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This paper mainly addresses the guaranteed-performance consensus problem of multi-agent systems (MASs), under the energy constraints and Lipschitz nonlinear dynamics. First, to reach consensus under energy constraints, the consensus protocol and the quadratic performance function are proposed. Afterward, by using the properties of nonsingular transformation matrix and Lipschitz's nonlinear representation, the consensus function conditions are expressed. Moreover, considering the guaranteed-performance cost, this paper designs the quadratic performance function derived from the Riccati inequality and Lyapunov–Krasovskii theorem. Eventually, numerical simulations are implemented to validate the theoretical results. [ABSTRACT FROM AUTHOR]

This study addressed the oscillation problems of half‐linear differential equations with periodic damping. The solution space of any linear equation is homogeneous and additive. Generally, by contrast, the solution space of half‐linear differential equations is homogeneous but not additive. Numerous oscillation and nonoscillation theorems have been devised for half‐linear differential equations featuring periodic functions as coefficients. However, in certain cases, such as applying Mathieu‐type differential equations to control engineering, which is a typical example of the Hill equation, some oscillation theorems cannot be applied. In this study, we established oscillation and nonoscillation theorems for half‐linear Hill‐type differential equations with periodic damping. To prove the results, we used the Riccati technique and the composite function method, which focuses on the composite function of the indefinite integral of the coefficients of the target equation and an appropriate multivalued continuously differentiable function. Furthermore, we discuss the special case of the oscillation constant of a damped half‐linear Mathieu equation. [ABSTRACT FROM AUTHOR]

Inspired by a recent work due to J.-Y. Wu (Potential Anal 58:203–223, 2023), we prove several new compactness criteria for complete Riemannian manifolds via integral radial m-Bakry–Émery Ricci curvatures when m is a positive constant, a negative constant, or infinity. Our results not only generalize the classical compactness criterion via Ricci curvature due to W. Ambrose (Duke Math. J. 24:345–348, 1957), but also generalize a compactness criterion via integral radial Bakry–Émery Ricci curvature due to J.-Y. Wu (Potential Anal 58:203–223, 2023). The key ingredients in proving our results are Riccati inequalities obtained from Bochner–Weitzenböck formulas via m-Bakry–Émery Ricci curvatures and suitable choices of growth conditions on integral radial m-Bakry–Émery Ricci curvatures, potential functions, and norms of potential vector fields. [ABSTRACT FROM AUTHOR]

The characterization of the solution set for a class of algebraic Riccati inequalities is studied. This class arises in the passivity analysis of linear time-invariant control systems. Eigenvalue perturbation theory for the Hamiltonian matrix associated with the Riccati inequality is used to analyze the extremal points of the solution set. [ABSTRACT FROM AUTHOR]

By using some line integrals in terms of the radial m -Bakry–Émery and radial m -modified Bakry–Émery Ricci curvatures, we give various compactness criteria for complete Riemannian manifolds when m is a positive constant, a negative constant, or infinity. Our results not only guarantee the compactness of complete Riemannian manifolds allowing the presence of negative amounts of the radial m -Bakry–Émery and radial m -modified Bakry–Émery Ricci curvatures, but also give new Myers-type theorems via m -Bakry–Émery and m -modified Bakry–Émery Ricci curvatures even the line integrals are reduced to pointwise positive lower bounds on the m -Bakry–Émery and m -modified Bakry–Émery Ricci curvatures. The key ingredients in proving our results are Riccati inequalities obtained from Bochner–Weitzenböck formulas via m -Bakry–Émery and m -modified Bakry–Émery Ricci curvatures. [ABSTRACT FROM AUTHOR]

Single-input, single-output (SISO) nonlinear systems have problems with sectorial dead zone nonlinearities, noise, uncertainties, approximation errors, and external disturbances. Therefore, we developed an interval Type-2 neural network fuzzy adaptive controller (IT2-NNFAC) for satisfactory H-infinity (H8) tracking performance to solve the problems of the SISO system. To adjust the parameters of the proposed IT2-NNFAC, a structure of the fuzzy logic inference system and online adaptive laws are adopted, which are based on the Lyapunov stability criterion and Riccati inequality. All systems with the proposed IT2-NNFAC attenuate the effect of external disturbances on tracking errors at any specified level. In the proposed IT2-NNFAC, all the signals in the closed-loop system guarantee uniform and ultimate boundedness and satisfactory tracking performance with the proper Lyapunov stability criterion and Riccati inequality. H8 tracking responses and the resilience and efficacy of the proposed IT2-NNFAC were proved by testing a mass spring damper system with sectorial dead zone nonlinearities, uncertainties, and external disturbances. [ABSTRACT FROM AUTHOR]

