Showing papers in "Journal of Dynamical and Control Systems in 2019"

A General Decay for a Weakly Nonlinearly Damped Porous System

Abstract: In this paper, we consider a one-dimensional porous system damped with a single weakly nonlinear feedback. Without imposing any restrictive growth assumption near the origin on the damping term, we establish an explicit and general decay rate, using a multiplier method and some properties of convex functions in case of the same speed of propagation in the two equations of the system. The result is new and opens more research areas into porous-elastic system.

General and Optimal Decay Result for a Viscoelastic Problem with Nonlinear Boundary Feedback

Abstract: In this paper, we consider a viscoleastic equation with a nonlinear feedback localized on a part of the boundary and a relaxation function satisfying g′(t) ≤−ξ(t)G(g(t)). We establish an explicit and general decay rate results, using the multiplier method and some properties of the convex functions. Our results are obtained without imposing any restrictive growth assumption on the damping term. This work generalizes and improves earlier results in the literature, in particular those of Messaoudi (Topological Methods in Nonlinear Analysis 51(2):413–427, 2018), Messaoudi and Mustafa (Nonlinear Analysis: Theory Methods & Applications 72(9–10):3602–3611, 2010), Mustafa (Mathematical Methods in the Applied Sciences 41(1): 192–204, 2018) and Wu (Zeitschrift fur angewandte Mathematik und Physik 63(1):65–106, 2012).

A Sequential Quadratic Hamiltonian Method for Solving Parabolic Optimal Control Problems with Discontinuous Cost Functionals

Abstract: A sequential quadratic Hamiltonian (SQH) method for solving control-constrained parabolic optimal control problems with continuous and discontinuous non-convex cost functionals is investigated The solution to these problems is characterised by the Pontryagin’s maximum principle, which is also the starting point for the development of a sequential quadratic Hamiltonian scheme In a general setting that includes discontinuous and non-convex cost functionals, it is proved that the SQH method is well defined; however, convergence to an optimal solution is proved only in the smooth case Results of numerical experiments are presented that successfully validate the proposed optimisation framework and demonstrate its effectiveness and large applicability

Asymptotic Stability for the Second Order Evolution Equation with Memory

Abstract: In this paper, we consider an abstract viscoelastic equation. We use memory-type damping with a general assumption on the relaxation function and establish explicit energy decay result from which we can recover the optimal exponential and polynomial rates. Our result generalizes the earlier related results in the literature.

Approximate Controllability for Navier–Stokes Equations in 3D Rectangles Under Lions Boundary Conditions

Abstract: The 3D Navier–Stokes system, under Lions boundary conditions, is proven to be approximately controllable provided a suitable saturating set does exist. An explicit saturating set for 3D rectangles is given.

On the Practical h-stability of Nonlinear Systems of Differential Equations

Abstract: In this paper, we present a new notion of stability for nonlinear systems of differential equations called practical h-stability. Necessary and sufficient conditions for practical h-stability are given using the Lyapunov theory. Our original results generalize well-known fundamental results: practical exponential stability, practical asymptotic stability, and practical stability for nonlinear time-varying systems. In addition, these results are used to study the practical h-stability of two important classes of nonlinear systems, namely perturbed and cascaded systems. The last part is devoted to the study of the problem of practical h-stabilization for certain classes of nonlinear systems.

Privileged Coordinates and Nilpotent Approximation of Carnot Manifolds, I. General Results

Abstract: In this paper, we attempt to give a systematic account on privileged coordinates and nilpotent approximation of Carnot manifolds. By a Carnot manifold, it is meant a manifold with a distinguished filtration of subbundles of the tangent bundle which is compatible with the Lie bracket of vector fields. This paper lies down the background for its sequel (Choi and Ponge 2017) by clarifying a few points on privileged coordinates and the nilpotent approximation of Carnot manifolds. In particular, we give a description of all the systems of privileged coordinates at a given point. We also give an algebraic characterization of all nilpotent groups that appear as the nilpotent approximation at a given point. In fact, given a nilpotent group $G$ satisfying this algebraic characterization, we exhibit all the changes of variables that transform a given system of privileged coordinates into another system of privileged coordinates in which the nilpotent approximation is given by $G$ .

