Evolution Equations In Hilbert Spaces Research Articles

Minimal time sliding mode control for evolution equations in Hilbert spaces

This work is concerned with the time optimal control problem for evolution equations in Hilbert spaces. The attention is focused on the maximum principle for the time optimal controllers having the dimension smaller that of the state system, in particular for minimal time sliding mode controllers, which is one of the novelties of this paper. We provide the characterization of the controllers by the optimality conditions determined for some general cases. The proofs rely on a set of hypotheses meant to cover a large class of applications. Examples of control problems governed by parabolic equations with potential and drift terms, porous media equation or reaction-diffusion systems with linear and nonlinear perturbations, describing real world processes, are presented at the end.

Global existence of solutions to some semilinear integro-differential evolution equations with sign-varying kernels

AbstractIn this paper, we study the global existence of solutions to some semilinear integro-differential evolution equations in Hilbert spaces with sign-varying kernels. By treating the problem through fixed point theory and energy method, we obtain the global existence theorem on [0, +∞) of mild and strong solution to the evolution equations with memory effects for oscillating and sign-varying kernels. This theorem generalizes and improves some previous existence results.

Decay rates for second order evolution equations in Hilbert spaces with nonlinear time-dependent damping

The paper is concerned with the Cauchy problem for second order hyperbolic evolution equations with nonlinear source in a Hilbert space, under the effect of nonlinear time-dependent damping. With the help of the method of weighted energy integral, we obtain explicit decay rate estimates for the solutions of the equation in terms of the damping coefficient and two nonlinear exponents. Specialized to the case of linear, time-independent damping, we recover the corresponding decay rates originally obtained in [ 3 ] via a different way. Moreover, examples are given to show how to apply our abstract results to concrete problems concerning damped wave equations, integro-differential damped equations, as well as damped plate equations.

Asymptotic model of linearly visco-elastic Kelvin\u2013Voigt type plates via Trotter theory

We confirm the study (Licht in C. R., Méc. 341:697–700, 2013) devoted to the quasi-static response for a visco-elastic Kelvin–Voigt plate whose thickness goes to zero. For each thickness parameter, the quasi-static response is given by a system of partial differential equations with initial and boundary conditions. Reformulating scaled systems into a family of evolution equations in Hilbert spaces of possible states with finite energy, we use Trotter theory of convergence of semi-groups of linear operators to identify the asymptotic behavior of the system. The asymptotic model we obtain and the genuine one have the same structure except an occurrence of a new state variable. Eliminating the new state variable from our asymptotic model leads to the asymptotic model in (Licht in C. R., Méc. 341:697–700, 2013) which involves an integro-differential system.

A class of stochastic hyperbolic evolution equations via weighted pseudo almost periodic coefficients and optimal controls

SummaryThis paper introduces the concept of the piecewise weighted pseudo periodicity for stochastic processes. In addition, it establishes a new composition theorem for weighted pseudo almost periodic functions under the non‐Lipschitz conditions. Using this composition theorem, evolution family together with a fixed‐point theorem for condensing maps, we investigate the existence of p‐mean piecewise weighted pseudo almost periodic mild solutions and optimal mild solutions for a class of impulsive stochastic hyperbolic evolution equations in Hilbert spaces. Then, the existence conditions of optimal pairs of systems governed by the nonlinear impulsive stochastic evolution equations are presented. Finally, an example is provided to illustrate the obtained theory.

NEWTON S METHOD FOR STOCHASTIC FUNCTIONAL EVOLUTION EQUATIONS IN HILBERT SPACES

We apply Newton’s method to stochastic functional evolution equations in Hilbert spaces using semigroup methods. The first-order convergence is based on our generalization of the Gronwall-type inequality. We also establish a second-order convergence in a probabilistic sense.

Unified error analysis for nonconforming space discretizations of wave-type equations

AbstractThis paper provides a unified error analysis for nonconforming space discretizations of linear wave equations in the time domain. We propose a framework that studies wave equations as first-order evolution equations in Hilbert spaces and their space discretizations as differential equations in finite-dimensional Hilbert spaces. A lift operator maps the semidiscrete solution from the approximation space to the continuous space. Our main results are a priori error bounds in terms of interpolation, data and conformity errors of the method. Such error bounds are the key to the systematic derivation of convergence rates for a large class of problems. To show that this approach significantly eases the proof of new convergence rates, we apply it to an isoparametric finite element discretization of the wave equation with acoustic boundary conditions in a smooth domain. Moreover, our results reproduce known convergence rates for already investigated conforming and nonconforming space discretizations in a concise and unified way. The examples discussed in this paper comprise discontinuous Galerkin discretizations of Maxwell’s equations and finite elements with mass lumping for the acoustic wave equation.

Pseudo Almost Periodicity and Its Applications to Impulsive Nonautonomous Partial Functional Stochastic Evolution Equations

Abstract In this paper, we establish a new composition theorem for pseudo almost periodic functions under non-Lipschitz conditions. We apply this new composition theorem together with a fixed-point theorem for condensing maps to investigate the existence of p $p$ -mean piecewise pseudo almost periodic mild solutions for a class of impulsive nonautonomous partial functional stochastic evolution equations in Hilbert spaces, and then, the exponential stability of p $p$ -mean piecewise pseudo almost periodic mild solutions is studied. Finally, an example is given to illustrate our results.

English

We establish the existence of solutions for evolution equations in Hilbert spaces with anti-periodic boundary conditions. The energies associated to these evolution equations are quadratic forms. Our approach is based on application of the Schaefer fixed-point theorem combined with the continuity method.

