Semilinear Evolution Equations Research Articles

Global existence and attractivity for Riemann-Liouville fractional semilinear evolution equations involving weakly singular integral inequalities

In this paper, we obtain several results on the global existence, uniqueness and attractivity for fractional evolution equations involving the Riemann-Liouville type by exploiting some results on weakly singular integral inequalities in Banach spaces. Some boundedness conditions of the nonlinear term are considered to obtain the main results that generalize and improve some well-known works.

Global existence and blowup of solutions for nonlinear Love equation

In this paper, we are concerned with the nonlinear Love equation, which describes some sort of “propagation process” in physics. Firstly, we establish the local existence and uniqueness of the solution by using the contraction mapping principle and the regularity theory for the elliptic problem. Secondly, we construct a family of potential wells to obtain a threshold for the existence of global solutions and blowup in a finite time of the solution in both cases with subcritical and critical initial energy. Additionally, we provide the decay rate of the global solution and estimate the lifespan of a blow‐up solution. Moreover, at the supercritical initial energy level, we provide a sufficient condition for initial data leading to blow‐up results. This paper, along with the previous work of A. N. Carvalho, J. Cholewa, Local well‐posedness, asymptotic behavior, and asymptotic bootstrapping for a class of semilinear evolution equations of the second order in time, Trans. Amer. Math. Soc. 361 (5) (2009) 2567‐2586, provides us with a comprehensive and systematic study of the dynamic behavior of the solution to the nonlinear Love equation.

On observer compensator design for non-autonomous control semi-linear evolution equations

This paper investigates the Luenberger observer design problem for non-autonomous control semilinear evolution equations with disturbances in Banach spaces. Then, the practicalstabilization problem of the system is solved, yielding a compensator based on the Luenberger observer by using integral inequalities of the Gronwall type. Sufficient conditions of the controller and observer problem are satisfied, we show that the proposed controller with estimatedstate feedback from the proposed practical Luenberger observer will achieve global practical stabilization. We develop novel ideas and techniques, which present the further development of mathematical control theory. Furthermore, an example is given to show the applicability of our theoretical results.

The move from Fujita type exponent to a shift of it for a class of semilinear evolution equations with time-dependent damping

The move from Fujita type exponent to a shift of it for a class of semilinear evolution equations with time-dependent damping

Well-posedness and properties of the flow for semilinear evolution equations

AbstractWe derive conditions for well-posedness of semilinear evolution equations with unbounded input operators. Based on this, we provide sufficient conditions for such properties of the flow map as Lipschitz continuity, bounded-implies-continuation property, boundedness of reachability sets, etc. These properties represent a basic toolbox for stability and robustness analysis of semilinear boundary control systems. We cover systems governed by general $$C_0$$ C 0 -semigroups, and analytic semigroups that may have both boundary and distributed disturbances. We illustrate our findings on an example of a Burgers’ equation with nonlinear local dynamics and both distributed and boundary disturbances.

Process-Controllability of Semilinear Evolution Equations and Applications

Process-Controllability of Semilinear Evolution Equations and Applications

On growth and instability for semilinear evolution equations: an abstract approach

AbstractWe propose a new approach to the study of (nonlinear) growth and instability for semilinear abstract evolution equations with compact nonlinearities. We show, in particular, that compact nonlinear perturbations of linear evolution equations can be treated as linear ones as far as the growth of their solutions is concerned. We obtain exponential lower bounds of solutions for initial values from a dense set in resolvent or spectral terms. The abstract results are applied, in particular, to the study of energy growth for semilinear backward damped wave equations.

Stabilization and asymptotic stability of the Caputo–Fabrizio fractional-order linear and semilinear evolution equations

We discuss the asymptotic stability of the fractional-order linear and semilinear evolution equations involving a Caputo–Fabrizio fractional derivative for fractional-order α∈(0,1) with a non-singular kernel. First, by using the concepts of an equilibrium point, it is proved that an autonomous Caputo–Fabrizio system admits only a constant solution and a new concept of a global solution for the Caputo–Fabrizio system is introduced. Laplace transform and Grönwall inequality are used to derive the local and global asymptotic stability conditions. By constructing a suitable linear feedback control and using our main results, we stabilize the Caputo–Fabrizio fractional-order linear and semilinear evolution equations. At the end, by using the stabilization result, we stabilize a fractional-order chaotic system to support the obtained results.

