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.
