Differential Inclusion Systems Research Articles

Trajectory approximately controllability and optimal control for noninstantaneous impulsive inclusions without compactness

In this paper, a noninstantaneous impulsive differential inclusion model is established based on the heating phenomenon of the rod. The controllability problem for this system governed by a semilinear differential inclusion with noninstantaneous impulses is studied in a Banach space and in this differential inclusion system we assume that the semigroup generated by the linear part of the inclusion is not compact. We suppose that the set-valued nonlinearity satisfies a regularity condition expressed in terms of the Hausdorff measure of noncompactness and some sufficient conditions for approximately controllability for both upper and almost lower semicontinuous types of nonlinearity are presented. Also we discuss existence and the stability of optimal control. As an application, the controllability for a differential inclusion system governed by a heat equation is considered.

On Caputo\u2013Hadamard type coupled systems of nonconvex fractional differential inclusions

This research article is mainly concerned with the existence of solutions for a coupled Caputo–Hadamard of nonconvex fractional differential inclusions equipped with boundary conditions. We derive our main result by applying Mizoguchi–Takahashi’s fixed point theorem with the help of mathcal{P}-function characterizations.

CONTROL SYSTEM DEPENDING ON A PARAMETER

A nonlinear control system depending on a parameter is considered in a finite-dimensional Euclidean space and on a finite time interval. The dependence on the parameter of the reachable sets and integral funnels of the corresponding differential inclusion system is studied. Under certain conditions on the control system, the degree of this dependence on the parameter is estimated. Problems of targeting integral funnels to a target set in the presence of an obstacle in strict and soft settings are considered. An algorithm for the numerical solution of this problem in the soft setting has been developed. An estimate of the error of the developed algorithm is obtained. An example of solving a specific problem for a control system in a two-dimensional phase space is given.

Systems of Simultaneous Differential Inclusions Implying Function Containment

An important problem in complex analysis is to determine properties of the image of an analytic function p defined on the unit disc U from an inclusion or containment relation involving several of the derivatives of p. Results dealing with differential inclusions have led to the development of the field of Differential Subordinations, while results dealing with differential containments have led to the development of the field of Differential Superordinations. In this article, the authors consider a mixed problem consisting of special differential inclusions implying a corresponding containment of the form D[p](U)⊂Ω⇒Δ⊂p(U), where Ω and Δ are sets in C, and D is a differential operator such that D[p] is an analytic function defined on U. We carry out this research by considering the more general case involving a system of two simultaneous differential operators in two unknown functions.

Filippov trajectories and clustering in the Kuramoto model with singular couplings

We study the synchronization of a generalized Kuramoto system in which the coupling weights are determined by the phase differences between oscillators. We employ the fast-learning regime in a Hebbian-like plasticity rule so that the interaction between oscillators is enhanced by the approach of phases. First, we study the well-posedness problem for the singular weighted Kuramoto systems in which the Lipschitz continuity fails to hold. We present the dynamics of the system equipped with singular weights in all the subcritical, critical and supercritical regimes of the singularity. A key fact is that solutions in the most singular cases must be considered in Filippov’s sense. We characterize sticking of phases in the subcritical and critical case and we exhibit a continuation criterion for classical solutions after any collision state in the supercritical regime. Second, we prove that strong solutions to these systems of differential inclusions can be recovered as singular limits of regular weights.We also study the emergence of synchronous dynamics for the singular and regular weighted Kuramoto models.

On some coupled systems of fractional differential inclusions

On some coupled systems of fractional differential inclusions

Existence results for coupled system of nonlinear differential equations and inclusions involving sequential derivatives of fractional order

<abstract><p>In this article, we investigate new results of existence and uniqueness for systems of nonlinear coupled differential equations and inclusions involving Caputo-type sequential derivatives of fractional order and along with new kinds of coupled discrete (multi-points) and fractional integral (Riemann-Liouville) boundary conditions. Our investigation is mainly based on the theorems of Schaefer, Banach, Covitz-Nadler, and nonlinear alternatives for Kakutani. The validity of the obtained results is demonstrated by numerical examples.</p></abstract>

Robust Adaptive Control for a Class of T-S Fuzzy Nonlinear Systems with Discontinuous Multiple Uncertainties and Abruptly Changing Actuator Faults

