The main objective of this paper is to study the optimal distributed control for a model of homogeneous incompressible two-phase flows. We apply the well-posedness and regularity results to establish the existence of optimal controls as well as the first-order necessary optimality conditions for an associated optimal control problem in which a distributed control is applied to the fluid.

We concern the asymptotic behaviour of the dynamical systems induced by nonlinear Lasota equation. We study chaoticity in the sense of Devaney and strong stability of the system. In many articles authors consider the properties of the linear version of the equation. By the construction of the operator in the separable space, we can formulate the relations between the solutions of the linear equation

The goal of this paper is to investigate Gevrey properties of formal solutions of certain generalized linear partial differential equations with variable coefficients. In particular, we extend the notion of moment partial differential equations to include differential operators that are not connected with kernel functions. Using the modified version of Nagumo norms and the properties of the Newton

This paper studies the concept of prolongational controllability for control affine system with piecewise constant controls. The purpose is to show the relationship between the controllability by prolongations and the notion of weak uniform attraction, presenting criteria for controllability and controllability by prolongations. The central result assures that a control affine system is controllable

In the following text we introduce specification property (stroboscopical property) for dynamical systems on uniform space. We focus on two classes of dynamical systems: generalized shifts and dynamical systems with Alexandroff compactification of a discrete space as phase space. We prove that for a discrete finite topological space X with at least two elements, a nonempty set Γ and a self-map φ :

In this paper, we introduce geometric tools to study the families of rational vector fields of a given degree over \(\mathbb {C}\mathbb {P}^{1}\). To a generic vector field of such a parametric family, we associate several geometric objects: a periodgon, a star domain, and a translation surface. These objects generalize objects with the same name introduced in previous works on polynomial vector fields

Using the averaging theory of second order, we study the limit cycles which bifurcate from a zero-Hopf equilibrium point of polynomial vector fields with cubic nonlinearities in \(\mathbb {R}^{n}\). We prove that there are at least 3n− 2 limit cycles bifurcating from such zero-Hopf equilibrium points. Moreover, we provide an example in dimension 6 showing that this number of limit cycles is reached

This paper deals with the controllability of the second-grade fluids, a class of non-Newtonian of differential type, on a two-dimensional torus. Using the method of Agrachev and Sarychev (J. Math Fluid Mech., 7(1):108–52 (2005)), Agrachev and Sarychev (Commun Math Phys., 265(3):673–97 (2006)), and of Shirikyan (Commun Math Phys., 266(1):123–51 (2006)), we prove that the system of second-grade fluids

In this paper, we introduce the notions of topological stability, shadowing, and weak shadowing properties for actions of some finitely generated groups on non-compact metric spaces which are dynamical properties and equivalent to the classical definitions in case of compact metric spaces. Also, we extend Walter’s stability theorem to group actions on locally compact metric spaces. We show that action

We perform a global dynamical analysis of a modified Nosé-Hoover oscillator, obtained as the perturbation of an integrable differential system. Using this new approach for studying such an oscillator, in the integrable cases, we give a complete description of the solutions in the phase space, including the dynamics at infinity via the Poincaré compactification. Then using the averaging theory, we prove

In this article, we give the existence of a smooth stable manifold which is invariant under the semiflows of the delay differential equation \( x^{\prime }= Ax(t) + Lx_t + f(t,x_t, \lambda ) \), with the assumption that the corresponding linear differential equation admits a nonuniform exponential dichotomy and the perturbation f(t, xt, λ) is small and smooth enough. We also show that the obtained

We consider the existence of affine-periodic solutions to the nonlinear ordinary differential equation: $\label {ae} x^{\prime }=f(t,x) $(0.1) in ℝn, where f is continuous and ensures the existence and uniqueness of solutions with respect to initial conditions, and there exist T > 0 and Q ∈ GL(n) such that: $\label {me} f(t+T,x)=Qf(t,Q^{-1}x) \quad \forall (t,x)\in \mathbb R\times \mathbb R^{n}. $(0

A quenching phenomenon of a degenerate parabolic equation with nonlinear source and singular boundary condition in one-dimensional space is investigated. We establish the results that quenching will occur in a finite time on the boundary x = 0 or x = 1, respectively. And the blowing up of ut at the quenching point and quenching rate estimates are discussed.

