2024/09/16 by L. V. Fardigola, Fardigola, Larissa, Kateryna Khalina +1 · 1 citation
Engineering · Mathematics · #Stability and Controllability of Differential Equations #Numerical methods in inverse problems #Differential Equations and Boundary Problems
paper · pdf · doi:10.48550/arxiv.2409.10169
In the paper, the problems of controllability and approximate controllability are studied for the control system wt=Δw, wx1(0,x2,t)=u(t)δ(x2), x1>0, x2∈\mathbb R, t∈(0,T), where u∈ L^∞(0,T) is a control. To this aid, it is investigated the set RT(0)⊂ L2((0,+∞)×\mathbb R) of its end states which are reachable from 0. It is established that a function f\inRT(0) can be represented in the form f(x)=g(|x|2) a.e. in (0,+∞)×\mathbb R where g∈ L2(0,+∞). In fact, we reduce the problem dealing with functions from L2((0,+∞)×\mathbb R) to a problem dealing with functions from L2(0,+∞). Both a necessary and sufficient condition for controllability and a sufficient condition for approximate controllability in a given time T under a control u bounded by a given constant are obtained in terms of solvability of a Markov power moment problem. Using the Laguerre functions (forming an orthonormal basis of L2(0,+∞)), necessary and sufficient conditions for approximate controllability and numerical solutions to the approximate controllability problem are obtained. It is also shown that there is no initial state that is null-controllable in a given time T. The results are illustrated by an example.