Pathwise uniqueness for stochastic evolution equations with Hölder drift and stable Lévy noise

We prove the pathwise uniqueness of solutions to stochastic evolution equations in Hilbert spaces with the $$\alpha $$ -stable Lévy noise and a bounded $$\beta $$ -Hölder continuous drift term. The proof is based on the regularity results of resolvent equations associated to Kolmogorov operators.

Partial-approximate controllability of nonlocal fractional evolution equations via approximating method

In this paper we study partial-approximate controllability of semilinear nonlocal fractional evolution equations in Hilbert spaces. By using fractional calculus, variational approach and approximating technique, we give the approximate problem of the control system and get the compactness of approximate solution set. Then new sufficient conditions for the partial-approximate controllability of the control system are obtained when the compactness conditions or Lipschitz conditions for the nonlocal function are not required. Finally, we apply our abstract results to the partial-approximate controllability of the semilinear heat equation and delay equation.

The regularity of fractional stochastic evolution equations in Hilbert space

ABSTRACTIn this paper, we concentrate on the regularity of the trajectories of the mild solution for the fractional evolution equations on the Hilbert space. By giving some critical estimates on the α-times resolvent generated by the operator A, we prove that under the condition of , the trajectories of the mild solution are Hölder continuous both on the Hilbert space and on the domain of the fractional power of the operator with some appropriate index.

On Homogeneous Finite-Time Control for Linear Evolution Equation in Hilbert Space

Based on the notion of generalized homogeneity, a new algorithm of feedback control design is developed for a plant modeled by a linear evolution equation in a Hilbert space with a possibly unbounded operator. The designed control law steers any solution of the closed-loop system to zero in a finite time. A method of homogeneous extension is presented in order to make the developed control design principles to be applicable for evolution systems with nonhomogeneous operators. The design scheme is demonstrated for the heat equation with the control input distributed on the segment ${[}0,1{]}$ .

Approximate controllability of fractional neutral stochastic evolution equations in Hilbert spaces with fractional Brownian motion

ABSTRACTIn this paper, the approximate controllability for Sobolev-type fractional neutral stochastic evolution equations with fractional stochastic nonlocal conditions and fractional Brownian motion in a Hilbert space are studied. The results are obtained by using semigroup theory, fractional calculus, stochastic integrals for fractional Brownian motion, Banach's fixed point theorem, and methods adopted directly from deterministic control problems for the main results. Finally, an example is given to illustrate the application of our result.

Dynamic and quasi-electromagnetostatic evolution of a thermoelectromagnetoelastic body

A standard technique of evolution equations in Hilbert spaces of possible states with finite energy supplies results of existence and uniqueness for the dynamic evolution of a thermoelectromagnetoelastic body and for its ‘‘quasi-electromagnetostatic approximation’’ whose relevance is established through a convergence result as a parameter, accounting for the ratio of the speed of elastic wave propagation to the celerity of the light, goes to zero.

A new and improved analysis of the time domain boundary integral operators for the acoustic wave equation

We present a novel analysis of the boundary integral operators associated to the wave equation. The analysis is done entirely in the time-domain by employing tools from abstract evolution equations in Hilbert spaces and semi-group theory. We prove a single general theorem from which well-posedness and regularity of the solutions for several boundary integral formulations can be deduced as particular cases. By careful choices of continuous and discrete spaces, we are able to provide a concise analysis for various direct and indirect formulations, both at the continuous level and for their Galerkin-in-space semi-discretizations. Some of the results here are improvements on previously known results, while other results are equivalent to those in the literature. The methodology presented here greatly simplifies the analysis of the operators of the Calderon projector for the wave equation and can be generalized for other relevant boundary integral equations.

Existence and exponential stability of pseudo almost periodic solutions for impulsive nonautonomous partial stochastic evolution equations

In this paper, the concept of p-mean piecewise pseudo almost periodic for stochastic processes is first introduced. Using the exponential dichotomy techniques and a fixed point strategy with stochastic analysis theory, we establish the existence of p-mean piecewise pseudo almost periodic mild solutions for a class of impulsive nonautonomous partial stochastic evolution equations in Hilbert spaces. Moreover, the exponential stability of p-mean piecewise pseudo almost periodic mild solutions is investigated. Finally, an example is provided to illustrate the obtained theory.

Controllability of neutral stochastic integro-differential evolution equations driven by a fractional Brownian motion

We establish sufficient conditions for the controllability of a certain class of neutral stochastic functional integro-differential evolution equations in Hilbert spaces. The results are obtained using semigroup theory, resolvent operators and a fixed-point technique. An application to neutral partial integro-differential stochastic equations perturbed by fractional Brownian motion is given.

Backward uniqueness of stochastic parabolic like equations driven by Gaussian multiplicative noise

One proves here the backward uniqueness of solutions to stochastic semilinear parabolic equations and also for the tamed Navier–Stokes equations driven by linearly multiplicative Gaussian noises. Applications to approximate controllability of nonlinear stochastic parabolic equations with initial controllers are given. The method of proof relies on the logarithmic convexity property known to hold for solutions to linear evolution equations in Hilbert spaces with self-adjoint principal part.

About periodicity of impulsive evolution equations through fixed point theory

By processing the problem through fixed point theory and propagator theory, we investigate the periodicity of solutions to a class of impulsive evolution equations in Hilbert spaces and establish some existence theorems for periodic solutions. Moreover, the asymptotic stability of periodic solutions is obtained under suitable conditions. As one will see, the concept of an impulsive propagator is introduced for the first time in the paper.