Existence of Affine-Periodic Solutions to Newton Affine-Periodic Systems

Abstract: In this paper, we prove that every Newton affine-periodic system admits an affine-periodic solution via the lower and upper solutions method and the homotopy invariance of Leray-Schauder degree. Furthermore, we give some specific examples about oscillators to illustrate our main results.

Necessary First- and Second-Order Optimality Conditions in Discrete Systems with a Delay in Control

Abstract: In the paper, an optimal control problem with a delay in control is considered. Suggesting a new approach, Euler-type necessary optimality conditions and the linearized discrete maximum principle are established. Also, the second-order necessary optimality conditions (a) based on the second variation of the objective functional and (b) for quasi-singular controls are obtained. An example to illustrate the richness of content of the suggested approach is presented.

On constrictions of phase-lock areas in model of overdamped Josephson effect and transition matrix of double confluent Heun equation

Abstract: In 1973, B. Josephson received a Nobel Prize for discovering a new fundamental effect concerning a Josephson junction,—a system of two superconductors separated by a very narrow dielectric: there could exist a supercurrent tunneling through this junction. We will discuss the model of the overdamped Josephson junction, which is given by a family of first-order nonlinear ordinary differential equations on two-torus depending on three parameters: a fixed parameter ω (the frequency); a pair of variable parameters (B, A) that are called respectively the abscissa, and the ordinate. It is important to study the rotation number of the system as a function ρ = ρ(B, A) and to describe the phase-lock areas: its level sets Lr = {ρ = r} with non-empty interiors. They were studied by V.M.Buchstaber, O.V.Karpov, S.I.Tertychnyi, who observed in their joint paper in 2010 that the phase-lock areas exist only for integer values of the rotation number. It is known that each phase-lock area is a garland of infinitely many bounded domains going to infinity in the vertical direction; each two subsequent domains are separated by one point, which is called constriction (provided that it does not lie in the abscissa axis). Those points of intersection of the boundary ∂Lr of the phase-lock area Lr with the line Λr = {B = rω} (which is called its axis) that are not constrictions are called simple intersections. It is known that our family of dynamical systems is related to appropriate family of double–confluent Heun equations with the same parameters via Buchtaber–Tertychnyi construction. Simple intersections correspond to some of those parameter values for which the corresponding “conjugate” double-confluent Heun equation has a polynomial solution (follows from results of a joint paper of V.M. Buchstaber and S.I.Tertychnyi and a joint paper of V.M. Buchstaber and the author). There is a conjecture stating that all the constrictions of every phase-lock areaLrlie in its axis Λr. This conjecture was studied and partially proved in a joint paper of the author with V.A.Kleptsyn, D.A.Filimonov, and I.V.Schurov. Another conjecture states that for any two subsequent constrictions in Lr with positive ordinates, the interval between them also lies in Lr. In this paper, we present new results partially confirming both conjectures. The main result states that for every $r \in \mathbb {Z}\setminus \{0\}$ the phase-lock area Lr contains the infinite interval of the axis Λr issued upwards from the simple intersection in ∂Lr ∩Λr with the biggest possible ordinate. The proof is done by studying the complexification of the system under question, which is the projectivization of a family of systems of second-order linear equations with two irregular non-resonant singular points at zero and at infinity. We obtain new results on the transition matrix between appropriate canonical solution bases of the linear system; on its behavior as a function of parameters. A key result, which implies the main result of the paper, states that the off-diagonal terms of the transition matrix are both non-zero at each constriction. We also show that their ratio is real at the constrictions. We reduce the above conjectures on constrictions to the conjecture on negativity of the ratio of the latter off-diagonal terms at each constriction.

