This paper is devoted to the null controllability of two backward stochastic heat equations coupled exclusively through the second state variables and governed by a single control force. Under some condition on the second coupling matrix, this result is obtained by proving a suitable observability estimate for the adjoint system of the controlled system.
In this paper, we provide a Gauss-Bonnet-Chern formula for α -Grushin manifolds. We establish a result that yields a geometric invariant linking asymptotically the geometry and topology of manifolds that exhibit a not necessarily connected submanifold of singularities, that we shall call singular set. This invariant is obtained by means of the classical Gauss-Bonnet-Chern formula for manifolds with boundary and asymptotic contraction of the tubular neighborhood of the singular set. As a consequence, a Gauss-Bonnet formula for 2D almost-Riemannian structures can be recovered.
For a classical linear control system on ℝ^n, in this paper we provide necessary and sufficient conditions for controllability and the existence of control sets with nonempty interiors. We assume that the control range is compact and convex. Our results build upon classical findings by removing the requirement that the origin must be within the interior of the control range. Additionally, we extend these properties to a family of affine control systems under the same type of control constraint.
This paper focuses on the stabilization for a one-dimensional anti-stable wave equation with nonlinear boundary condition and unknown disturbance. With only two boundary outputs, we construct a state observer, an observer-based disturbance estimator and a boundary controller based on these observers. By resorting to the method of backstepping and active disturbance rejection control, it is shown that the feedback controller based on the state observer and disturbance estimator can stabilize the control plant and, at the same time, ensure that all subsystems involved are uniformly bounded. The well-posedness of the closed-loop system is rigorously analyzed by means of operator semigroup theory and the Galerkin approximation method. Finally, numerical simulations are carried out to illustrate the effectiveness of the proposed control strategy.
This paper develops systematic methods to investigate the existence and feedback control of a new class of evolutionary equations governed by operators satisfying condition (M). First, by applying the Rothe time-discretization technique together with a surjectivity theorem for (M)-type operators, we derive several existence results for the evolutionary problem. Next, under suitable assumptions, we establish the existence of feasible state–control pairs for the associated feedback control system.
This article examines the approximate controllability of a class of fractional differential equations with nonlocal conditions in Banach spaces, focusing on fractional orders q ∈ (1,2) . We derive explicit and verifiable criteria ensuring the approximate controllability of the associated control system. The analysis combines the framework of resolvent operators, key techniques from fractional calculus, and Krasnosel’skii’s fixed point theorem, assuming that the corresponding linear system already satisfies approximate controllability. The results obtained not only refine but also generalize several existing contributions in the literature. An illustrative example is included to demonstrate the effectiveness and applicability of the proposed theoretical approach.
This paper investigates the bifurcation of limit cycles in sixth-order polynomial differential systems of the form {[ ẋ=U_1(x,y)+U_6(x,y),; ẏ=V_1(x,y)+V_6(x,y), ]. where U_i and V_i are homogeneous polynomials of degree i. Using the second-order averaging method, we derive explicit conditions for the existence of limit cycles emanating from both the origin and a perturbed linear center. The analysis involves a transformation of the system into an Abel differential equation. We provide closed-form approximate expressions for the limit cycles in polar coordinates, detailing their global shape. The theoretical results are validated through two specific examples, confirming the effectiveness of the approach and contributing to the qualitative analysis of high-order polynomial systems.
We provide a global analysis of the classical Brusselator system, classifying all possible configurations of equilibria, limit cycles, and nullclines. We study the infinite dynamics through the Poincaré compactification, analyze the bifurcation diagram for the finite local dynamics and draw the global phase portraits in the Poincaré disc for different ranges of the parameters. Our results extend local studies to a global perspective, yielding a complete characterization of the system’s dynamics.
In this paper, we investigate the h-stability of a class of the time-changed differential equation with impulsive in Banach spaces. By establishing novel time-changed retarded Gronwall-like inequalities with jumps, we derive explicit sufficient conditions for the global uniform h-stability of the mild solution. We also analyze two illustrative examples to show the interest and usefulness of the main results.
This article deals with two problems related with germs of foliations induced by real–analytic vector fields in (ℝ ^2,0) having singularity at 0 of order n, n≥ 2 . We will denote this class of vector fields by 𝒜_n . The goal of the first problem is to obtain a set of real–analytic invariants, called real–analytic Thom’s invariants, that allow us to achieve the real–analytic classification of vector fields satisfying genericity assumptions in a subclass 𝒜^0_n of 𝒜_n . The second problem we deal with is called the realization problem, which consists in constructing a real analytic vector field having as real–analytic invariants a suitable group G of conformal mappings and a collection {α _ij} of (n-3)(n-2)/2 real constants that are given in advance.