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In this study, a simple hyperplane design method based on the discrete‐time Riccati equation is proposed, and the inter‐relations are investigated for the basic requirement on sliding surface selection, the assumption of stabilisability, the Riccati inequality and the standard Riccati equation. A state and disturbance observer is embedded in discrete‐time sliding mode control (DSMC) to achieve its applicability when only partial system states can be measured. In order to improve the performance of the system in control updating times, a novel event‐triggered DSMC with a state and disturbance observer is proposed. It is shown that the proposed method achieves a quasi sliding mode with a small boundary layer. Simulation examples are presented to show the effectiveness and advantages of the proposed design schemes. [ABSTRACT FROM AUTHOR]

Recently there has been an increasing interest in studying $p(t)$-Laplacian equations, an example of which is given in the following form (|u'(t)|^{p(t)-2}u'(t))'+c(t)|u(t)|^{q(t)-2}u(t)= 0, \quad t>0. In particular, the first study of sufficient conditions for oscillatory solution of $p(t)$-Laplacian equations was made by Zhang (2007), but to our knowledge, there has not been a paper which gives the oscillatory conditions by utilizing Riccati inequality. Therefore, we establish sufficient conditions for oscillatory solution of nonlinear differential equations with $p(t)$-Laplacian via Riccati method. The results obtained are new and rare, except for a work of Zhang (2007). We present more detailed results than Zhang (2007).

This paper deals with the second-order linear differential equation x ″ + a ( t ) x ′ + b ( t ) x = 0 , where a and b are periodic coefficients. The main purpose is to present new criteria which guarantee that all nontrivial solutions are nonoscillatory and that those are oscillatory. Our nonoscillation theorem and oscillation theorem are proved by using the Riccati technique. In our theorem, the composite function of an indefinite integral of b and a suitable multiple-valued continuously differentiable function are focused, and the composite function of them plays an important role. The results obtained here include a result by Kwong and Wong [15] and a result by Sugie and Matsumura [26] . An application to a equation of Whittaker–Hill type is given to show the usefulness of our results. Finally, simulations are also attached to illustrate that our oscillation criterion is sharp. [ABSTRACT FROM AUTHOR]

In this paper, a new agent-based method is proposed to address the speed synchronization problem in the network-connected motors. In this study, DC motor is used, driven by a buck chopper circuit. In the proposed method, the consensus protocol of the leader-following multi-agent system is modified, in order to make consensus on the speed among multiple motors in the network, so that they can attain synchronous speed. In order to have a stable system, a common Lyapunov function is developed such that consensus is said to be reached if the ith agent is controllable and observable. MATLAB is used for the purpose of simulation, and results obtained authorize the proposed methodology. [ABSTRACT FROM AUTHOR]

This study investigates the consensus problem for general linear multi‐agent systems (MAS) via observer‐based dynamic output feedback event‐triggered control. Assume that the relative state information is unknown for each agent and only the relative output feedback between neighbouring agents can be measured. To solve the consensus problem, the authors develop a novel observer‐based event‐triggered approach to design a dynamic relative output feedback control. Two event‐triggering time sequences are constructed. One is the relative output to the observer and another is the observer error to the controller. One can prove that under the proposed control strategies, the consensus problem can be solved via linear matrix inequalities as well as Riccati inequality if the communication graph of the MAS is connected. Furthermore, the Zeno‐behaviour of triggering time sequences can be avoided. Finally, the effectiveness of the proposed strategies is illustrated by a numerical example. [ABSTRACT FROM AUTHOR]

We present an extension of a recent characterisation of diagonal Riccati stability and, using this, extend a result of Kraaijevanger on diagonal Lyapunov stability to Riccati stability of time-delay systems. We also describe a class of transformations that preserve the property of being diagonally Riccati stable and apply these two results to provide novel stability results for classes of time-delay systems. [ABSTRACT FROM AUTHOR]

We consider the Newton–Kleinman method for strongly stabilizable infinite-dimensional systems. Under certain assumptions, the maximal self-adjoint solution to the associated control algebraic Riccati equation is constructed. The constructed solution is also the maximal solution to the corresponding control algebraic Riccati inequality. [ABSTRACT FROM AUTHOR]

In this article, we consider the third-order nonlinear differential equations with functional arguments. By using the Riccati inequality, we find conditions for all solutions to be oscillatory.

In this paper, we consider the mixed nonlinear hyperbolic equations with forcing term via Riccati inequality. Some sufficient conditions for the oscillation are derived by using Young inequality and integral averaging method.

In this article, we study a Riccati inequality that appears in the theory of partial differential equations with p-Laplacian. Our results allow to compare existence and nonexistence of positive solutions for Riccati type inequalities which are associated with equations with different powers p.

