Showing papers in "Evolution equations and control theory in 2023"

On the effect of perturbations in first-order optimization methods with inertia and Hessian driven damping

Abstract: <p style='text-indent:20px;'>Second-order continuous-time dissipative dynamical systems with viscous and Hessian driven damping have inspired effective first-order algorithms for solving convex optimization problems. While preserving the fast convergence properties of the Nesterov-type acceleration, the Hessian driven damping makes it possible to significantly attenuate the oscillations. To study the stability of these algorithms with respect to perturbations, we analyze the behaviour of the corresponding continuous systems when the gradient computation is subject to exogenous additive errors. We provide a quantitative analysis of the asymptotic behaviour of two types of systems, those with implicit and explicit Hessian driven damping. We consider convex, strongly convex, and non-smooth objective functions defined on a real Hilbert space and show that, depending on the formulation, different integrability conditions on the perturbations are sufficient to maintain the convergence rates of the systems. We highlight the differences between the implicit and explicit Hessian damping, and in particular point out that the assumptions on the objective and perturbations needed in the implicit case are more stringent than in the explicit case.</p>

Qualitative analysis of certain reaction-diffusion systems of the FitzHugh-Nagumo type

Abstract: This article aims to provide insights into the qualitative analysis of some nonlinear Reaction-Diffusion (RD) systems arising in Neuroscience. We first introduce a non-homogeneous FitzHugh-Nagumo (nhFHN) featuring excitability and oscillatory properties. Then, we discuss the qualitative analysis of a toy model related to nhFHN. In particular, we focus on the convergence of solutions of the toy model toward different solutions (fixed point, periodic) and show the existence of a cascade of Hopf bifurcations. Finally, we connect this analysis to the nhFHN system.

Stability of a damped wave equation on an infinite star-shaped network

Abstract: <p style='text-indent:20px;'>In this paper, we study the stability of an infinite star-shaped network of a linear viscous damped wave equation. We prove that, under some conditions, the whole system is asymptotically stable. Moreover we give a decay rate of the energy of the solution. Our technique is based on a frequency domain method.</p>

Existence and regularity in inverse source problem for fractional reaction-subdiffusion equation perturbed by locally Lipschitz sources

Abstract: <p style='text-indent:20px;'>In this paper, we consider an inverse problem of determining a space-dependent source in the time fractional reaction-subdiffusion equation involving locally Lipschitz perturbations, where the additional measurements take place at the terminal time which are allowed to be nonlinearly dependent on the state. By providing regularity estimates on both time and space of resolvent operator and using local estimates on Hilbert scales, we establish some results on the existence and uniqueness of solutions and the Lipschitz type stability of solution map of the problem under consideration. In addition, when the input data take more regular values, we obtain results on regularity in time of solution for both the direct linear problem and the inverse problem above.</p>

Existence and robustness results of attractors for partially-damped piezoelectric beams

Abstract: This paper deals with a fully-dynamic piezoelectric beam model with a nonlinear force acting on the longitudinal displacements of the beam. The equations of motion follows a system of non-compactly coupled wave equation. As the magnetic effects are discarded by setting the magnetic permeability $ \mu $ to zero, the equations are fully decoupled to a single wave equation (electrostatic/quasi-static beam model). The existence of smooth finite-dimensional global attractors is proved by considering a damping injection to only one of the equations. Next, the existence of a finite set of determining functionals for the long-time behavior of the system is proved. Finally, to transition from the fully-dynamic beam model to the commonly-used electrostatic/quasi-static beam model, a singular limit problem $ \mu \to 0 $ is considered. This analysis allows to compare the attractors from one model to another in the sense of the upper semi-continuity of the attractor $ \mathcal{A}_{\mu} $ as $ \mu \to 0 $.