This paper investigates the local analytic classification of germs of pairs of singular holomorphic foliations at the origin in dimension two. We focus on reduced, non-degenerate foliations that share common separatrices, a setting where the interplay between analytic and geometric structures becomes particularly significant. Our approach combines analytic techniques with geometric intuition to emphasize the role of the tangency divisor in shaping local dynamics. In the specific case where the tangency divisor is simple, we establish a necessary and sufficient condition for conjugacy between pairs of foliations. This result provides a precise equivalence between the classification theorem for germs of pairs of foliations and the decomposition theorem for germs of diffeomorphisms. By clarifying when the classification of pairs reduces to the independent classification of each foliation, our work contributes to a deeper understanding of foliation theory and its local invariants. Furthermore, the framework developed here opens new perspectives for exploring normal forms of such pairs and for identifying related analytic invariants.
In this paper, we establish a new Carleman estimate for the Korteweg-de Vries equation in order to solve an inverse problem retrieving a coefficient of the third-order term from interior measurements. We prove the local stability result for this inverse problem, the proof of the result relies on a new Carleman estimate.
We prove the null controllability of a boundary controlled 1D linear parabolic partial integro-differential equation (PIDE) with state space L^2(0,1) over the time interval [0, T] for any T>0 . We first suppose that the initial state u_0 of the PIDE is a continuous function. We choose an integer n≥ 3 and discretize the spatial derivative and the integral term in the PIDE to obtain an n^th -order ordinary differential equation (ODE) in time. Using the flatness technique, we construct an input signal f_n which transfers this ODE from the initial state obtained by discretizing u_0 to the zero final state over the time interval [0, T]. We prove that f_n converges to a limiting function f as n→∞ and that this input f transfers the PIDE from u_0 to zero over the time interval [0, T]. Combining this result with the regularizing effect of the PIDE, which guarantees that the state of the PIDE is a continuous function for all time t>0 provided the initial state is in L^2(0,1) and its input is zero, yields a proof for the null controllability of the PIDE.
In this paper, we consider a swelling porous-elastic system subject to the effect of boundary frictional dampings with time-dependent coefficients. We use the multiplier method to establish explicit formulae for the energy decay rates, from which the usual exponential and polynomial estimates are only special cases. Our results combine the generality and optimality and improve earlier related results in the literature.
We establish three variational principles for the upper metric mean dimension with potential of level sets of continuous maps in terms of the entropy of partitions and Katok’s entropy for dynamical systems exhibiting the specification property. Moreover, we apply our results to investigate the metric mean dimension of suspension flows. As a byproduct, we establish certain properties of suspension flows and prove a measure-theoretic metric mean dimension version of Abramov’s formula.
We analyze the relationship between the time-optimal control problem for linear and nonlinear (affine in control) systems and the classical Markov moment problem. The paper surveys results on solving linear time-optimal control problems using ideas from classical moment theory and discusses the extension of this approach to nonlinear systems.
One kind of approach, which we called “equivalent cost functional method” is introduced by Yu (ESAIM Control Optim Calc Var. 2013;19:78–90) in the setup of Hamilton system, is applied to get a solvability of a stochastic linear quadratic (SLQ for short) optimal control problem with random jumps and indefinite control weight costs in a finite time horizon. Our analysis is featured by some equivalent cost functionals which enable us to transform the indefinite SLQ problems to positive-definite case, it is remarkable that the solvability of the former is rather complicated than the latter. Consequently, the indefinite SLQ optimal control problem with random jumps is discussed and an explicit state feedback representation of the indefinite SLQ optimal control problem with random jumps is given by the solution of associated indefinite stochastic Riccati equation(SRE for short).
This work investigates the approximate controllability and free-time approximate controllability of a generalized semilinear Benjamin–Bona–Mahony type dynamic equation defined on homogeneous time scales, subject to homogeneous Dirichlet boundary conditions. To accomplish this, the problem is framed within an abstract setting, employing the C_0 -semigroup theory on time scales. Moreover, we apply a technique introduced by Bashirov et al. [1, 2], which enables us to avoid relying on fixed point theorems.
This paper primarily investigates the (r, s)-sensitivity and its stronger variants in dynamical systems. We first prove that chain mixing systems with the shadowing property exhibit strong multi- β -n-sensitivity and strong β -n-sensitivity under the condition of surjection. We improve Theorem 5(i) in T.K.S. Moothathu. J Differ Equ Appl. (18), establishing the equivalence between: (i) (X, T) is sensitive. (ii) (X, T) has 𝒫 -(r, s)-sensitive pairs almost everywhere for every r,s∈ℕ . (iii) (X, T) is 𝒫 -(r, s)-sensitive for every r,s∈ℕ . Furthermore, we demonstrate that multi-transitivity is strictly stronger than β -n-sensitivity for any, and investigate how multi-(r, s)-sensitivity and (r, s)-sensitivity are transmitted to the hyperspace and product dynamical systems. Finally, we define strong β -shadowing property and prove that it is equivalent to the classical shadowing property.
This paper’s focus is on singular infinite-dimensional systems with stochastic perturbations. Under some assumptions on the consistency of the initial condition, we discuss the existence and uniqueness of the solution. Then we determine the necessary and sufficient conditions for exponential L^2 -stability. We present new findings on the robust stability of stochastically perturbed singular systems in Hilbert space. We derive new results regarding the stability radius. We obtain these via a class of operator Lyapunov equations.