Local (Sub)-Finslerian Geometry for the Maximum Norms in Dimension 2

Abstract: We consider specific sub-Finslerian structures in the neighborhood of 0 in $\mathbb {R}^{2}$ , defined by fixing a family of vector fields (F1,F2) and considering the norm defined on the non-constant rank distribution Δ = vect{F1, F2} by $$|G|=\inf_{u}\{\max\{|u_{1}|,|u_{2}|\} | G=u_{1} F_{1}+u_{2} F_{2}\}. $$ If F1 and F2 are not proportional at p, then we obtain a Finslerian structure; if not, the structure is sub-Finslerian on a distribution with non-constant rank. We are interested in the study of the local geometry of these Finslerian and sub-Finslerian structures: generic properties, normal form, short geodesics, cut locus, switching locus, and small spheres.

Control Problem for a Magneto-Micropolar Flow with Mixed Boundary Conditions for the Velocity Field

Abstract: We prove the existence and uniqueness of weak solutions of the stationary magneto-micropolar equations with mixed boundary conditions for velocity, including Navier slip condition. We study an optimal boundary control problem associated to weak solutions of these equations. By using the Lagrange multipliers method, we obtain first-order necessary conditions from which we derive an optimality system.

Privileged Coordinates and Nilpotent Approximation for Carnot Manifolds, II. Carnot Coordinates

Abstract: This paper is a sequel of Choi and Ponge (J Dyn Control Syst 25:109–157, 2019) and deals with privileged coordinates and nilpotent approximation of Carnot manifolds. By a Carnot manifold, it is meant a manifold equipped with a filtration by subbundles of the tangent bundle which is compatible with the Lie bracket of vector fields. In this paper, we single out a special class of privileged coordinates in which the nilpotent approximation at a given point of a Carnot manifold is given by its tangent group. We call these coordinates Carnot coordinates. Examples of Carnot coordinates include Darboux coordinates on contact manifolds and the canonical coordinates of the first kind of Goodman and Rothschild-Stein. By converting the privileged coordinate of Bellaiche into Carnot coordinates, we obtain an effective construction of Carnot coordinates, which we call e-Carnot coordinates. They form the building block of all systems of Carnot coordinates. On a graded nilpotent Lie group, they are given by the group law of the group. For general Carnot manifolds, they depend smoothly on the base point. Moreover, in Carnot coordinates at a given point, they are osculated in a very precise manner by the group law of the tangent group at the point.

The Stokes Phenomenon for Some Moment Partial Differential Equations

Abstract: We study the Stokes phenomenon for the solutions of general homogeneous linear moment partial differential equations with constant coefficients in two complex variables under condition that the Cauchy data are holomorphic on the complex plane but finitely many singular or branching points with the appropriate growth condition at the infinity. The main tools are the theory of summability and multisummability, and the theory of hyperfunctions. Using them, we describe Stokes lines, anti-Stokes lines, jumps across Stokes lines, and a maximal family of solutions.

Optimality of Broken Extremals

Abstract: In this paper, we analyse the optimality of broken Pontryagin extremal for an n-dimensional affine control system with a control parameter, taking values in a k-dimensional closed ball. We prove the optimality of broken normal extremals in many cases that include (but not are exhausted by) the cases of the involutive driftless part of the system for any n > k and of the contact driftless part for n = 3 and k = 2.

Nonlinear Multivalued Periodic Systems

Abstract: We consider a first-order periodic system involving a time-dependent maximal monotone map, a subdifferential term, and a multivalued perturbation F(t, x). We prove existence theorems for the “convex” problem (that is, F is convex valued and for the “nonconvex” problem (that is, F is nonconvex valued). Also, we establish the existence of extremal trajectories (that is, solutions when the multivalued perturbation F(t, x) is replaced by ext F(t, x), the extreme points of F(t, x)). Also, we show that every solution of the convex problem can be approximated uniformly by certain extremal trajectories (“strong relaxation” theorem). Finally, we illustrate our result by examining a nonlinear periodic feedback control system.

Infinitely Many Rotating Periodic Solutions for Second-Order Hamiltonian Systems

Abstract: In this paper, we consider a class of second-order Hamiltonian system in $\mathbb {R}^{N}$ with combined nonlinearities. We will study the multiplicity of rotating periodic solutions, i.e., $x(t+T)=Qx(t)$ with $T>0$ and Q is an $N\times N$ orthogonal matrix. In the case $Q^{k} eq I_{N}$ for any positive integer k, such a rotating periodic solution is just a quasi-periodic solution; In the case $Q^{k}=I_{N}$ for some positive integer k, such a rotating periodic solution is just a subharmonic solution. We will use the Fountain Theorem and its dual form to obtain two sequences of rotating periodic solutions with the corresponding energy tending to infinity and zero respectively.