In the complex environment, the suddenly changing structural parameters and abrupt actuator failures are often encountered, and the negligence or unproper handling method may induce undesired or unacceptable results. In this paper, taking the suddenly changing structural parameters and abrupt actuator failures into consideration, we focus on the robust adaptive control design for a class of heterogeneous Takagi–Sugeno (T-S) fuzzy nonlinear systems subjected to discontinuous multiple uncertainties. The key point is that the switch modes not only vary with the system time but also vary with the system states, and the intrinsic heterogeneous characteristics make it difficult to design stable controllers. Firstly, the concepts of differential inclusion are introduced to describe the heterogeneous fuzzy systems. Meanwhile, a fundamental lemma is provided to demonstrate the criteria of the boundness for a Filippov solution. Then, by using the set-valued Lie derivative of the Lyapunov function and introducing a vector of specific continuous functions, the closed-loop T-S fuzzy differential inclusion systems are proved to be ultimately bounded. The sufficient conditions for system stability are derived in term of linear matrix inequalities (LMIs), which can be solved directly. Finally, a numerical example is provided to illustrate the effectiveness of the proposed control algorithm.

Fixed-time synchronization analysis for discontinuous fuzzy inertial neural networks with parameter uncertainties

This paper aims to investigate the fixed-time synchronization analysis for discontinuous fuzzy inertial neural networks in the presence of parameter uncertainties. By using a new variable transformation and differential inclusions theory, we first establish two kinds of drive-response differential inclusion systems. By designing some novel discontinuous control inputs and using Lyapunov-Krasovskii functional approach, some sufficient criteria are derived for achieving fixed-time synchronization, and the corresponding setting times are estimated. The established results provide a new framework to deal with the inertial neural networks with fuzzy logics and discontinuous activation functions. Some previous works in the literature are extended and complement. Finally, two topical simulation examples are given to show the effectiveness of the developed main control schemes.

Iterative learning control for nonlinear differential inclusion systems

Subsequent to publication, errors were identified in the Funding information and Acknowledgements sections of Liu et al.1 These sections have been corrected below. National Natural Science Foundation of China, Grant/Award Number: (11661016; 61673045); Training Object of High Level and Innovative Talents of Guizhou Province, Grant/Award Number: ([2016]4006); Major Research Project of Innovative Group in Guizhou Education Department, Grant/Award Number: ([2018]012); the Slovak Research and Development Agency under the contract No. APVV-18-0308, and the Slovak Grant Agency VEGA No. 1/0358/20 and No. 2/0127/20. The authors thank the referees for their careful reading of the manuscript and insightful comments, which help to improve the quality of the paper. The authors thank the help from the editor too. This work was supported by the National Natural Science Foundation of China (grant numbers 11661016; 61673045), Training Object of High Level and Innovative Talents of Guizhou Province ([2016]4006), Major Research Project of Innovative Group in Guizhou Education Department ([2018]012), the Slovak Research and Development Agency under the contract No. APVV-18-0308, and the Slovak Grant Agency VEGA No. 1/0358/20 and No. 2/0127/20. The Publisher apologizes for this error.

Observer for differential inclusion systems with incremental quadratic constraints

This paper studies an exponential observer design method for a kind of differential inclusion systems. The set-valued term is monotone, while the nonlinear term satisfies incremental quadratic constraints. The extended-type observer framework for the system is proposed, and the existence conditions are obtained by the constraints of linear matrix inequalities and linear matrix equalities. Specially, the algorithm of computing the incremental multiplier matrix is also discussed. The availability of the proposed method is verified by three numerical examples.

Systems of simultaneous differential inequalities, inclusions and subordinations in the complex plane

There are many articles in the literature dealing with a first, second and third-order differential inequalities, inclusions or subordinations in the complex plane, none of which deals with systems of such topics. This article investigates systems of two second-order simultaneous differential inequalities, inclusions and subordinations in two complex functions p and q in the complex plane. A typical example of a system of differential inequalities is $$ \left\{ {\begin{array}{*{20}l} {\text{Re} [2p(z) + zp^{{\prime }} (z) + z^{2} p^{{\prime \prime }} (z) - q(z)] > 0,} \hfill \\ {\text{Re} [p(z) + 7zq^{{\prime }} (z)] > 4.} \hfill \\ \end{array} } \right. $$ The authors determine properties of the functions p and q satisfying some special systems of differential inequalities, and extend their results to differential inclusions and subordinations.