Asymptotic behavior of the wave equation with nonlocal weak damping, anti-damping and critical nonlinearity

Abstract: <p style='text-indent:20px;'>In this paper, we prove the existence of the global attractor for the wave equation with nonlocal weak damping, nonlocal anti-damping and critical nonlinearity.</p>

Sharp polynomial decay rate of multi-link hyperbolic-parabolic systems with different network configurations

Abstract: The asymptotic behavior of hyperbolic-parabolic coupled systems on some special networks (loop-type, chain-type, tree-type) are discussed. By a detailed spectral analysis, the asymptotic expressions of the spectrum of these three networks are obtained, and based on which, the lack of exponential decay is deduced for all these systems. Then using the frequency domain approach together with the asymptotic expressions of the spectrum, we derive the optimal polynomial decay rate of the systems under the smooth initial states. Moreover, some numerical simulations are given to support the results obtained.

Optimal control for stochastic differential equations and related Kolmogorov equations

Abstract: <p style='text-indent:20px;'>This paper concerns a stochastic optimal control problem with feedback Markov inputs. The problem is reduced to a deterministic optimal control problem for a Kolmogorov equation where the control for the deterministic problem is of open-loop type. The existence of an optimal control is proved for the deterministic control problem in a particular case. A maximum principle and some first order necessary optimality conditions are derived. Some examples and comments are discussed.</p>

An anisotropic tempered fractional $ p $-Laplacian model involving logarithmic nonlinearity

Abstract: In this paper, by introducing an anisotropic tempered fractional $ p $-Laplacian operator $ (-\Delta)_{p, \lambda}^{\beta/2, m} $, based on the anisotropic fractional Laplacian $ \Delta_{m}^{\beta/2} $ and the tempered one $ \Delta_{m}^{\beta/2, \lambda} $, which are studied by Deng et.al recently in [13], an anisotropic tempered fractional $ p $-Laplacian model involving logarithmic nonlinearity is considered. We first establish maximum principle and boundary estimate, which play a very crucial role in the later process. Then we obtain radial symmetry and monotonicity results by using the direct method of moving planes.

Controllability for the wave equation on graph with cycle and delta-prime vertex conditions

Abstract: Exact controllability for the wave equation on a metric graph consisting of a cycle and two attached edges is proven. One boundary and one internal control are used. At the internal vertices, delta-prime conditions are satisfied. As a second example, we examine a tripod controlled at the root and the junction, while the leaves are fixed. These examples are key to understanding controllability properties in general metric graphs.

Total approximate controllability of non-instantaneous impulsive fractional differential systems involving Riemann-Liouville derivatives of order $ \beta \in (1,2) $

Abstract: This artifact introduces the concept of total approximate controllability in order to seek approximate controllability of the impulsive systems at break points in addition to the final point. The chosen system of inspection is a nonlinear fractional differential system affected by non-instantaneous impulses. The system is governed by Riemann-Liouville derivatives of higher-order with fixed lower limits. The appropriate integral-type initial conditions are determined differently depending on the impulsive functions. Firstly, mild solution of the concerned system is constructed, followed by a sufficient set of assumptions required to manifest the existence, uniqueness, and controllability results. Next, the definition of total approximate controllability is interposed. Further, the total approximate controllability of the concerned fractional system is established using Riemann-Liouville fractional resolvent, appropriately defined interval-wise Nemytskii operators, and an iterative technique. Lastly, an illustration is put forward to validate the proposed methodology.

Boundary exact controllability for the longitudinal vibrations of a bar in memoriam to Professor Luiz Adauto Medeiros

Abstract: This paper is concerned with the exact controllability of a system that describes the small vibrations of a bar which is clamped at one and in the other end is glued a mass. To obtain the exact controllability of this system, we will use the HUM Method due to Lions [20].

A new regularity criterion for the 3D incompressible Boussinesq equations in terms of the middle eigenvalue of the strain tensor in the homogeneous Besov spaces with negative indices

Abstract: This paper is concerned with the logarithmically improved regularity criterion in terms of the middle eigenvalue of the strain tensor to the 3D Boussinesq equations in Besov spaces with negative indices. It is shown that a weak solution is regular on $ (0, T] $ provided that$ \begin{align*} \int_{0}^{T}\frac{\left\Vert \lambda _{2}^{+}(\cdot , t)\right\Vert _{\dot{B} _{\infty , \infty }^{-\delta }}^{\frac{2}{2-\delta }}}{\ln (e+\left\Vert u(\cdot , t)\right\Vert _{\dot{B}_{\infty , \infty }^{-\delta }})}dt<\infty . \end{align*} $for some $ 0<\delta <1 $. As a consequence, this result is some improvements of recent works [11,12] established by Neustupa-Penel and Miller.