On Exponential Splitting Methods for Semilinear Abstract Cauchy problems

Due to the seminal works of Hochbruck and Ostermann (Appl Numer Math 53(2–4):323–339, 2005, Acta Numer 19:209–286, 2010) exponential splittings are well established numerical methods utilizing operator semigroup theory for the treatment of semilinear evolution equations whose principal linear part involves a sectorial operator with angle greater than frac{pi }{2} (meaning essentially the holomorphy of the underlying semigroup). The present paper contributes to this subject by relaxing the sectoriality condition, but in turn requiring that the semigroup operators act consistently on an interpolation couple (or on a scale of Banach spaces). Our conditions (on the semigroup and on the semilinearity) are inspired by the approach of Kato (Math Z 187(4):471–480, 1984) to the local solvability of the Navier–Stokes equation, where the textrm{L}^p - textrm{L}^r-smoothing of the Stokes semigroup was fundamental. The present abstract operator theoretic result is applicable for this latter problem (as was already the result of Hochbruck and Ostermann), or more generally in the setting of Hochbruck and Ostermann (2005), but also allows the consideration of examples, such as non-analytic Ornstein–Uhlenbeck semigroups or the Navier–Stokes flow around rotating bodies.

Existence and stability of pullback exponential attractors for a nonautonomous semilinear evolution equation of second order

We consider a nonautonomous semilinear evolution problem that models some sort of propagation problem in nonlinear elastic rods and nonlinear ion-acoustic waves. We investigate the existence and stability of a family of pullback exponential attractors for our problem under suitable growth and dissipativeness conditions. Moreover, we also prove the upper and lower semicontinuity of this family of pullback exponential attractors at time zero. As a particular case, we obtain the existence of the pullback attractor in an appropriate space, we prove its upper semicontinuity and, lastly, we obtain a regularity result of this pullback attractor.

Input‐to‐state practical stability for nonautonomous nonlinear infinite‐dimensional systems

AbstractIn this article, we are concerned with the input‐to‐state practical stability (ISpS) of nonautonomous nonlinear infinite‐dimensional systems. Sufficient conditions of ISpS are provided based on Lyapunov functions and a nonlinear inequality. We characterize ISpS in terms of uniform practical asymptotic gain property. We show that for a class of admissible inputs the existence of an ISpS‐Lyapunov function implies the ISpS of a system in Banach spaces. These results are used to study the input‐to‐state practical stability of a certain class of nonautonomous semilinear evolution equations. Furthermore, we provide a small‐gain theorem for nonautonomous nonlinear interconnected systems. An example is used to illustrate the application of the developed method.

Mild solutions of semilinear evolution equation and their applications in second‐order hyperbolic PDE

The principal goal of this work is to solvability of the mild solution for the second‐order hyperbolic PDE with initial/boundary‐value problem with nonlocal condition of the form where is open. Our analysis relies on the technique of measure of noncompactness.

Positive almost periodic solutions of nonautonomous evolution equations and application to Lotka–Volterra systems

The aim of this paper is to establish some sufficient conditions ensuring the existence and uniqueness of positive (Bohr) almost periodic solutions to a class of semilinear evolution equations of the form: . We assume that the family of closed linear operators on a Banach lattice satisfies the “Acquistapace–Terreni” conditions, so that the associated evolution family is positive and has an exponential dichotomy on . The nonlinear term , acting on certain real interpolation spaces, is assumed to be almost periodic only in a weaker sense (i.e., in Stepanov's sense) with respect to , and Lipschitzian in bounded sets with respect to the second variable. Moreover, we prove a new composition result for Stepanov almost periodic functions by assuming only continuity of with respect to the second variable (see the condition Lemma 1‐(ii)). Finally, we provide an application to a system of Lotka–Volterra predator–prey type model with diffusion and time–dependent parameters in a generalized almost periodic environment.