This paper investigates the exponential observer design problem for one-sided Lipschitz nonlinear systems. A unified framework for designing both full-order and reduced-order exponential state observers is proposed. The developed design approach requires neither scaling of the one-sided Lipschitz constant nor the additional quadratically inner-bounded condition. It is shown that the synthesis conditions established include some known existing results as special cases and can reduce the intrinsic conservatism. For design purposes, we also formulate the observer synthesis conditions in a tractable LMI form or a Riccati-type inequality with equality constraints. Simulation results on a numerical example are given to illustrate the advantages and effectiveness of the proposed design scheme. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, we consider the nonlinear hyperbolic equations with forcing term. Some sufficient conditions for the oscillation are derived by using integral averaging method and a generalized Riccati technique. [ABSTRACT FROM AUTHOR]

We investigate the eventual sign changing for the solutions of the linear equation x ( α ) ′ + q ( t ) x = 0 , t ⩾ 0 , when the functional coefficient q satisfies the Kamenev-type restriction lim sup t → + ∞ 1 t ε ∫ t 0 t ( t - s ) ε q ( s ) ds = + ∞ for some ε > 2 , t 0 > 0 . The operator x ( α ) is the Caputo differential operator and α ∈ ( 0 , 1 ) . [ABSTRACT FROM AUTHOR]

Abstract: This paper studies both continuous and discrete time consensus problems for multi-agent systems with linear time-invariant agent dynamics over randomly switching topologies. The switching is governed by a time-homogeneous Markov process, whose state corresponds to a possible interaction topology among agents. Necessary and sufficient conditions are derived for achieving consensus under a common control protocol, respectively. It is shown that the effect of switching topologies on consensus is determined by the union of topologies associated with the positive recurrent states of the Markov process. Moreover, the effect of random link failures on discrete time consensus is investigated. The implications and relationships with the existing results are discussed. Finally, the theoretical results are validated via simulations. [Copyright &y& Elsevier]

Sufficient conditions are obtained for the oscillation of solutions of half-linear and super-half-linear elliptic inequalities withp(x)-Laplacian. The results obtained are new even for the one-dimensional case. [ABSTRACT FROM PUBLISHER]

Abstract: Riccati inequality for half-linear elliptic inequalities with -Laplacians is established, and oscillation criteria are derived by using the Riccati inequality. Our method is to reduce oscillation problems for half-linear elliptic inequalities with -Laplacians to one-dimensional Riccati inequality with variable exponents. Generalizations to more general elliptic inequalities with -Laplacians are also discussed. [Copyright &y& Elsevier]

In this article, we study a Riccati inequality that appears in the theory of partial differential equations with p-Laplacian. Our results allow to compare existence and nonexistence of positive solutions for Riccati type inequalities which are associated with equations with different powers p. [ABSTRACT FROM AUTHOR]

Abstract: This paper is devoted to the study of the oscillatory behavior of solutions of nonlinear hyperbolic equations with functional arguments by using integral averaging method and a generalized Riccati technique. First, we establish oscillation results for nonlinear hyperbolic equations by applying oscillation criteria for Riccati inequality. Secondly, we present interval oscillation results for nonlinear hyperbolic equations. These results improve and extend oscillation results of Cui and Xu . [ABSTRACT FROM AUTHOR]

Abstract: The leader-following consensus problem of higher order multi-agent systems is considered. In the system, the dynamics of each agent and the leader is a linear system. The control of each agent using local information is designed and detailed analysis of the leader-following consensus is presented for both fixed and switching interaction topologies, which describe the information exchange between the multi-agent systems. The design technique is based on algebraic graph theory, Riccati inequality and Lyapunov inequality. Simulations indicate the capabilities of the algorithms. [Copyright &y& Elsevier]

Abstract: A Picone identity is established for a class of half-linear elliptic operators, and Sturmian comparison and oscillation results are derived on the basis of the Picone identity. Wirtinger inequalities and Riccati inequalities are also obtained as corollaries of the Picone identity, and a new class of half-linear elliptic equations is introduced. [Copyright &y& Elsevier]

As for the robust controller design for the precision isolation platform that consists of perturbed identified parameters, an output feedback robust controller design method for a kind of parameter perturbation system is deduced based on the relationship between the robust performance of the parameter perturbation system and solution of the Riccati inequality. The main principle of this method is to transform the robust controller design into the general output feedback robust controller design of an auxiliary system with certain parameters, and the calculation is relatively easier. The simulation results show that the SDOF isolation platform supported by air springs can isolate high frequency vibration form the base, while the performance of isolating low frequency vibration and direct disturbance to the platform is not satisfactory. The closed loop system using the designed robust controller can effectively isolate the low frequency vibration from the base with the uncertain parameters in large range of perturbation, and it also exhibits good robustness as to the direct disturbance and measurement noise. [ABSTRACT FROM AUTHOR]