The Cauchy problem for the critical inhomogeneous nonlinear Schrödinger equation in $ H^{s}(\mathbb R^{n}) $

Abstract: In this paper, we study the Cauchy problem for the critical inhomogeneous nonlinear Schrödinger (INLS) equation$ iu_{t} +\Delta u = |x|^{-b} f(u), \; u(0) = u_{0} \in H^{s} (\mathbb R^{n} ), $where $ n\ge3 $, $ 1< s<\frac{n}{2} $, $ 0<b<2 $ and $ f(u) $ is a nonlinear function that behaves like $ \lambda |u|^{\sigma } u $ with $ \lambda \in \mathbb C $ and $ \sigma = \frac{4-2b}{n-2s} $. We establish the local well-posedness as well as the small data global well-posedness and scattering in $ H^{s} (\mathbb R^{n} ) $ with $ 1<s<\frac{n}{2} $ for the critical INLS equation under some assumption on $ b $. To this end, we first establish various nonlinear estimates by using fractional Hardy inequality and then use the contraction mapping principle based on Strichartz estimates.

A new class of small initial data which may shift the critical power and lifespan estimates for the classical damped wave equations

Abstract: In this manuscript, lifespan of solutions to semilinear classical damped wave equations is investigated when the sum of initial position and speed is $ 0 $ pointwisely. The results in the existing literature are considered mainly in the case when the sum is positive. Here, it is shown that lifespan is extended with appropriate initial data with modified ODE approach. The result implies that the blowup mechanics for semilinear damped wave equations cannot be directly described by using the scaling property and the corresponding weak form approach. The results in this work leave different directions for further study of the lifespan behavior of the solutions to semilinear damped wave equations.

Cone-bounded feedback laws for linear second order systems

Abstract: The strong stabilizability of linear second order equation is investigated in this paper. It is supposed that there exists a linear feedback law that makes the origin of the closed-loop system globally asymptotically stable. Then this control is subject to a cone-bounded nonlinearity. Well-posedness and stability results of the closed-loop system under such (nonlinear) control are stated. The first result is proven by using nonlinear semigroups techniques and the Schauder fixed-point theorem and the second one is based on the result of Marx et al. [12]. Our results are then applied to the particular damped and undamped wave equation. Simulation results are presented to validate the theoretical results. Note that some of the results of this paper apply for a large class systems.

Null controllability of $ \psi$ - Hilfer implicit fractional integro-differential equations with $ \psi $-Hilfer fractional nonlocal conditions

Abstract: In this paper, we study exact null controllability of a new class of non local $ \psi - $ Hilfer implicit fractional integro-differential equations in Hilbert space. The results are obtained by using semigroup theory, $ \psi - $Hilfer fractional calculus and Banach's fixed point theorem. Finally, we provide an example to illustrate the applicability of our results.

A Newtonian approach to general black holes

Abstract: We show how it is possible to define and study black holes of arbitrary shapes in the framework of Newtonian mechanics.

Global smooth solutions in a three-dimensional cross-diffusive SIS epidemic model with saturated taxis at large densities

Abstract: This manuscript is concerned with the homogeneous Neumann initial-boundary problem for a susceptible-infected-susceptible model involving chemotactic movement of susceptibles away from infected individuals, as well as a mass action infection mechanism in its full quadratic strength. By constructing a quasi-energy functional on the basis of a suitable exploitation of a zero-order dissipative term together with the impact of diffusion, under an assumption on asymptotic smallness of the prescribed tactic sensitivity at large cell-densities a result on global existence of classical solutions emanating from initial data of arbitrary size is derived.

On the Cauchy problem for the Hartree type semilinear Schrödinger equation in the de Sitter spacetime

Abstract: The Cauchy problem for the Hartree type semilinear Schrödinger equation is considered in the de Sitter spacetime. Global solutions and blowing-up solutions are considered, and the effects of the spatial expansion and contraction on the solutions are studied.