We address the output feedback stabilization for a Euler–Bernoulli beam equation with boundary moment control and disturbance. The stabilization of this system has been studied in Guo et al. (J Dyn Control Syst. 2014;20:539–58), where the controller is based on full state feedback. In order to derive the output feedback controller, we design a new disturbance estimator to estimate the total disturbance

This paper is concerned with the blow-up property of solutions to an initial boundary value problem for a fourth-order parabolic equation with a general nonlinearity. It is shown, under certain conditions on the initial data, that the solutions to this problem blow up in finite time, using differential inequalities. Moreover, upper and lower bounds for the blow-up time are derived when blow-up occurs

In this paper, we study an optimal control problem for nonlocal Cahn-Hilliard-Brinkman system, which models phase separation of binary fluids in porous media. The system evolves with regular potential in a two-dimensional bounded domain. We extend the existence of weak solution results for the system to prove the existence of strong solution under extra assumptions on the forcing term and initial datum

The well-known Poincaré-Bendixson theorem tells us that the structure of the omega-limit sets of planar autonomous differential equations can be described by fixed points, limit cycles, or finite number of fixed points together with homoclinic and heteroclinic orbits connecting them. However, this is very different for planar nonautonomous differential equations. In this paper, we study the omega-limit

In this paper, practical stochastic uniform input-to-state stability (PSUISS) for perturbed and perturbed triangular systems depending on parameter is investigated. We present sufficient conditions for which each of this notion is preserved under triangular interconnection. Finally, an example with simulation is provided to demonstrate the applicability of our results.

We describe a method to analyze and decompose the dynamics of a bilinear control system subject to symmetries. The method is based on the concept of generalized Young symmetrizers of representation theory. It naturally applies to the situation where the system evolves on a tensor product space and there exists a finite group of symmetries for the dynamics which interchanges the various factors. This

This paper is concerned with the front-like entire solutions for a Lotka-Volterra weak competition system with nonlocal dispersal. Here, an entire solution means a classical solution defined for all space and time variables. This system has traveling wavefronts and enjoys the comparison principle. Based on these traveling wavefronts, we construct some super- and sub-solutions. Then, by using the comparison

We prove that every expansive continuous action with the weak specification property of an amenable group G on a compact Hausdorff space X has the Myhill property, i.e., every pre-injective continuous self-mapping of X commuting with the action of G on X is surjective. This extends a result previously obtained by Hanfeng Li in the case when X is metrizable.

In this paper, we consider a heat equation with mixed boundary conditions in a two-dimensional domain with a reentrant corner. This allows the solution to exhibit singularities near the corner as well as at the points where the mixed boundary conditions meet. The aim of this work is to establish a Carleman inequality by constructing a convenient weight function. As a consequence, we prove some control

In this paper, we study the approximate controllability of a class of semilinear systems governed by coupled degenerate parabolic equations. The equations may be weakly degenerate and strongly degenerate on a portion of the lateral boundary. We prove that the control systems are approximately controllable by using the Kakutani fixed point theorem and the controls can be taken to be of quasi bang-bang

We consider finitely generated free semigroup actions on (X, d) and generalize Boshernitzan’s quantitative recurrence theorem to general free semigroup actions. Let G be a finitely generated free semigroup endowed with a Bernoulli probability measure \(\mathbb P_{\underline {a}}\) and \(\mathbb S\) be the corresponding continuous semigroup continuous semigroup action. Assume that, for some α > 0, the

Sufficient optimality conditions for optimal control problems involving isoperimetric and mixed inequality and equality constraints are derived. The main novelty of our approach is the fact that, for such problems, discontinuous and singular solutions can be detected. In other words, our result can deal with solutions for which the proposed optimal control is not continuous, but only essentially bounded

We deal with piecewise-smooth differential systems \(\dot {z}=X(z), z=(x,y)\in \mathbb {R}\times \mathbb {R}^{n-1},\) with switching occurring in a codimension one smooth surface Σ. A regularization of X is a 1-parameter family of smooth vector fields Xδ,δ > 0, satisfying that Xδ converges pointwise to X in \(\mathbb {R}^{n}\setminus {\Sigma }\), when \(\delta \rightarrow 0\). The regularized system

In this paper, we study the pointwise controllability of the one-dimensional degenerate heat equation. Necessary and sufficient conditions for approximate and null controllability are proved. Our approach is mainly based on the moment method developed by Fattorini and Russell.

