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This paper concerns a controllability problem for blowup points on heat equation. It can be described as follows: In the absence of control, the solution to the linear heat system globally exists in a bounded domain $Omega$. While, for a given time $T>0$ and a point $a$ in this domain, we find a feedback control, which is acted on an internal subset $omega$ of this domain, such that the corresponding solution to this system blows up at time $T$ and holds unique point $a$. We show that $ain omega$ can be the unique blowup point of the corresponding solution with a certain feedback control, and for any feedback control, $ain Omegasetminus overline{omega}$ could not be the unique blowup point.
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The aim of this paper is to perform a Stackelberg strategy to control parabolic equations. We have one control, textit{the leader}, that is responsible for a null controllability property; additionally, we have a control textit{the follower} that solves a robust control objective. That means, that we seek for a saddle point of a cost functional. In this way, the follower control is not sensitive to a broad class of external disturbances. As far as we know, the idea of combining robustness with a Stackelberg strategy is new in literature
Observability inequalities on lattice points are established for non-negative solutions of the heat equation with potentials in the whole space. As applications, some controllability results of heat equations are derived by the above-mentioned observability inequalities.
We address the following problem: given a Riemannian manifold $(M,g)$ and small parameters $varepsilon>0$ and $v>0$, is it possible to find $T>0$ and an absolutely continuous map $x:[0,T]rightarrow M, tmapsto x(t)$ satisfying $|dot{x}|_{infty}leq v$ and such that any geodesic of $(M,g)$ traveled at speed $1$ meets the open ball $B_g(x(t),varepsilon)subset M$ within time $T$? Our main motivation comes from the control of the wave equation: our results show that the controllability of the wave equation in any dimension of space can be improved by allowing the domain of control to move adequately, even very slowly. We first prove that, in any Riemannian manifold $(M,g)$ satisfying a geodesic recurrence condition (GRC), our problem has a positive answer for any $varepsilon>0$ and $v>0$, and we give examples of Riemannian manifolds $(M,g)$ for which (GRC) is satisfied. Then, we build an explicit example of a domain $Xsubsetmathbb{R}^2$ (with flat metric) containing convex obstacles, not satisfying (GRC), for which our problem has a negative answer if $varepsilon$ and $v$ are small enough, i.e., no sufficiently small ball moving sufficiently slowly can catch all geodesics of $X$.
In this paper we study the internal exact controllability for a second order linear evolution equation defined in a two-component domain. On the interface we prescribe a jump of the solution proportional to the conormal derivatives, meanwhile a homogeneous Dirichlet condition is imposed on the exterior boundary. Due to the geometry of the domain, we apply controls through two regions which are neighborhoods of a part of the external boundary and of the whole interface, respectively. Our approach to internal exact controllability consists in proving an observability inequality by using the Lagrange multipliers method. Eventually we apply the Hilbert Uniqueness Method, introduced by J.-L. Lions, which leads to the construction of the exact control through the solution of an adjoint problem. Finally we find a lower bound for the control time depending not only on the geometry of our domain and on the matrix of coefficients of our problem but also on the coefficient of proportionality of the jump with respect to the conormal derivatives.
We are concerned about the controllability of a general linear hyperbolic system of the form $partial_t w (t, x) = Sigma(x) partial_x w (t, x) + gamma C(x) w(t, x) $ ($gamma in mR$) in one space dimension using boundary controls on one side. More precisely, we establish the optimal time for the null and exact controllability of the hyperbolic system for generic $gamma$. We also present examples which yield that the generic requirement is necessary. In the case of constant $Sigma$ and of two positive directions, we prove that the null-controllability is attained for any time greater than the optimal time for all $gamma in mR$ and for all $C$ which is analytic if the slowest negative direction can be alerted by {it both} positive directions. We also show that the null-controllability is attained at the optimal time by a feedback law when $C equiv 0$. Our approach is based on the backstepping method paying a special attention on the construction of the kernel and the selection of controls.
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