Distributed parameter systems (DPS) comprise an important class of engineering systems ranging from "traditional" such as tubular reactors, to cutting edge processes such as nano-scale coatings. DPS have been studied extensively and significant advances have been noted, enabling their accurate simulation. To this end a variety of tools have been developed. However, extending these advances for systems design is not a trivial task . Rigorous design and operation policies entail systematic procedures for optimisation and control. These tasks are "upper-level" and utilize existing models and simulators. The higher the accuracy of the underlying models, the more the design procedure benefits. However, employing such models in the context of conventional algorithms may lead to inefficient formulations. The optimisation and control of DPS is a challenging task. These systems are typically discretised over a computational mesh, leading to large-scale problems. Handling the resulting large-scale systems may prove to be an intimidating task and requires special methodologies. Furthermore, it is often the case that the underlying physical phenomena span various temporal and spatial scales, thus complicating the analysis. Stiffness may also potentially be exhibited in the (nonlinear) models of such phenomena. The objective of this work is to design reliable and practical procedures for the optimisation and control of DPS. It has been observed in many systems of engineering interest that although they are described by infinite-dimensional Partial Differential Equations (PDEs) resulting in large discretisation problems, their behaviour has a finite number of significant components , as a result of their dissipative nature. This property has been exploited in various systematic model reduction techniques. Of key importance in this work is the identification of a low-dimensional dominant subspace for the system. This subspace is heuristically found to correspond to part of the eigenspectrum of the system and can therefore be identified efficiently using iterative matrix-free techniques. In this light, only low-dimensional Jacobians and Hessian matrices are involved in the formulation of the proposed algorithms, which are projections of the original matrices onto appropriate low-dimensional subspaces, computed efficiently with directional perturbations.The optimisation algorithm presented employs a 2-step projection scheme, firstly onto the dominant subspace of the system (corresponding to the right-most eigenvalues of the linearised system) and secondly onto the subspace of decision variables. This algorithm is inspired by reduced Hessian Sequential Quadratic Programming methods and therefore locates a local optimum of the nonlinear programming problem given by solving a sequence of reduced quadratic programming (QP) subproblems . This optimisation algorithm is appropriate for systems with a relatively small number of decision variables. Inequality constraints can be accommodated following a penalty-based strategy which aggregates all constraints using an appropriate function , or by employing a partial reduction technique in which only equality constraints are considered for the reduction and the inequalities are linearised and passed on to the QP subproblem . The control algorithm presented is based on the online adaptive construction of low-order linear models used in the context of a linear Model Predictive Control (MPC) algorithm , in which the discrete-time state-space model is recomputed at every sampling time in a receding horizon fashion. Successive linearisation around the current state on the closed-loop trajectory is combined with model reduction, resulting in an efficient procedure for the computation of reduced linearised models, projected onto the dominant subspace of the system. In this case, this subspace corresponds to the eigenvalues of largest magnitude of the discretised dynamical system. Control actions are computed from low-order QP problems solved efficiently online.The optimisation and control algorithms presented may employ input/output simulators (such as commercial packages) extending their use to upper-level tasks. They are also suitable for systems governed by microscopic rules, the equations of which do not exist in closed form. Illustrative case studies are presented, based on tubular reactor models, which exhibit rich parametric behaviour.