We present enumerative results for holomorphic foliations by curves on \(\mathbb {P}^{n}\), n ≥ 3, with singularities of positive dimension. Some of the results presented improve previous ones due to Corrêa et al. (Annales de l’institut Fourier, 64(4):1781–1805, 2014) and Costa (Ann Fac Sci Toulouse, Math (6), 15(2):297–321, 2006). We also present an enumerative result bounding the number of isolated

We treat the periodic trajectory tracking problem: given a periodic trajectory of a control-affine, left-invariant driftless system in a compact and connected Lie group G and an initial condition in G, find another trajectory of the system satisfying the initial condition given and that asymptotically tracks the periodic trajectory. We solve this problem locally (for initial conditions in a neighborhood

We give a complete algebraic characterization of the integrable weak saddles for planar trigonometric Liénard systems. The main tools used in our proof are the classification of the weak saddles on planar analytic Liénard systems and the characterization of some subfields of the quotient field of the ring of trigonometric polynomials.

The linear time-optimal problem for some classes of linear systems with a multidimensional control is considered. The general position condition, which guarantees the uniqueness of the optimal control, is not assumed to be satisfied. We introduce a vector moment min-problem, which is a further development of the moment min-problem proposed by V.I. Korobov and G.M. Sklyar in 1987 for solving linear

We consider a nonlinear periodic problem driven by a nonhomogeneous differential operator, which includes as a particular case the scalar p-Laplacian. We assume that the reaction is a Carathéodory function which admits time-dependent zeros of constant sign. No growth control near ±∞ is imposed on the reaction. Using variational methods coupled with suitable truncation and comparison techniques, we

We show that an analogue of the ball-box theorem holds true for a class of corank 1, non-differentiable tangent subbundles that satisfy a geometric condition. We also we give examples of such bundles and give an application to dynamical systems.

In this paper, finite time blow-up property of solutions to a p-Kirchhoff-type parabolic equation with general nonlinearity is considered. Some sufficient conditions are given for the weak solutions to blow up in finite time. An upper bound for the blow-up time is also derived. The results partially generalize some recent ones reported by Han and Li (Comput Math Appl. 2018;75:3283–3297).

In this paper, we study a class of fast diffusion p-Laplace equation with singular potential in a bounded smooth domain with homogeneous Dirichlet boundary condition. By using energy estimates, Hardy-Littlewood-Sobolev inequality, and some ordinary differential inequalities, we get the solution of the equation exists globally. Moreover, the conditions of extinction and non-extinction are studied. The

We study local control of a mechanism with the growth vector (4,7). We study controllability and extremal trajectories of the nilpotent approximation as an example of the control theory on a Lie group. We provide solutions to the system and show examples of local extremal trajectories.

In this paper, we consider the output feedback exponential stabilization problem of ODE-Schrödinger cascade systems with the external disturbance. We propose a new extended state observer (ESO) that estimates both state and disturbance by the three output signals, then design a stabilizing control law by utilizing the backstepping technique. The resulting closed-loop system is shown to be exponentially

In this paper, we deal with the multiplicity results for Kirchhoff equations with Hardy-Littlewood-Sobolev critical nonlinearity in \(\mathbb {R}^{N}\). By using the second concentration-compactness principle and concentration-compactness principle at infinity to prove that the (PS)c condition holds locally and together with the new version of symmetric mountain pass theorem of Kajikiya, we prove that

We analyze controlled mass transportation plans with free end-time that minimize the transport cost induced by the generating function of a Lagrangian within a bounded domain, in addition to costs incurred as export and import tariffs at entry and exit points on the boundary. We exhibit a dual variational principle à la Kantorovich that takes into consideration the additional tariffs. We then show

This paper investigates global strong stabilization of time-delayed bilinear systems on a real Hilbert state space. Sufficient conditions for strong global stabilization are formulated in terms of observability like assumptions. An explicit decay rate of the stabilized state is provided. Illustrative examples are presented.