Solvability of non-semicontinuous systems of Stieltjes differential inclusions and equations

We prove an existence result for systems of differential inclusions driven by multivalued mappings which need not assume closed or convex values everywhere, and need not be semicontinuous everywhere. Moreover, we consider differentiation with respect to a nondecreasing function, thus covering discrete, continuous and impulsive problems under a unique formulation. We emphasize that our existence result appears to be new even when the derivator is the identity, i.e. when derivatives are considered in the usual sense. We also apply our existence theorem for inclusions to derive a new existence result for discontinuous Stieltjes differential equations. Examples are given to illustrate the main results.

Existence of Solutions for Some Coupled Systems of Fractional Differential Inclusions

We study two coupled systems of nonconvex fractional differential inclusions with certain nonlocal boundary conditions and we prove the existence of solutions in the case when the set-valued maps are Lipschitz in the state variables.

Iterative learning control for nonlinear differential inclusion systems

SummaryIn this paper, we propose an iterative learning control strategy to track a desired trajectory for a class of uncertain systems governed by nonlinear differential inclusions. By imposing Lipschitz continuous condition on a set‐valued mapping described by a closure of the convex hull of a set and using D‐type and PD‐type updating laws with initial iterative learning, we establish the iterative learning process and give a new convergence analysis with the help of Steiner‐type selector. Finally, numerical examples are provided to verify the effectiveness of the proposed method with suitable selection of set‐valued mappings. An application to the speed control of robotic fish is also given.

A study of coupled systems of mixed order fractional differential equations and inclusions with coupled integral fractional boundary conditions

In this work we investigate existence and uniqueness of solutions for new coupled systems of mixed order fractional differential equations and inclusions supplemented with coupled nonlocal fractional boundary conditions. We apply the Leray–Schauder alternative and the Banach contraction mapping principle to obtain the existence and uniqueness results, while in the multi-valued case we use the nonlinear alternative for Kakutani maps and Covitz and Nadler’s fixed point theorem.

On Fixed Points of Multivalued Mappings in Spaces with a Vector-Valued Metric

Nadler’s theorem on a fixed point of a multivalued mapping is extended to spaces with a vector-valued metric. A vector-valued metric is understood as a mapping with the properties of a usual metric and values in a linear normed ordered space. We prove an analog of Nadler’s theorem and apply it to a system of integral inclusions in a space of summable functions. Then we study a boundary value problem with multivalued conditions for systems of functional differential inclusions by means of reduction to a system of integral inclusions. Conditions for the existence of solutions are obtained and estimates of the solutions are given. The existence conditions do not contain the convexity requirement for the values of the multivalued function generating a Nemytskii operator.

Observer design for stochastic one-sided Lipschitz Lur'e differential inclusion system

ABSTRACTThis paper presents an observer design for a stochastic one-sided Lipschitz Lur'e differential inclusion system. The feature of the system is that a nonlinear term satisfies one-sided Lipschitz condition and elements of a transition rates matrix are partially unknown. First, a framework of a Luenberger-type observer with stochastic switching parameters is presented. Then, the conditions which guarantee that the error system is exponentially stable in mean square are derived. Finally, the simulation with a rotor system is conducted to prove the effectiveness of the proposed observer design.

Topological method for coupled systems of impulsive stochastic semilinear differential inclusions with fractional Brownian motion

Topological method for coupled systems of impulsive stochastic semilinear differential inclusions with fractional Brownian motion

Interval State Estimation of Linear Multicellular Systems

Linear multicellular system is a type of differential inclusion system, which can be deemed as an extension of linear control system with set‐valued mapping. As an important issue in existing control systems, interval state estimation has been widely applied in engineering practices. Over the years, the objects of the studies on interval state estimation have been extended from the initial linear time‐invariant systems to linear time‐varying systems, chaotic systems, feedback linearization systems, and nonlinear Lipschitz systems. However, there is no report on the design of interval observer for linear multicellular system. To make up for this gap, this chapter attempts to explore the design of an interval observer for linear multicellular systems.