Trapped Modes in Linear Quantum Stochastic Networks with Delays

Networks of open quantum systems with feedback have become an active area of research for applications such as quantum control, quantum communication and coherent information processing. A canonical formalism for the interconnection of open quantum systems using quantum stochastic differential equations (QSDEs) has been developed by Gough, James and co-workers and has been used to develop practical modeling approaches for complex quantum optical, microwave and optomechanical circuits/networks. In this paper we fill a significant gap in existing methodology by showing how trapped modes resulting from feedback via coupled channels with finite propagation delays can be identified systematically in a given passive linear network. Our method is based on the Blaschke-Potapov multiplicative factorization theorem for inner matrix-valued functions, which has been applied in the past to analog electronic networks. Our results provide a basis for extending the Quantum Hardware Description Language (QHDL) framework for automated quantum network model construction (Tezak \textit{et al.} in Philos. Trans. R. Soc. A, Math. Phys. Eng. Sci. 370(1979):5270-5290, to efficiently treat scenarios in which each interconnection of components has an associated signal propagation time delay

Dynamical analysis of particular classes of linear time-delay singular control systems defined over finite and infinite time interval

U disertaciji su razmatrani problemi dinamicke analize posebnih klasa singularnih sistema sa cistim vremenskim kašnjenjem prisutnim u stanju sistema, kao i njihovo ponašanje na konacnom i beskonacnom vremenskom intervalu. Pružen je presek savremenih koncepata stabilnosti, prednosti jednih nad drugima i posebno su obraeni tzv neljapunovski koncepti: stabilnost na konacnom vremenskom intervalu i koncept prakticne stabilnosti. Nadograene su osnovne definicije stabilnosti. Isrpno je izložen hronološki sistematican pregled osnovnih koncepata stabilnosti, polazeci od ljapunovske metodologije, kao osnove na kojoj se zasniva dinamicka analiza sistema. Ukazano je na istorijski razvoj i nastanak ideja i rezultata u ovoj oblasti i na taj nacin su izvedene i smernice daljih istraživanja otvorenih problema. U disertaciji su sistemi tretirani sa stanovišta dva savremena pristupa: deskriptivnog i LMI, odnosno sa pozicija linearnih matricnih nejednakosti, koja se svodi na metode konveksne optimizacije. Izvedeni su i saopšteni novi rezultati. Izložen je prilaz koji se bazira na kvaziljapunovskim funkcijama za dobijanje uslova prakticne i stabilnosti na konacnom vremenskom intervalu posebne klase singularnih sistema sa cistim vremenskim kašnjenje, u stanju sistema. Pokazano je da, polazeci od pretpostavke da agregacione funkcije ne moraju da budu odreene po znaku i da njihovi izvodi duž trajektorija sistema ne moraju da budu negativno odrreene funkcije, uz pomoc deskriptivnog prilaza se mogu dobiti novi kriterijumi za ocenu neljapunovske stabilnosti. Kombinovanjem rezultata sa ljapunovskim prilazom, izvedeni su o uslovi atraktivne prakticne stabilnosti. Drugi doprinos je odreivanje dovoljnih uslova stabilnosti na konacnom vremenskom intervalu iste klase sistema pomocu savremenih LMI metoda. Dobijeni i prezentovani rezultati imaju prakticnu primenu u savremenoj teoriji i praksi upravljanja i mogu se primeniti na sve klase proucavanih sistema, pod uslovom da su dostupni verodostojni matematicki modeli. Verifikacija rezultata je izvedena kroz numericke primereIn this thesis the problems of dynamical analysis of particular class of singular control systems with time delays are considered, as well as their behavior on finite and infinite time intervals. Emphasis has been put on the peculiar properties of singular ad descriptor systems, concerning the existence and uniqueness of the solutions, the problems of impulsive behavior, consistent initial conditions and causality of the system itself. On overview of the modern stability frameworks has been presented, starting from the classical Lyapunov ideas and extending through so called non-lyapunov concepts: finite time stability and practical stability in particular. A historical overview of ideas, concepts and results has been presented and the key contributions have been highlighted through key papers from the modern literature. This dissertation follows two main lines of research: the descriptive approach and the LMI (linear matrix inequalities) methodology, the latter being known to reduce control tasks to convex optimization problems, thus making them easily solvable by numerical computation. New results are presented. A new approach, based on lyapunov-like functions, is used in order to establish new sufficient conditions of practical and finite time interval stability of a particular class of singular time delay systems. Another new result is based on the modern LMI approach and gives new sufficient conditions for finite time stability. The obtained results are numerically verified and have great practical value, as they are easy to compute and less restrictive and conservative than their predecessors