On the Exact Controllability of a Semilinear Evolution Equation with an Unbounded Operator

On the Exact Controllability of a Semilinear Evolution Equation with an Unbounded Operator

Compensator Design Via the Separation Principle for a Class of Semilinear Evolution Equations

Compensator Design Via the Separation Principle for a Class of Semilinear Evolution Equations

Weighted input-to-output practical stability of non-autonomous infinite-dimensional systems with disturbances

In this paper, we introduce a new concept of weighted input-to-output practical stability (WIOpS) and weighted integral-input-to-output practically stable (WiIOpS) for non-autonomous infinite-dimensional systems with disturbances. By using the notion of practical stable scalar functions, sufficient conditions for WIOpS and WiIOpS are derived. As a result, the robust global asymptotic output practical stability criteria of non-autonomous infinite-dimensional systems with zero input is also established via an indefinite Lyapunov function. Thus, we study the UWISpS of non-autonomous nonlinear evolution equations. Moreover, we discuss UWISS and UWiISS for linear non-autonomous infinite-dimensional control systems. A feedback law is provided for a class of semi-linear evolution equations by which the closed-loop system is uniform input-to-state practical stable (UISpS) with respect to disturbances acting in the input. Two examples are given throughout the paper to illustrate the theoretical results.

Stabilization of non-autonomous infinite-dimensional systems depending on a parameter

ABSTRACT In this paper, we investigate the practical stabilization of non-autonomous infinite-dimensional systems depending on a parameter. By the Lyapunov method, several sufficient conditions are provided to obtain the practical stabilization of two classes of non-autonomous semilinear evolution equations in Hilbert spaces. We propose some classes of memoryless state linear and nonlinear feedback controllers. Numerical examples are given to illustrate the validity of the proposed theoretical results.

Compensator design via the separation principle for a class of semilinear evolution equations

UDC 517.9 We establish a compensator design via the separation principle in the practical sense for a class of semilinear evolution equations in Hilbert spaces. Under a restriction imposed on the perturbation, which is bounded by an integrable function, we propose a nonlinear time-varying practical Luenberger observer to estimate the states of the system and prove that the Luenberger observer based on the linear controller stabilizes the system. An illustrative example is given to demonstrate the applicability of our theoretical results.

A Simplified Newton Method to Generate Snapshots for POD Models of Semilinear Optimal Control Problems

In PDE-constrained optimization, proper orthogonal decomposition (POD) provides a surrogate model of a (potentially expensive) PDE discretization, on which optimization iterations are executed. Because POD models usually provide good approximation quality only locally, they have to be updated during optimization. Updating the POD model is usually expensive, however, and therefore often impossible in a model-predictive control (MPC) context. Thus, reduced models of mediocre quality might be accepted. We take the view of a simplified Newton method for solving semilinear evolution equations to derive an algorithm that can serve as an offline phase to produce a POD model. Approaches that build the POD model with impulse response snapshots can be regarded as the first Newton step in this context. In particular, POD models that are based on impulse response snapshots are extended by adding a second simplified Newton step. This procedure improves the approximation quality of the POD model significantly by introducing a moderate amount of extra computational costs during optimization or the MPC loop. We illustrate our findings with an example satisfying our assumptions.

Approximate controllability for semilinear second-order neutral evolution equations with infinite delay

In this article, we study the existence of mild solutions and the approximate controllability for a class of systems governed by neutral equations of second-order with infinite delay in infinite-dimensional Hilbert spaces. The mild solution and approximate controllability are achieved by constructing the fundamental solution for the associated linear equation and assuming that the linear system is approximately controllable. The discussion is based on the fundamental solution theory and Rothe's fixed point theorem. In addition, an example is given to illustrate our main conclusion.