Analytical approximation of the maximal invariant ellipsoid for discrete-time linear systems with saturated optimal control is established, which is less conservative than existing computationally un-intensive results. Simultaneously, necessary and sufficient conditions for such approximation being equal to the real maximal invariant ellipsoid is presented. All results are given analytically and can easily be implemented in practice. An illustrative example is given to show the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we establish several new results regarding the positivity and nonnegativity of discrete quadratic functionals associated with discrete symplectic systems. In particular, we derive (i) the Riccati inequality for the positivity of with separated endpoints, (ii) a characterization of the nonnegativity of for the case of general (jointly varying) endpoints, and (iii) several perturbation-type inequalities regarding the nonnegativity of with zero endpoints. Some of these results are new even for the special case of discrete Hamiltonian systems. [Copyright &y& Elsevier]

This paper discusses the problem of robust H∞ control for singular impulsive uncertain systems. Some new fundamental properties of singular systems with impulse effect are derived. Based on the Riccati inequality approach, sufficient conditions for robust H∞ criteria of the corresponding singular impulsive closed-loop systems are established. A simple approach to the design of a robust impulsive controller is then presented. Finally, a design example and simulation results are given to demonstrate the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

In this paper we consider the design problem of output feedback H∞ controllers for sampled-data fuzzy systems. We first transfer them into equivalent jump fuzzy systems. We establish the so-called Bounded Real Lemma for jump fuzzy systems and give a design method of γ-suboptimal output feedback H∞ controllers in terms of two Riccati inequalities with jumps. We then apply the main results to the sampled-data fuzzy systems and obtain a design method of γ-suboptimal output feedback H∞ controllers. We give a numerical example and construct a γ-suboptimal output feedback H∞ controller. [Copyright &y& Elsevier]

Some oscillation criteria are given for the partial differential equation with p-Laplacian and the nonlinearity of the Emden–Fowler typediv(∥∇u∥p−2∇u)+c(x)|u|p−2u=0, p>1.The results are extension of integral averaging technique of Kamenev and improve the result of Usami. [Copyright &y& Elsevier]

This paper studies the problem of stochastic stability and disturbance attenuation for a class of linear continuous-time uncertain systems with Markovian jumping parameters. The uncertainties are assumed to be nonlinear and state, control and external disturbance dependent. A sufficient condition is provided to solve the above problem. An H∞ controller is designed such that the resulting closed-loop system is stochastically stable and has a disturbance attenuation γ for all admissible uncertainties. It is shown that the control law is in terms of the solutions of a set of coupled Riccati inequalities. A numerical example is included to demonstrate the potential of the proposed technique. [Copyright &y& Elsevier]

We consider the problem of parametrizing the set of all nonsquare minimal spectral factors of a rational matrix function taking positive semidefinite values on the imaginary axis. [Copyright &y& Elsevier]

Thispaper deals with the H∞ filtering problemfor linear discrete-time two-dimensional (2-D) systems describedby the Roesser model. It firstly establishes a version of thebounded real lemma to give a sufficient condition for quantificationof the H∞ noise attenuation for 2-D systems.Based on the bounded real lemma, the H∞ filteringproblem is investigated for filters of an observer-based structureor a general state equation form and the solutions are obtainedin terms of Riccati inequalities or linear matrix inequalities.The design approach is illustrated by an example of a stationaryfield in image processing. [ABSTRACT FROM AUTHOR]

By using some line integrals in terms of the m -Bakry–Émery and m -modified Bakry–Émery Ricci curvatures, we give various compactness criteria for complete Riemannian manifolds when m is a positive constant, a negative constant, or infinity. Our results generalize previous Myers-type compactness criteria obtained by M. Fernández-López and E. García-Río, M. Limoncu, Z. Qian, G. Wei and W. Wylie, J.-Y. Wu, and W. Wylie, as well as a previous Ambrose-type compactness criterion obtained by K. Kuwae and X.-D. Li. The key ingredients in proving our results are Riccati inequalities obtained from Bochner–Weitzenböck formulas via m -Bakry–Émery and m -modified Bakry–Émery Ricci curvatures. [ABSTRACT FROM AUTHOR]

The robust stability of linear systems with unstructured, structured or non-linear perturbations is investigated in a unifying framework. New sufficient conditions in terms of matrix Riccati inequalities are obtained. Some previously known results are generalized. [ABSTRACT FROM AUTHOR]