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Register a new userNumerical approximation of the averaged controllability for the wave equation with unknown velocity of propagation
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We propose a numerical method to approximate the exact averaged boundary control of a family of wave equations depending on an unknown parameter sigma. More precisely the control, independent of sigma, that drives an initial data to a family of final states at time t = T, whose average in sigma is given. The idea is to project the control problem in the finite dimensional space generated by the first N eigenfunctions of the Laplace operator. The resulting discrete control problem has solution whenever the continuous one has it, and we give a convergence result of the discrete controls to the continuous one. The method is illustrated with several examples in 1-d and 2-d in a square domain.
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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 $X$.
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.
The semilinear beam equation with impulses, memory and delay is considered. We obtain the approximate controllability. This is done by employing a technique that avoids fixed point theorems and pulling back the control solution to a fixed curve in a short time interval. Demonstrating, once again, that the controllability of a system is robust under the influence of impulses and delays.
Solutions to the stochastic wave equation on the unit sphere are approximated by spectral methods. Strong, weak, and almost sure convergence rates for the proposed numerical schemes are provided and shown to depend only on the smoothness of the driving noise and the initial conditions. Numerical experiments confirm the theoretical rates. The developed numerical method is extended to stochastic wave equations on higher-dimensional spheres and to the free stochastic Schrodinger equation on the unit sphere.
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
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