Nonlinear Differential Inequalities Research Articles

An anti-windup INDI fault-tolerant control scheme for flying wing aircraft with actuator faults

In this paper, an anti-windup incremental nonlinear dynamic inversion (INDI) fault-tolerant scheme is proposed for flying wing aircraft with actuator faults, actuator saturation and uncertainties of aerodynamic parameters. An optimal anti-windup compensator based on nonlinear partial differential inequalities is used to compensate the actuator saturation. INDI is used to control the fault system and compensate the uncertainties of the flight dynamics. Control allocation strategy is designed in consideration of the control scheme and configuration of the control surfaces. The proposed control method can guarantee the bounded tracking of the reference signals. Simulation results are given to show the effectiveness of the proposed method.

Second order differential variational inequalities involving anti-periodic boundary value conditions

This paper deals with a new class of second order nonlinear differential variational inequalities. Firstly, we study the properties of a solution set for a variational inequality by using the KKM technique and a monotonicity property. Then, we prove the existence of solutions to a second order nonlinear differential variational inequality with anti-periodic boundary value conditions not assuming the Lipschitz continuity of the nonlinear term. Our approaches, which are essentially different from previous research methods in differential variational inequalities, are based on the topological degree theory and the Filippov implicit function lemma.

Nonlinear impulsive differential and integral inequalities with nonlocal jump conditions

Some new nonlinear impulsive differential and integral inequalities with nonlocal integral jump conditions are presented in this paper. Using the method of mathematical induction, we obtain a new upper bound estimation of certain differential and integral inequalities; these inequalities have both nonlocal integral jump and weakly singular kernels. Finally, we give two examples of these inequalities in estimating solutions of certain equations with Riemann–Liouville fractional integral conditions.

Weighted Rellich type inequalities related to Baouendi-Grushin operators

We find a simple sufficient criterion on a pair of nonnegative weight functions a ( x , y ) a\left ( x,y\right ) and b ( x , y ) b\left ( x,y\right ) in R m + k \mathbb {R} ^{m+k} so that the general weighted L p L^{p} Rellich type inequality ∫ R m + k a ( x , y ) | Δ γ u ( x , y ) | p d x d y ≥ ∫ R m + k b ( x , y ) | u ( x , y ) | p d x d y \begin{equation*} \int _{\mathbb {\mathbb {R}}^{m+k}}a\left ( x,y\right ) \left \vert \Delta _{\gamma }u\left ( x,y\right ) \right \vert ^{p}dxdy\geq \int _{\mathbb {\mathbb {R}}^{m+k}}b\left ( x,y\right ) \left \vert u\left ( x,y\right ) \right \vert ^{p}dxdy \end{equation*} holds for all u ∈ C 0 ∞ ( R m + k ) u\in C_{0}^{\infty }(\mathbb {R}^{m+k}) . Here Δ γ = Δ x + | x | 2 γ Δ y \Delta _{\gamma }=\Delta _{x}+|x|^{2\gamma }\Delta _{y} is the Baouendi-Grushin operator, γ > 0 , \gamma >0, m , k ≥ 1 m,k\geq 1 and p > 1 p>1 . It is important to point out here that our approach is constructive in the sense that it allows us to retrieve already established weighted sharp Rellich type inequalities as well as to get other new results with an explicit constant on R m + k . \mathbb {\mathbb {R}}^{m+k}. We also obtain a sharp L p L^{p} Rellich type inequality that connects first to second order derivatives and several new two-weight Rellich type inequalities with remainder terms on smooth bounded domains Ω \Omega in R m + k \mathbb {\mathbb {R}}^{m+k} via a nonlinear differential inequality.

New criterion for asymptotic stability of time-varying dynamical systems

In this paper, we establish some new sufficient conditions for uniform global asymptotic stability for certain classes of nonlinear systems. Lyapunov approach is applied to study exponential stability and stabilization of time-varying systems. Sufficient conditions are obtained based on new nonlinear differential inequalities. Moreover, some examples are treated and an application to control systems is given.

