Sufficient Conditions For Existence Research Articles

Distributed Minimum-Energy Containment Control of Continuous-Time Multi-Agent Systems by Inverse Optimal Control

Dear Editor, This letter focuses on the distributed optimal containment control of continuous-time multi-agent systems (CTMASs) with respect to the minimum-energy performance index over fixed topology. To achieve this, we firstly investigate the optimal containment control problem using the inverse optimal control method, where all states of followers asymptotically converge to the convex hull spanned by the leaders while some quadratic performance indexes get minimized. A sufficient condition for existence of the distributed optimal containment control protocol is derived. By introducing the parametric algebraic Riccati equation (PARE), it is strictly proved that the global performance index can be used to approximate the standard minimum-energy performance index as the parameters tends to infinity. In consequence, the standard minimum-energy cooperative containment control can be solved by local steady state feedback protocols. Finally, numerical examples are given to demonstrate the effectiveness of theoretical results.

On angular limits of quasiregular mappings

We investigate Lindelöf and Koebe type boundary behavior results for bounded quasiregular mappings in n-dimensional Euclidean space. Our results give sufficient conditions for the existence of non-tangential limits at a boundary point.

Uniform bounds of families of analytic semigroups and Lyapunov Linear Stability of planar fronts

AbstractWe study families of analytic semigroups, acting on a Banach space, and depending on a parameter, and give sufficient conditions for existence of uniform with respect to the parameter norm bounds using spectral properties of the respective semigroup generators. In particular, we use estimates of the resolvent operators of the generators along vertical segments to estimate the growth/decay rate of the norm for the family of analytic semigroups. These results are applied to prove the Lyapunov linear stability of planar traveling waves of systems of reaction–diffusion equations, and the bidomain equation, important in electrophysiology.

Proportional integral observer‐based robust fault‐tolerant consensus for nonlinear two‐time‐scale‐agent systems with heterogeneous sensor faults

AbstractThis article studies the fault‐tolerant consensus issue of a class of nonlinear two‐time‐scale‐agent systems with heterogeneous sensor faults and external disturbances. First, the decentralized proportional integral (PI) observer for each agent is constructed to estimate the states and sensor faults simultaneously. Then, a distributed adaptive protocol based on the decentralized PI observer is designed to guarantee the consensus of the underlying systems. Some sufficient existence conditions are given in terms of linear matrix inequalities (LMIs) for the decentralized PI observer and the fault tolerant consensus protocol. Note that performance index is employed to attenuate the influence of sensor faults and disturbances on state estimation and the consensus of the underlying systems. Finally, the effectiveness and superiority of the proposed method are demonstrated by an application example and a numerical example.

Quasilinear Lane–Emden type systems with sub-natural growth terms

Global pointwise estimates are obtained for quasilinear Lane–Emden-type systems involving measures in the “sublinear growth” rate. We give necessary and sufficient conditions for existence expressed in terms of Wolff’s potential. Our approach is based on recent advances due to Kilpeläinen and Malý in the potential theory. This method enables us to treat several problems, such as equations involving general quasilinear operators and fractional Laplacian.

An analysis on asymptotic stability of Hilfer fractional stochastic evolution equations with infinite delay

This article deals with the existence and asymptotic stability in the p-th moment of a mild solution for Hilfer fractional stochastic delay differential equations in Hilbert spaces. Our main results are obtained by fractional calculus, semigroup theory, stochastic analysis, and the Banach fixed point theorem. Further, a new set of sufficient conditions for existence and asymptotic stability in the p-th moment is derived with the help of a fixed point technique. Additionally, we provide an example to demonstrate the reliability of our results.

Necessary and sufficient condition for existence for a case of eigenvalues of multiplicity two

We establish necessary and sufficient condition for existence of solutions for a class of semilinear Dirichlet problems with the linear part at resonance at eigenvalues of multiplicity two. The result is applied to give a condition for unboundness of all solutions of the corresponding semilinear heat equation.
 For more information see https://ejde.math.txstate.edu/Volumes/2024/09/abstr.html

Observer‐based fault tolerant control for a class of nonlinear multi‐agent systems with sensor faults

AbstractThis article focuses on the robust fault tolerant control (FTC) problem for a class of Lipschitz nonlinear multi‐agent systems(MASs) subject to sensor faults. Firstly, sensor faults are transformed into actuator faults via introducing a new intermediate auxiliary state variable, and a distributed adaptive fault estimation observer is designed to estimate the state information and the concerned faults by using the relative output estimation error. Then, the sufficient existence conditions for the observer to satisfy the robust performance index are given. Thirdly, based on the results of observer design, a new design method of dynamic output feedback controller is proposed to implement consensus of MASs and ensure the desired disturbance rejection performance. Finally, the simulation results are presented to illustrate the effectiveness of the proposed method.

ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF TWO-PART DIFFERENTIAL EQUATIONS WITH EXPONENTIAL NONLINEARITY

For a two-term nonautonomous ordinary differential equation of the fourth order with an exponential nonlinearity of the form y(4) = α0p0(t)[1 + r(t)]eσy where α0 ∈ {−1,1}, σ ≠ 0, the function p0(t) is continuous or continuously differentiable and nonzero in some left neighbourhood of ω (ω ≤ +∞), r(t) is a continuous function such that limt↑ωr(t) = 0, the asymptotic behaviour at t↑ω of one class of Pω(Y0, λ0)-solutions is studied. For this equation, in [1], the necessary and sufficient conditions for the existence of such solutions were obtained in the case when λ0 ∈ R∖ {0, 1/2, 2/3, 1}. The proof of sufficient conditions for existence was carried out under some additional conditions that are quite strict. The aim of this paper is to improve the results obtained in [1] for sufficient conditions of existence. An attempt is made to extend the results of this paper to conditions that are less stringent. In contrast to [1], the proof of the main result in this paper assumes that there is a finite or infinite limit limt↑ω πω(t)q’(t). The equation under study is reduced to a system of equations for which it is necessary to determine the existence of solutions vanishing at infinity. This fact is established using the known results of [2]. The question of the number of solutions of the equation with the found asymptotic images is also solved.

The space of Hardy-weights for quasilinear equations: Maz’ya-type characterization and sufficient conditions for existence of minimizers

The space of Hardy-weights for quasilinear equations: Maz’ya-type characterization and sufficient conditions for existence of minimizers

On polylinear differential realization of the determined dynamic chaos in the class of higher order equations with delay

The investigation has defined the characteristic criterion (and its modification) of solvability of the problem of differential realization of the bundle of controlled trajectory curves of determined chaotic dynamic processes in the class of bilinear non-autonomous ordinary second- and higher-order differential equations (with and without delay) in the separable Hilbert space. The problem statement under consideration belongs to the type of converse problems for the additive combination of nonstationary linear and bilinear operators of the evolution equation in the Hilbert space. The constructions of tensor products of the Hilbert spaces, structures of lattices with an orthocomplement, the theory of extension of M2 -operators and the functional apparatus of the entropy Relay Ritz operator represent the basis of this theory. It has been shown that in the case of the finite bundle of the controlled trajectory curves the existence of the property of sub-linearity of the given operator allows one to obtain sufficient conditions of existence of such realizations. Side by side with solving the main problems, grounded are topological-group conditions of continuity of projectivization of the Relay Ritz operator with computing the fundamental group (Poincare group) of its compact image. The results obtained give incentives for the development of the quantitative theory of converse problems of higher-order multilinear evolution equations with the operators of generalized delay describing, for example, differential modeling of nonlinear Van der Pol oscillators or Lorentz strange attractors.

Global existence and blow-up in higher-dimensional Patlak-Keller-Segel system for multi populations

This work is concerned with a degenerate parabolic-elliptic Patlak-Keller-Segel system for multi populations in space dimensions d≥3. We first provide a sufficient condition for the global existence of weak solution to the Cauchy problem. Then the global results are obtained both for any large initial data in the sub-critical case and for small initial data in the super-critical case. Finally, the finite-time blow-up solutions are constructed for large initial data in the super-critical case.

Constrained evolutionary variational–hemivariational inequalities with application to fluid flow model

We study a new class of abstract evolution variational–hemivariational inequalities with constraints in the framework of evolution triples of spaces. Existence of solution is established based on the so-called mixed equilibrium problem formulation. Then, we apply this result to a mathematical analysis of the unsteady Oseen model for a generalized Newtonian incompressible fluid. A variational–hemivariational inequality for the flow problem is derived and sufficient conditions for existence of weak solutions are obtained. The mixed boundary conditions involve a unilateral boundary condition, the Navier slip condition, a nonmonotone version of the nonlinear Navier–Fujita slip condition, and the threshold slip and leak condition of frictional type.

