相关论文: Constructive exact control of semilinear 1D heat e…

We consider the optimal control of singular nonlinear partial differential equation which is the distributional formulation of the multiphase Stefan type free boundary problem for the general second order parabolic equation. Boundary heat…

We study the controllability to trajectories, under positivity constraints on the control or the state, of a one-dimensional heat equation involving the fractional Laplace operator $ (-\partial_x^2)^s$ (with $0<s<1$) on the interval…

In the paper, the problems of controllability and approximate controllability are studied for the control system $w_t=\frac{1}{\rho}\left(kw_x\right)_x+\gamma w$, $\left.\left(\sqrt{\frac{k}{\rho}}w_x\right)\right|_{x=0}=u$, $x>0$,…

We derive in a direct and rather straightforward way the null controllability of the N-dimensional heat equation in a bounded cylinder with boundary control at one end of the cylinder. We use the so-called flatness approach, which consists…

In this note we study the control problem for the heat equation on $\mathbb{R}^d$, $d\geq 1$, with control set $\omega\subset\mathbb{R}^d$. We provide a necessary and sufficient condition (called $(\gamma, a)$-\emph{thickness}) on $\omega$…

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…

We consider a fully-discrete approximations of 1-D heat equation with dynamic boundary conditions for which we provide a controllability result. The proof of this result is based on a relaxed observability inequality for the corresponding…

We are interested in the determination of the reachable states for the boundary control of the one-dimensional heat equation. We consider either one or two boundary controls. We show that reachable states associated with square integrable…

Let $\Omega\subset\mathbb R^N$ be a bounded open set with Lipschitz continuous boundary $\Gamma$. Let $\gamma>0$, $\delta\ge 0$ be real numbers and $\beta$ a nonnegative measurable function in $L^\infty(\Gamma)$. Using some suitable…

We consider the integral definition of the fractional Laplacian and analyze a linear-quadratic optimal control problem for the so-called fractional heat equation; control constraints are also considered. We derive existence and uniqueness…

We investigate approximate null-controllability for semi-discrete heat equations on the lattice $h\mathbb{Z}^d$ with a potential. By establishing spectral inequalities for the discrete Schr{\"o}dinger operator $P_h = -\Delta_h + V$ on…

We consider the null-controllability of a non-local heat equation by interior $L^2(\Omega)$ controls. We confirm a conjecture of Lissy and Zuazua by showing that it is enough to assume that the kernel $k(x,\xi)$ is symmetric and…

We survey recent results on the control problem for the heat equation on unbounded and large bounded domains. First we formulate new uncertainty relations, respectively spectral inequalities. Then we present an abstract control cost…

We consider linear control problems for the heat equation of the form $\dot f (t) = -Hf (t) + \mathbf{1}_D u (t)$, $f (0) \in \ell_2 (X,m)$, where $H$ is the weighted Laplacian on a discrete graph $(X,b,m)$, and where $D \subseteq X$ is…

In this paper we consider the optimal control of semilinear fractional PDEs with both spectral and integral fractional diffusion operators of order $2s$ with $s \in (0,1)$. We first prove the boundedness of solutions to both semilinear…

In this article we study a controllability problem for a parabolic and a hyperbolic partial differential equations in which the control is the shape of the domain where the equation holds. The quantity to be controlled is the trace of the…

In this paper, we are concerned with the boundary controllability of heat equation with dynamic boundary conditions. More precisely, we prove that the equation is null controllable at any positive time by means of a boundary control…

We consider the following exponential reaction-diffusion equation involving a nonlinear gradient term: $$\partial_t U = \Delta U + \alpha|\nabla U|^2 + e^U,\quad (x, t)\in\mathbb{R}^N\times[0,T), \quad \alpha > -1.$$ We construct for this…

We consider the energy super critical semilinear heat equation $$\partial_t u=\Delta u+u^{p}, \ \ x\in \mathbb R^3, \ \ p>5.$$ We first revisit the construction of radially symmetric backward self similar solutions and propose a bifurcation…

We consider the control problem for the generalized heat equation for a Schroedinger operator on a domain with a reflection symmetry with respect to a hyperplane. We show that if this system is null-controllable, then so is the system on…