.

Results on partial approximate controllability of fractional control systems in Hilbert spaces with conformable derivatives

This paper investigates partial approximate controllability of fractional linear and semi-linear control systems involving the conformable derivative. First, we derive sufficient and necessary conditions for partial approximate controllability of fractional linear systems over a small interval. Then, we apply a new approach, different from typical methods involving complexity inequalities and fixed point theorems, to obtain sufficient conditions for partial approximate controllability of semi-linear fractional control systems, assuming that the linear part is partially approximately controllable. The effectiveness of these results is demonstrated through an illustrative example.

On the Inherent Distributional Robustness of Stochastic and Nominal Model Predictive Control

We define a notion of distributional robustness, via the Wasserstein metric, for closed-loop systems subject to errors in the disturbance distribution used to construct the controller. We then establish sufficient conditions for stochastic model predictive control (SMPC) to satisfy this definition of distributional robustness and establish a similar notion of distributional robustness for economic applications of SMPC. These results address incorrectly or unmodeled disturbances, demonstrate the efficacy of scenario optimization as a means to approximate and solve the SMPC problem, and unify the descriptions of robustness for stochastic and nominal model predictive control. This definition of distributional robustness for closed-loop systems is general and can be applied to other stochastic optimal control algorithms and, potentially, the developing field of distributionally robust control.

Formation containment control of multi‐agent systems with actuator and sensor faults

AbstractThis article investigates the fault‐tolerant formation containment control of multi‐agent systems with actuator and sensor faults. First, adaptive descriptor observers are proposed for the followers to estimate system states, actuator faults, and sensor faults simultaneously. Then, distributed fault‐tolerant formation containment controllers are designed based on the estimated values and sufficient conditions for the formation containment control are identified using Lyapunov method in the time domain. Finally, the effectiveness and advantages of the proposed method are demonstrated through numerical simulations.

Processing the Controllability of Control Systems with Distinct Fractional Derivatives via Kalman Filter and Gramian Matrix

In this paper, we investigate the controllability conditions of linear control systems involving distinct local fractional derivatives. Sufficient conditions for controllability using Kalman rank conditions and the Gramian matrix are presented. We show that the controllability of the local fractional system can be determined by the invertibility of the Gramian matrix and the full rank of the Kalman matrix. We also show that the local fractional system involving distinct orders is controllable if and only if the Gramian matrix is invertible. Illustrative examples and an application are provided to demonstrate the validity of the theoretical findings.

Controllability of Multilayer Networked Sampled-Data Systems.

The controllability analysis of networked systems is challenging due to their high dimensionality and complex structure. The influence of sampling on network controllability is rarely studied, making it an important topic to explore. In this article, the state controllability of multilayer networked sampled-data systems is studied, considering the deep network structure, multidimensional node dynamics, various inner couplings, and sampling patterns. Necessary and/or sufficient controllability conditions are proposed and validated by numerical and practical examples, requiring less computation than the classic Kalman criterion. Single-rate and multirate sampling patterns are analyzed, showing that adjusting the sampling rate of local channels can affect the controllability of the overall system. It is shown that the pathological sampling of single-node systems can be eliminated by an appropriate design of interlayer structures and inner couplings. In the case of systems with drive-response mode, the overall system may not lose controllability even when the response layer is uncontrollable. The results demonstrate that mutually coupled factors collectively affect the controllability of the multilayer networked sampled-data system.

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.