Global Attractor for a Class of Sixth-Order Viscous Cahn-Hilliard Equation in an Unbounded Domain

Abstract: In this paper, we consider the existence of global attractor for a class of sixth-order Cahn-Hilliard equation with a nonlinear diffusion and viscous effects in an infinite domain. Due to the noncompactness of operators, we use weighted Sobolev spaces to prove that the semigroup generated by the equation has the global attractor in a suitable space.

The Quenching Behavior for a Quasilinear Parabolic Equation with Singular Source and Boundary Flux

Abstract: In this paper, we consider the quenching behavior for a one-dimensional quasilinear parabolic equation with singular source and boundary flux. We show that the quenching occurs only on the boundary in finite time, and derive the lower and upper bounds of quenching rate. Moreover, we give the exact bounds of quenching rate of special examples.

Pseudo Almost Periodic Solutions to Impulsive Non-autonomous Stochastic Differential Equations with Unbounded Delay and its Optimal Control

Abstract: This paper is concerned with the pseudo almost periodic in distribution mild solutions for impulsive non-autonomous stochastic differential equations with unbounded delay and optimal controls in Hilbert spaces. Firstly, a suitable pseudo almost periodic in distribution mild solutions is introduced. The existence of pseudo almost periodic in distribution mild solutions are proved by means of a fixed-point theorem for condensing maps combined with stochastic analysis theory and evolution family. Secondly, the existence of optimal pairs of system governed by impulsive non-autonomous stochastic differential equations is also presented. Finally, an example is given for demonstration.

Invisibility in Billiards is Impossible in an Infinite Number of Directions

Abstract: We show that the maximum number of directions of invisibility in a planar billiard defined in the exterior of a piecewise smooth body is at most finite.

Approximate Controllability for Nonlocal Fractional Propagation Systems of Sobolev Type

Abstract: This paper is concerned with the approximate controllability of a class of fractional propagation systems of Sobolev type with nonlocal conditions in Hilbert spaces. By utilizing the theory of propagation family and techniques of measures of noncompactness, we firstly prove existence of the mild solutions and compactness of solutions set in Banach spaces under weaker conditions, i.e., the compactness condition of propagation family is instead of norm continuous in the sense of uniform operator topology. Secondly, we establish the principle for linear fractional propagation systems is approximately controllable in Hilbert spaces. Then, we present an interesting criteria for approximate controllability of semilinear fractional propagation systems in Hilbert spaces by requiring the corresponding linear fractional propagation systems is approximately controllable and assuming that the resolvent set of the operator pair is compact. Finally, a fractional PDEs model is used to demonstrate that how to check the conditions in the above abstract theorems, In particular, a principle for approximate controllability of linear problem is verified by using the property of Mittag-Leffler function.

New Family of Centers of Planar Polynomial Differential Systems of Arbitrary Even Degree

Abstract: The problem of distinguishing between a focus and a center is one of the classical problems in the qualitative theory of planar differential systems. In this paper, we provide a new family of centers of polynomial differential systems of arbitrary even degree. Moreover, we classify the global phase portraits in the Poincare disc of the centers of this family having degree 2, 4, and 6.

Stability of Traveling Wavefronts for a Nonlocal Dispersal System with Delay

Abstract: This paper is concerned with a nonlocal epidemic model arising from the spread of an epidemic by oral-fecal transmission. Comparing with the previous works, here we extend the model in Capasso and Maddalena, Nonlinear Phenom Math Sci. 41:207–217 (1982) by including a spatial convolution term and a discrete delay term corresponding to the dispersal of bacteria in the environment and the latent period of the virus, respectively. Besides existence and asymptotic behavior, the main part of the paper is devoted to the stability of the traveling wavefronts under some monostable assumptions. By using a comparison theorem and the weighted energy method with a suitably selected weight function, we show that all the non-critical traveling waves are exponentially stable. Finally, we apply our results to a specific epidemic model and discuss the effect of time delay on the stability of wavefront.