This paper addresses near viability of a set-valued map graph \(\mathcal {G}\) with respect to a quasi-autonomous fully nonlinear differential inclusion of the form y′(t) ∈ Ay(t) + F(t,y(t)). We introduce a new notion of A-quasi-tangency when A is a nonlinear m-dissipative set-valued operator. We give necessary and sufficient conditions for \(\mathcal {G}\) to be near viable with respect to the previous

In this paper, the adaptive stabilization of a nonlinear moving string of “Kirchhoff” type with high gain adaptive output feedback type is investigated. The model that we have adopted is more general than the one adopted in the previous studies. The concept of this type of control allows the system to be dissipative. We show that the gain function is bounded and then we prove under this result that

In this paper, we study the existence and stability of traveling waves of infinite-dimensional lattice differential equations with time delay, where the equation may be not quasi-monotone. Firstly, by applying Schauder’s fixed point theorem, we get the existence of traveling waves with the speed c > c∗ (here c∗ is the minimal wave speed). Using a limiting argument, the existence of traveling waves

We study a Klein-Gordon-Maxwell system, in a bounded spatial domain, under Neumann boundary conditions on the electric potential. We allow a nonconstant coupling coefficient. For sufficiently small data, we find infinitely many standing waves.

A new concept called backward controller for an evolution process is introduced, which is the minimal compact set controlling all fibers of the pullback attractor in the past. The existence criteria are established by using the backward flattening compactness and backward equi-continuity of the system if the phase space is all of continuous vector-valued functions. Both backward controller and backward

We prove differentiability of certain linear combinations of the Lyapunov spectra of a flow on a principal bundle of a semi-simple Lie group. The specific linear combinations that yield differentiability are determined by the finest Morse decomposition on the flag bundles. Differentiability is taken with respect to a differentiable structure on the gauge group, which is a Banach-Lie group.

We establish an existence and uniqueness result for quasilinear parabolic systems of the form \(\frac {\partial u}{\partial t}-\text {div} \sigma (x,t,Du)=f\) in Q, where the source term f is assumed to be in \(W^{-1,x}E_{\overline {M}}(Q;\mathbb {R}^{m})\). The proof is based on the theory of Young measures which permits to identify weak limits.

The main goal of this paper is to present a piecewise smooth vector field tangent to S2 without equilibrium points. Moreover, we will show that this vector field contains a chaotic non-trivial minimal set.

In this paper, we consider a family of dynamical systems on the same compact metric space. We then consider the dynamics given when the given flow shifts between these different flows at regular time intervals. We further require that shifts be allowed by a given directed graph. We then define a type of set, called a chain set, that exhibits many similar properties to chain transitive sets of flows

The paper presents the solution for the existence of analytic solutions for some generalized Lane–Emden (LE) equation. Such a solution exists on the unit interval, in which endpoints are singularities of the proposed perturbed LE equation. The solution has many possible applications and one of the examples was provided.

We consider a thin-film equation modeling the epitaxial growth of nanoscale thin films. By exploiting the boundary conditions and the variational structure of the equation, we look for conditions on initial data that ensure the solution exists globally or blows up in finite time. Moreover, for global solution, we establish the exponential decays of solutions and energy functional, and give the concrete

In this paper, the existence and multiplicity of solutions of Kirchhoff type problems with critical nonlinearity is considered in \(\mathbb {R}^{3}: -\varepsilon ^{2}\left (a+b\displaystyle {\int }_{\mathbb {R}^{3}}|\nabla u|^{2}dx\right ){\Delta } u + V(x)u -\varepsilon ^{2}{\Delta }(u^{2})u = K(x)|u|^{22^{\ast }-2}u + h(x,u)\), \((t, x) \in \mathbb {R} \times \mathbb {R}^{3}\). Under suitable assumptions