相关论文: An optimal spectral inequality for degenerate oper…

The decomposition or approximation of a linear operator on a matrix space as a sum of Kronecker products plays an important role in matrix equations and low-rank modeling. The approximation problem in Frobenius norm admits a well-known…

In this paper, a quadratic optimal control problem is considered for second-order parabolic PDEs with homogeneous Dirichlet boundary conditions, in which the "point" control function (depending only on time) constitutes a source term. These…

We consider an optimal control problem with tracking-type cost functional constrained by the Cattaneo equation, which is a well-known model for delayed heat transfer. In particular, we are interested the asymptotic behaviour of the optimal…

We consider the simplest optimal control problem with one nonregular mixed inequality constraint, i.e. when its gradient in the control can vanish on the zero surface. Using the Dubovitskii--Milyutin theorem on the approximate separation of…

We propose a time domain decomposition approach to optimal control of partial differential equations (PDEs) based on semigroup theoretic methods. We formulate the optimality system consisting of two coupled forward-backward PDEs, the state…

Over the past two decades, the controllability of several examples of parabolic-hyperbolic systems has been investigated. The present article is the beginning of an attempt to find a unified framework that encompasses and generalizes the…

We study the minimum energy null-controllability problem for differential equations with point-wise delays. For the equations of both neutral and retarded type we reduce the problem of finding the optimal control to a Volterra integral…

We establish the null controllability for linear stochastic fourth order parabolic equations. Utilizing the duality argument, the null controllability is reduced to the observability for backward fourth order stochastic parabolic equations,…

In this paper we classify all positive extremal functions to a sharp weighted Sobolev inequality on the upper half space, which involves divergent operators with degeneracy on the boundary. As an application of the results, we can derive a…

We study one-dimensional random Jacobi operators corresponding to strictly ergodic dynamical systems. In this context, we characterize the spectrum of these operators by non-uniformity of the transfer matrices and the set where the Lyapunov…

We consider the linear degenerate wave equation, on the interval $(0, 1)$ $$ w_{tt} - (x^\alpha w_x)_x = p(t) \mu (x) w, $$ with bilinear control $p$ and Neumann boundary conditions. We study the controllability of this nonlinear control…

In this paper, we study the separability and spectral properties of singular degenerate elliptic equations in vector valued spaces. We prove that a realization operator by this equation with some boundary conditions is separable and…

In this work we study the global approximate multiplicative controllability for the linear degenerate parabolic Cauchy-Neumann problem $$ \{{array}{l} \displaystyle{v_t-(a(x) v_x)_x =\alpha (t,x)v\,\,\qquad {in} \qquad Q_T…

We consider operators of the form ${\mathcal L}=-L-V$, where $L$ is an elliptic operator and $V$ is a singular potential, defined on a smooth bounded domain $\Omega\subset \R^n$ with Dirichlet boundary conditions. We allow the boundary of…

Recently finite-dimensional observer-based controllers were introduced for the 1D heat equation, where at least one of the observation or control operators is bounded. In this paper, for the first time, we manage with such controllers for…

Motivated by Fredholm theory, we develop a framework to establish the convergence of spectral methods for operator equations $\mathcal L u = f$. The framework posits the existence of a left-Fredholm regulator for $\mathcal L$ and the…

The energy of solutions of the scalar damped wave equation decays uniformly exponentially fast when the geometric control condition is satisfied. A theorem of Lebeau [leb93] gives an expression of this exponential decay rate in terms of the…

An asymptotic equality of the form $\operatorname{Tr}_{L^2} e^{-t(L+V)}=Ct^{-\alpha}+o(t^{-\alpha})$ as $t\rightarrow 0$ is given for the trace of the heat semigroup generated by operators on compact manifolds of the form…

We deal with a degenerate model in divergence form describing the dynamics of a population depending on time, on age and on space. We assume that the degeneracy occurs in the interior of the spatial domain and we focus on null…

This paper deals with the numerical computation of null controls for the linear heat equation. The goal is to compute approximations of controls that drive the solution from a prescribed initial state to zero at a given positive time. In…