General weighted Hardy type inequalities related to Baouendi-Grushin operators

ABSTRACTIn this paper, we derive a sufficient condition on a pair of nonnegative weight functions and w in so that the general weighted Hardy type inequality with a remainder termholds for all . Here and is the sub-elliptic gradient. It is worth emphasizing here that our unifying method may be readily used to recover most of the previously known sharp weighted Hardy and Heisenberg–Pauli–Weyl type inequalities as well as to construct other new inequalities with an explicit constant. Furthermore, we also obtain new results on two-weight Hardy type inequalities with remainder terms on smooth bounded domains in via a non-linear partial differential inequality.

Global existence and estimates of the solutions to nonlinear integral equations

It is proved that a class of nonlinear integral equations of the Volterra-Hammerstein type has a global solution, that is, solutions defined for all \(t\ge 0\), and estimates of these solutions as \(t\to \infty\) are obtained. The argument uses a nonlinear differential inequality which was proved by the author and has broadapplications.

On the nonexistence of solutions of some nonlinear inequalities with nonintegral powers of the Laplace operator

Some results about the nonexistence of nontrivial nonnegative solutions for some classes of nonlinear partial differential inequalities containing nonintegral powers of the Laplace operator are obtained.

Blow-up of solutions of some nonlinear inequalities in unbounded domains

Results concerning the blow-up of nontrivial nonnegative solutions are obtained for several classes of nonlinear partial differential inequalities and systems in unbounded domains with coefficients having singularities near the boundary of the domain.

Non-existence of solutions for nonlinear differential inequalities with singularities on the boundary

In this paper, we prove some new Liouville theorems for nonlinear differential inequalities with two gradient terms and singular variable coefficients. The proofs are based on the test function method developed by Mitidieri and Pohozaev.

Multi-valued stochastic differential equations driven by G-Brownian motion and related stochastic control problems

ABSTRACTIn this paper, we prove the existence and uniqueness of a solution for a class of multi-valued stochastic differential equations driven by G-Brownian motion (MSDEG) by means of the Yosida approximation method. Moreover, we set up an optimality principle of stochastic control problem and prove the value function of the control problem is the unique viscosity solution of a class of nonlinear partial differential variational inequalities.

Nonlinear impulsive differential and integral inequalities with integral jump conditions

Some new nonlinear impulsive differential inequalities and integral inequalities with integral jump conditions for discontinuous functions are established using the method of successive iteration. These jump conditions at a discontinuous point are related to the integral conditions of the past state, which can be used in the qualitative analysis of the solutions to certain nonlinear impulsive differential systems.

Generalized Differential Balancing for Variationally Symmetric Systems

Abstract: In this paper, we introduce the notion of variational symmetry for nonlinear prolonged systems (a variational extension of a nonlinear system) by using corresponding variational systems and by developing the recently introduced generalized differential balancing method for this type of system. Generalized differential balancing is proposed for model reduction of nonlinear systems in the contraction framework and is based on solutions to two nonlinear differential matrix inequalities. Our aim of this paper is to show that for variationally symmetric systems, these two inequalities reduce to one inequality. Finally, we show that variational symmetry is preserved under generalized balanced truncation if the variational Hankel singular values are distinct.

Stability of Solutions to Generalized Forchheimer Equations of any Degree

The dynamics of generalized Forchheimer equations for slightly compressible fluids are tudied by means of initial boundary value problem for the pressure. We prove that the solutions continuously depend on the boundary data and the Forchheimer polynomials. New bounds for the solutions are established in the Lα-norm for all α ≥ 1, and then are used to improve estimates for their spatial and time derivatives. New Poincare–Sobolev inequalities and nonlinear Gronwall type estimates for nonlinear differential inequalities are utilized to achieve better asymptotic bounds. The methods developed can be applied to other degenerate parabolic equations. Bibliography: 25 titles.

A constructive approach to positive solutions of Δpu + f(u,∇u)≤0 on Riemannian manifolds

Grigor'yan–Sun in [6] (with p=2) and Sun in [10] (with p>1) proved that ifsupr≫1⁡vol(B(x0,r))rpσp−σ−1(ln⁡r)p−1p−σ−1<∞ then the only non-negative weak solution of Δpu+uσ≤0 on a complete Riemannian manifold is identically 0; moreover, the powers of r and ln⁡r are sharp. In this note, we present a constructive approach to the sharpness, which is flexible enough to treat the sharpness for Δpu+f(u,∇u)≤0. Our construction is based on a perturbation of the fundamental solution to the p-Laplace equation, and we believe that the ideas introduced here are applicable to other nonlinear differential inequalities on manifolds.