Les événements extrêmes sont des phénomènes, souvent considérés catastrophiques, qui se produisent dans la queue d'une distribution généralement en s'écartant d'une décroissance attendue exponentielle. En optique, ces événements ont été étudié dans le contexte des fibres, où ils ont été amplement analysés, comme des vagues scélérates, par analogie bien connue entre l'optique et l'hydrodynamique à travers l'équation de Schroedinger non linéaire. Avec le développement et l'élargissement du domaine, l'étude des événements extrêmes a été étendue à des systèmes dissipatifs avec ou sans degrés spatiaux de liberté.Dans cette thèse on se concentre sur l'étude des événements extrêmes dans trois différents types de systèmes optiques actifs et dissipatifs, présentant chacun un ou deux degrés spatiales de liberté, soit dans le plan transversal (perpendiculaire à la direction de propagation de la lumière) soit dans la direction de propagation. Des structures localisées de nature différente constituent une solution possible importante dans chacun des systèmes étudiés ; leurs interactions autant que leurs rôles dans la formation des événements extrêmes ont donc été analysés en détails. Dans le premier système, un laser à semiconducteur monolithique (VCSEL) à large surface avec un absorbant saturable, on présente la formation d'événements extrêmes dans le plan transversal à deux dimensions de l'intensité du champ électrique. En particulier, on met en évidence la liaison entre ces objets et les solitons de cavité, soit stationnaires soit oscillatoires, aussi présents dans le système. Dans le deuxième système, un laser multimodal spatialement étendu dans la direction de propagation avec injection optique, on analyse l'interaction et la fusion des solitons de phase, des structures localisées qui se propagent dans la cavité en transportant une rotation de phase de 2π. Les événements extrêmes ont été étudié dans deux configurations : une première où ils émergent de la collision des solitons de phase avec des autres structures transitoires transportant une charge chirale négative, et une deuxième où des événements d'intensité élevée émergent d'un régime instable de motif en rouleau où les solitons de cavité ne sont pas des solutions stables. Dans les deux systèmes, on examine le rôle de la chiralité dans la formation des événements extrêmes. Dans le troisième système, un laser à semi-conducteur avec injection optique, on étudie dans les détails l'interaction des solitons de cavité dans le plan transversal, décrits comme deux particules soumises à un potentiel d'interaction décroissant exponentiellement avec la distance entre les deux objets : une analogie possible avec les matériaux hydrophobes a été proposée. Des résultats préliminaires présentant des événements extrêmes spatiotemporels dans ce système sont aussi donnés. / Extreme events are phenomena, often considered as catastrophic, that occur in the tail of a distribution usually deviating from an expected, exponential decay. In optics, these events were first studied in the context of fibers, where they have been extensively analyzed, as optical rogue waves, in light of the well known analogy between optics and hydrodynamics, through the nonlinear Schroedinger equation. With the development and the broadening of the field, extreme events have been also studied in dissipative optical systems with or without spatial degrees of freedom. In this Thesis we focused on the study of extreme events in three different active and dissipative optical systems, each presenting one or two spatial degrees of freedom, either in the transverse plane, perpendicular to the direction of propagation of light, or in the propagation direction. Localized structures of different nature represent an important possible solution in each one of the systems here studied, hence their interaction and the role played in the formation of extreme events have been also investigated into details. In the first system, a monolithic broad-area semiconductor laser (VCSEL) with an intracavity saturable absorber, we report on the occurrence of extreme events in the 2D transverse plane of the electric field intensity. In particular we highlight the connection between these objects and cavity solitons, both stationary and oscillatory, also present in the system. In the second system, a highly multimode laser with optical injection spatially extended along the propagation direction, we analyze the interaction and merging of phase solitons, localized structures propagating along the cavity carrying a 2π phase rotation. Extreme events have been investigated in two configurations: a first one where they emerge from the collision of phase solitons with other transient structures carrying a negative chiral charge, and a second one where high-peak events emerge from an unstable roll regime where phase solitons are not a stable solution. In both these systems we investigate the role of chirality in the extreme event formation. In the third system, a broad-area semiconductor laser (VCSEL) with optical injection, we study into details the interaction of cavity solitons in the transverse plane, described as two particles subjected to an interaction potential exponentially decreasing with the distance between the two objects: a possible analogy with hydrophobic materials is here suggested. Some preliminary results showing spatiotemporal extreme events in this system are also given.

