| Abstract | Ovaj se rad bavi linearnim sustavima običnih diferencijalnih jednadžbi koji su parametrizirani elementom \(\nu \in K\), gdje je \(K\) kompaktan skup. U primjenama nas zanima konstrukcija ulazne funkcije \(u_{\nu}\) koja za zadan \(\nu \in K\) dovodi funkciju stanja do \(x(T)=x^{1}\), za unaprijed određenu vrijednost \(x^{1}\). Takva je konstrukcija prezentirana u Teoremu 1.5.3. Numerički, da bismo došli do takve funkcije je potrebno riješiti linearan sustav (2.11), što može biti vremenski zahtjevno za velike sustave. Ideja je pronaći konačno reprezentativnih ulaznih funkcija \(u_{1},...,u_{n}\) (koje razapinju prostor \(V_{n} \subset (L^{2}[0,T])^{m}\)) pomoću kojih se za proizvoljan \(\nu \in K\) može naći aproksimacija \(u^{*}\) iz \(V_{n}\) egzaktne ulazne funkcije \(u_{\nu}\) za čiju pripadnu funkciju stanja vrijedi \[||x(T)-x^{1}||<\epsilon,\] za unaprijed definiran \(\epsilon\). Dakle, želimo ubrzati proces traženja ulazne funkcije za dan \(\nu\) i pritom žrtvujemo preciznost, tj. više nemamo egzaktnu upravljivost nego aproksimativnu upravljivost. Na kraju testiramo uvedene algoritme na primjerima valne jednadžbe i jednadžbe provođenja. |
| Abstract (english) | In this thesis, we analyse linear systems of ordinary differential equations, which are parametrised by \(\nu \in K\), where \(K\)is a compact set (in our case, \(K \subset \mathbb{R}^{n}\), for some \(n\)). In applications, we want to construct a function \(u_{\nu}\), called an input function, which for a given \(\nu \in K\), drives the state function \(x(t)\) to state \(x^{1}\) at \(t=T\), where \(x^{1}\) is determined beforehand. Such a construction is described in the proof of Theorem 1.5.3. Numerically, to construct such a function, we need to solve a linear system of equations (2.11), which can take a long time, and the matrix can be ill-conditioned. So, we wish to determine a finite number of representative input functions \(u_{1},...,u_{n}\) (which span a space \(V_{n} \subset (L^{2}[0,T])^{m}\)) such that for every \(\nu \in K\), we can approximate the relevant input function in the space \(V_{n}\) in the way that the following holds: \[||x(T)-x^{1}||<\epsilon,\] where \(\epsilon\) is fixed. So, we wish to speed up the process of find the input function, but we lose the precision by requiring approximate controllability instead of exact controllability. In the last chapter, the proposed method is shown on two examples: for the wave equation and for the heat equation. |