General decay rates for the wave equation with mixed-type damping mechanisms on unbounded domain with finite measure

This paper is concerned with the study of the uniform decay rates of the energy associated with the wave equation subject to a locally distributed viscoelastic dissipation and a nonlinear frictional damping $$u_{tt}- \Delta u+ \int_0^t g(t-s){\rm div}[a(x)\nabla u(s)]\,{\rm d}s + b(x) f(u_t)=0\,\quad {\rm on} \quad \Omega\times]0,\infty[,$$ where \({\Omega\subset\mathbb{R}^n, n\geq 2}\) is an unbounded open set with finite measure and unbounded smooth boundary \({\partial\Omega = \Gamma}\). Supposing that the localization functions satisfy the “competitive” assumption \({a(x)+b(x)\geq\delta>0}\) for all \({x\in \Omega}\) and the relaxation function g satisfies certain nonlinear differential inequalities introduced by Lasiecka et al. (J Math Phys 54(3):031504, 2013), we extend to our considered domain the prior results of Cavalcanti and Oquendo (SIAM J Control Optim 42(4):1310–1324, 2003). In addition, while in Cavalcanti and Oquendo (2003) the authors just consider exponential and polynomial decay rate estimates, in the present article general decay rate estimates are obtained.

Blow-up of solutions of some nonlinear inequalities with singularities on unbounded sets

Results on the blow-up of nontrivial nonnegative solutions for several classes of nonlinear partial differential inequalities with singularities on unbounded sets are obtained.

Finite-time H∞ filtering for non-linear stochastic systems

This paper describes the robust H∞ filtering analysis and the synthesis of general non-linear stochastic systems with finite settling time. We assume that the system dynamic is modelled by Itô-type stochastic differential equations of which the state and the measurement are corrupted by state-dependent noises and exogenous disturbances. A sufficient condition for non-linear stochastic systems to have the finite-time H∞ performance with gain less than or equal to a prescribed positive number is established in terms of a certain Hamilton–Jacobi inequality. Based on this result, the existence of a finite-time H∞ filter is given for the general non-linear stochastic system by a second-order non-linear partial differential inequality, and the filter can be obtained by solving this inequality. The effectiveness of the obtained result is illustrated by a numerical example.

A nonlinear nonautonomous delay differential inequality for dissipativity of Lotka-Volterra functional differential equations

In this paper, a Lotka-Volterra functional differential equation is considered. By establishing a nonlinear nonautonomous delay differential inequality and using a generalized Barb� alat's lemma, we obtain some new sufficient conditions ensuring the dissipativity of the Lotka-Volterra functional differential equation.

Intrinsic decay rate estimates for the wave equation with competing viscoelastic and frictional dissipative effects

Wave equation defined on a compact Riemannian manifold $(M, \mathfrak{g})$ subject to a combination of locally distributed viscoelastic and frictional dissipations is discussed. The viscoelastic dissipation is active on the support of $a(x)$ while the frictional damping affects the portion of the manifold quantified by the support of $b(x)$ where both $a(x)$ and $b(x)$ are smooth functions. Assuming that $a(x) + b(x) \geq \delta >0 $ for all $x\in M$ and that the relaxation function satisfies certain nonlinear differential inequality, it is shown that the solutions decay according to the law dictated by the decay rates corresponding to the slowest damping. In the special case when the viscoelastic effect is active on the entire domain and the frictional dissipation is differentiable at the origin, then the overall decay rates are dictated by the viscoelasticity. The obtained decay estimates are intrinsic without any prior quantification of decay rates of both viscoelastic and frictional dissipative effects. This particular topic has been motivated by influential paper of Fabrizio-Polidoro [15] where it was shown that viscoelasticity with poorly behaving relaxation kernel destroys exponential decay rates generated by linear frictional dissipation. In this paper we extend these considerations to: (i) nonlinear dissipation with unquantified growth at the origin (frictional) and infinity (viscoelastic) , (ii) more general geometric settings that accommodate competing nature of frictional and viscoelastic damping.