In this thesis the path integral formalism is applied to the calculation of the dynamics of dissipative quantum systems. The time evolution of a system of bilinearly coupled bosonic modes is treated using the real-time path integral technique in coherent-state representation. This method is applied to a damped harmonic oscillator within the Caldeira-Leggett model. In order to get the stationary trajectories the corresponding Lagrangian function is diagonalized and then the path integrals are evaluated by means of the stationary-phase method. The time evolution of the reduced density matrix in the basis of coherent states is given in simple analytic form for weak system-bath coupling, i.e. the so-called rotating-wave terms can be evaluated exactly but the non-rotating-wave terms only in a perturbative manner. The validity range of the rotating-wave approximation is discussed from the viewpoint of spectral equations. In addition, it is shown that systems without initial system-bath correlations can exhibit initial jumps in the population dynamics even for rather weak dissipation. Only with initial correlations the classical trajectories for the system coordinate can be recovered. The path integral formalism in a combined phase-space and coherent-state representation is applied to the problem of curve-crossing dynamics. The system of interest is described by two coupled one-dimensional harmonic potential energy surfaces interacting with a heat bath. The mapping approach is used to rewrite the Lagrangian function of the electronic part of the system. Using the Feynman-Vernon influence-functional method the bath is eliminated whereas the non-Gaussian part of the path integral is treated using the perturbation theory in the small coordinate shift between potential energy surfaces. The vibrational and the population dynamics is considered in a lowest order of the perturbation. The dynamics of a Gaussian wave packet is analyzed along a one-dimensional reaction coordinate. Also the damping rate of coherence in the electronic part of the relevant system is evaluated within the ordinary and variational perturbation theory. The analytic expressions for the rate functions are obtained in the low and high temperature regimes.

In this thesis, we first propose a new scalar auxiliary variable (SAV) approach for general dissipative nonlinear systems. This new approach is half computational cost of the original SAV approach, can be extended to high order unconditionally energy stable backward differentiation formula (BDF) schemes and not restricted to the gradient flow structure. Rigorous error estimates for this new SAV approach are conducted for the Allen-Cahn and Cahn-Hilliard type equations from the BDF1 to the BDF5 schemes in a unified form. As an application of this new approach, we construct high order unconditionally stable, fully discrete schemes for the incompressible Navier-Stokes equation with periodic boundary condition. The corresponding error estimates for the fully discrete schemes are also reported. Secondly, by combining the new SAV approach with functional transformation, we propose a new method to construct high-order, linear, positivity/bound preserving and unconditionally energy stable schemes for general dissipative systems whose solutions are positivity/bound preserving. We apply this new method to second order equations: the Allen-Cahn equation with logarithm potential, the Poisson-Nernst-Planck equation and the Keller-Segel equations and fourth order equations: the thin film equation and the Cahn-Hilliard equation with logarithm potential. Ample numerical examples are provided to demonstrate the improved efficiency and accuracy of the proposed method.

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We develop the robust control theories of stopping-time nonlinear systems and switching-control nonlinear systems. We formulate a robust optimal stopping-time control problem for a state-space nonlinear system and give the connection between various notions of lower value function for the associated game (and storage function for the associated dissipative system) with solutions of the appropriate variational inequality (VI). We show that the stopping-time rule can be obtained by solving the VI in the viscosity sense. It also happens that a positive definite supersolution of the VI can be used for stability analysis. We also show how to solve the VI for some prototype examples with one-dimensional state space. For the robust optimal switching-control problem, we establish the Dynamic Programming Principle (DPP) for the lower value function of the associated game and employ it to derive the appropriate system of quasivariational inequalities (SQVI) for the lower value vector function. Moreover we formulate the problem in the <I>L</I>₂-gain/dissipative system framework. We show that, under appropriate assumptions, continuous switching-storage (vector) functions are characterized as viscosity supersolutions of the SQVI, and that the minimal such storage function is equal to the lower value function for the game. We show that the control strategy achieving the dissipative inequality is obtained by solving the SQVI in the viscosity sense; in fact this solution is also used to address stability analysis of the switching system. In addition we prove the comparison principle between a viscosity subsolution and a viscosity supersolution of the SQVI satisfying a boundary condition and use it to give an alternative derivation of the characterization of the lower value function. Finally we solve the SQVI for a simple one-dimensional example by a direct geometric construction. / Ph. D.