A Remark on the Invertibility of Semi-invertible Cocycles

Abstract: We observe that under certain conditions on the Lyapunov exponents, a semi-invertible cocycle is, indeed, invertible. As a consequence, if a semi-invertible cocycle generated by a Holder continuous map $A:\mathcal {M}\to M(d, \mathbb {R})$ over a hyperbolic system $f:\mathcal {M}\to \mathcal {M}$ satisfies a Livsic’s type condition, that is, if A(fn− 1(p)) ⋅… ⋅ A(f(p))A(p) = Id for every p ∈ Fix(fn), then the cocycle is invertible, meaning that $A(x)\in GL(d,\mathbb {R})$ for every $x\in \mathcal {M}$ , and a Livsic’s type theorem is satisfied.

Control Systems on the Engel Group

Abstract: We consider control affine systems, as well as cost-extended control systems, on the (four-dimensional) Engel group. Specifically, we classify the full-rank left-invariant control affine systems (under both detached feedback equivalence and strongly detached feedback equivalence). The cost-extended control systems with quadratic cost are then classified (under cost equivalence), as are their associated Hamilton-Poisson systems (up to affine isomorphism). In all cases, we exhibit a complete list of equivalence class representatives.

Optimization of Boundary Value Problems for Certain Higher-Order Differential Inclusions

Abstract: The present paper studies a new class of problems of optimal control theory with special differential inclusions described by higher-order linear differential operators (HLDOs). There arises a rather complicated problem with simultaneous determination of the HLDOs and a Mayer functional depending of high-order derivatives of searched functions. The sufficient conditions, containing both the Euler-Lagrange and Hamiltonian type inclusions and “transversality” conditions at the endpoints t = − 1, 0 and t = 1 are derived. One of the key features in the proof of sufficient conditions is the notion of locally adjoint mappings. Then, we demonstrate how these conditions can be transformed into Pontryagin’s maximum principle in some particular cases.

Stabilization of One-Dimensional Schrödinger Equation Under Joint Feedback Control with Delayed Observation

Abstract: In this paper, we consider a one-dimensional Schrodinger equation under a joint linear feedback control at an arbitrary internal point $\xi (0<\xi <1)$ . It is shown that the pointwise control system is asymptotically stable if $\xi $ is either an irrational number or a rational number satisfying $\xi eq 2l/(2m-1)$ for any positive integers $l, m (1\leq l \leq m-1)$ ; further, the system is exponentially stable if $\xi $ is a rational number satisfying $\xi eq 2l/(2m-1)$ for any positive integers $l, m (1\leq l \leq m-1)$ . Moreover, we consider the Schrodinger equation under a joint feedback control where the observation is suffered from a given time delay. A Luenberger observer is designed at the time interval when the observation signal is available, while a predictor is designed at the time interval when the observation signal is not available. A natural control law is constructed based on the estimated state. The closed-loop system is shown to be exponentially stable for the smooth initial value. Finally, numerical simulations demonstrate effectiveness of the dynamic output feedback controller.

Global Stabilization of the Navier-Stokes Equations Around an Unstable Steady State with Mixed Boundary Kinetic Energy Controller

Abstract: The paper, benefiting from techniques developed in Ngom et al. (Evol Equ Control Theory. 2015;4:89–106), presents a mixed (Dirichlet-Neumann) boundary feedback controller for stabilizing the Navier-Stokes equations around a prescribed steady state, in a bounded domain ${\Omega }$ . The Neumann part of the boundary controller is designed to be zero when the inflow vanishes, and to have the magnitude of the kinetic energy. Like in Ngom et al. (Evol Equ Control Theory. 2015;4:89–106), the present paper proves exponential decrease of the perturbation in $L^{2}$ , without blowup. In addition, it goes further than (Ngom et al., Evol Equ Control Theory. 2015;4:89–106) by proving, on the one hand, that the exponential convergence towards zero holds in $H^{1}$ , on the other hand, that the weak solution is unique when the computational domainis two-dimensional.