Controllability for Schrödinger type system with mixed dispersion on compact star graphs

Abstract: In this work we are concerned with solutions to the linear Schr\"odinger type system with mixed dispersion, the so-called biharmonic Schr\"odinger equation. Precisely, we are able to prove an exact control property for these solutions with the control in the energy space posed on an oriented star graph structure $\mathcal{G}$ for $T>T_{min}$, with $$T_{min}=\sqrt{ \frac{ \overline{L} (L^2+\pi^2)}{\pi^2\varepsilon(1- \overline{L} \varepsilon)}},$$ when the couplings and the controls appear only on the Neumann boundary conditions.

Partial approximate controllability results for fractional order stochastic evolution equations using approximation method

Abstract: The sufficient conditions for existence and partial approximate controllability of fractional stochastic evolution equations with nonlocal initial conditions have been discussed. The discussion is based on the variational method, fractional calculus, Schauder's fixed point theorem, and stochastic analysis. Contrary to the results available in the literature, the non-local function does not need to be compact or satisfy Lipschitz's condition. Moreover, the pertinent nonlinear function does not need to be uniformly bounded. At the end, an application has also been demonstrated.

Analysis and simulation of an adhesive contact problem governed by fractional differential hemivariational inequalities with history-dependent operator

Abstract: The main idea of this manuscript is to study an adhesive contact problem with long memory which is governed by a hemivariational inequality and a fractional differential equation. We first prove the existence of a unique solution of the fractional differential hemivariational inequality system. Subsequently, we consider a fully discrete scheme of this system and then focus on deriving error estimates for numerical solutions. To the tail of this manuscript, we present two numerical simulation examples for the adhesive contact problem, which provide numerical evidence to support our theoretical predictions.

Controllability of a simplified time-discrete stabilized Kuramoto-Sivashinsky system

Abstract: In this paper, we study some controllability and observability properties for a coupled system of time-discrete fourth- and second-order parabolic equations. This system can be regarded as a simplification of the well-known stabilized Kumamoto-Sivashinsky equation. Unlike the continuous case, we can prove only a relaxed observability inequality which yields a $ \phi(\triangle t) $-controllability result. This result tells that we cannot reach exactly zero but rather a small target whose size goes to 0 as the discretization parameter $ \triangle t $ goes to 0. The proof relies on a known Carleman estimate for second-order time-discrete parabolic operators and a new Carleman estimate for the time-discrete fourth-order equation.

Stabilization of an interconnected system of Schrödinger and wave equations with boundary coupling

Abstract: In this paper, we consider the Schrödinger equation coupled by the interface with a wave equation and with a boundary damping. The dissipation is acting on the wave equation through the Neumann boundary condition. We formulate the coupled system as an abstract evolution equation in an appropriate Hilbert space and use linear semigroup theory to show the well-posedness of the system. Then under some assumptions on the geometry of the spatial domain, we prove exponential stability of the solution. The proof of this result is based on a frequency domain approach which consists in verifying that the imaginary axis is included in the resolvent set of the system and analyzing the behavior of the resolvent operator of the system on the imaginary axis. The analysis of the resolvent is carried out by combining contradiction argument with the multipliers technique. This result extends Theorem 3.2 in [13] to multimensional spatial domains.

The continuous Redner-Ben-Avraham-Kahng coagulation system: Well-posedness and asymptotic behaviour

Abstract: This paper examines the existence of solutions to the continuous Redner-Ben-Avraham-Kahng coagulation system under specific growth conditions on unbounded coagulation kernels at infinity. Moreover, questions related to uniqueness and continuous dependence on the data are also addressed under additional restrictions. Finally, the large-time behaviour of solutions is also investigated

Effective filtering for slow-fast systems via Wong-Zakai approximation

Abstract: Nonlinear filtering problem has been widely investigated in diverse fields. One important topic related to it is how to formulate an effective approximation scheme for a given observation. By using Wong-Zakai approximation and random invariant manifold theory, we will propose an effective approximation result for a class of slow-fast systems with respect to filtering. We will firstly establish the smooth reduced system via random invariant manifold theory, and then show exponential attractive property of it. At last, we will prove the filtering of the smooth reduced system can well approximate that of the original system.