Modern services led growth and development in a structuralist dual economy: Long‐run implications of skilled labor constraint

AbstractMotivated by the South Asian experience, this paper examines the role of expansion of skilled labor force for modern services led growth and development in economies with low average educational attainment. The model economy consists of two capital‐using sectors—a service sector that employs only skilled labor and an industry sector that does not require skilled labor. The service sector represents modern skilled‐labor intensive services whereas the industry sector represents a typical South Asian industry. Supply of unskilled labor is unlimited but skilled labor is relatively scarce and grows at a finite rate. Increase in the skill premium only partially explains growth of skilled‐labor supply while the rest depends on autonomous factors, which may be influenced by education policies of the government. The main result specifies a set of necessary and sufficient conditions for existence of a unique steady state characterized by balanced sectoral growth at a rate determined by the autonomous part of skilled labor growth. A low wage share and stagnant investment conditions in the industry sector are conducive for both existence and local stability of this steady state. The model shows that supply‐side factors may completely determine steady state growth rates in structuralist models despite presence of unemployed resources.

Existence and uniqueness of limits at infinity for homogeneous Sobolev functions

We establish the existence and uniqueness of limits at infinity along infinite curves outside a zero modulus family for functions in a homogeneous Sobolev space under the assumption that the underlying space is equipped with a doubling measure which supports a Poincaré inequality. We also characterize the settings where this conclusion is nontrivial. Secondly, we introduce notions of weak polar coordinate systems and radial curves on metric measure spaces. Then sufficient and necessary conditions for existence of radial limits are given. As a consequence, we characterize the existence of radial limits in certain concrete settings.

Bacterial Competition in the Presence of a Virus in a Chemostat

We derive a mathematical model that describes the competition of two populations in a chemostat in the presence of a virus. We suppose that only one population is affected by the virus. We also suppose that the substrate is continuously added to the bioreactor. We obtain a model taking the form of an “SI” epidemic model using general increasing growth rates of bacteria on the substrate and a general increasing incidence rate for the viral infection. The stability of the steady states was carried out. The system can have multiple steady states with which we can determine the necessary and sufficient conditions for both existence and local stability. We exclude the possibility of periodic orbits and we prove the uniform persistence of both species. Finally, we give some numerical simulations that validate the obtained results.

A survey on local integrability and its regularity

In this survey paper, we summarize our results and also some related ones on local integrability of analytic autonomous differential systems near an equilibrium. The results are on necessary conditions related to existence of local analytic or meromorphic first integrals, on existence of analytic normalization of local analytically integrable system, and also on some sufficient conditions for existence of local analytic first integrals. Among which the results are also on regularity of the local first integrals, including analytic and Gevrey smoothness. We also present some open questions for further investigation.

A sufficient condition for global existence of the solution to nonlinear damped wave equations at arbitrary positive initial energy

AbstractWe consider the global existence of the solution for nonlinear damped wave equations at arbitrary positive initial energy, that is, . It is well known from a work of G. Filippo and S. Marco [Ann. Inst. H. Poincaré Anal. Non Linéaire, 2006], that which initial data with high energy make the model exists global solutions is still an interesting question. Here, we give it an affirmative answer. For this purpose, we suggest a unified approach, which is also valid where various forms of damping are present. Applying it, we show a sufficient condition to get the global existence of the solution for problem nonlinear damped wave equations.

On the Fisher‐KPP model with nonlocal nonlinear sources

AbstractThe Cauchy problem considered in this paper is the following: where . When the coefficient remains positive, (1) is analogous to It is well known that when , the local solution of (2) blows up in finite time as long as the initial value is nontrivial. The main aim of this paper is to obtain sufficient conditions for global existence of solutions to (1) for and thus it forms a contrast to (2). The exponent produces an exact balance in the mass‐invariant scaling of diffusion and growth in (1). In particular, we prove that if , and where is the sharp constant of the inequality , then (1) admits a global classical solution. Moreover, the long time behavior of the solution is also discussed. In addition, we prove that when , the problem is solvable globally in time without any restriction on the initial data.

Global Existence and Blow-up Solutions for a Parabolic Equation with Critical Nonlocal Interactions

AbstractIn this paper, we study the initial boundary value problem for the nonlocal parabolic equation with the Hardy–Littlewood–Sobolev critical exponent on a bounded domain. We are concerned with the long time behaviors of solutions when the initial energy is low, critical or high. More precisely, by using the modified potential well method, we obtain global existence and blow-up of solutions when the initial energy is low or critical, and it is proved that the global solutions are classical. Moreover, we obtain an upper bound of blow-up time for $$J_{\mu }(u_{0})<0$$ J μ ( u 0 ) < 0 and decay rate of $$H^{1}_{0}$$ H 0 1 and $$L^{2}$$ L 2 -norm of the global solutions. When the initial energy is high, we derive some sufficient conditions for global existence and blow-up of solutions. In addition, we are going to consider the asymptotic behavior of global solutions, which is similar to the Palais-Smale (PS for short) sequence of stationary equation.