Optimal Output Regulation for Square, Over-Actuated and Under-Actuated Linear Systems

This paper considers two different problems in trajectory tracking control for linear systems. First, if the control is not unique which is most input energy efficient. Second, if exact tracking is infeasible which control performs most accurately. These are typical challenges for over-actuated systems and for under-actuated systems, respectively. We formulate both goals as optimal output regulation problems. Then we contribute two new sets of regulator equations to output regulation theory that provide the desired solutions. A thorough study indicates solvability and uniqueness under weak assumptions. E.g., we can always determine the solution of the classical regulator equations that is most input energy efficient. This is of great value if there are infinitely many solutions. We derive our results by a linear quadratic tracking approach and establish a useful link to output regulation theory.Comment: 8 pages, 0 figures, final version to appear in IEEE Transactions on Automatic Contro

Efficient Algorithms for Optimal Control of Quantum Dynamics: The "Krotov'' Method unencumbered

Efficient algorithms for the discovery of optimal control designs for coherent control of quantum processes are of fundamental importance. One important class of algorithms are sequential update algorithms generally attributed to Krotov. Although widely and often successfully used, the associated theory is often involved and leaves many crucial questions unanswered, from the monotonicity and convergence of the algorithm to discretization effects, leading to the introduction of ad-hoc penalty terms and suboptimal update schemes detrimental to the performance of the algorithm. We present a general framework for sequential update algorithms including specific prescriptions for efficient update rules with inexpensive dynamic search length control, taking into account discretization effects and eliminating the need for ad-hoc penalty terms. The latter, while necessary to regularize the problem in the limit of infinite time resolution, i.e., the continuum limit, are shown to be undesirable and unnecessary in the practically relevant case of finite time resolution. Numerical examples show that the ideas underlying many of these results extend even beyond what can be rigorously proved.Comment: 19 pages, many figure

Finite-time behavior of inner systems

In this paper, we investigate how nonminimum phase characteristics of a dynamical system affect its controllability and tracking properties. For the class of linear time-invariant dynamical systems, these characteristics are determined by transmission zeros of the inner factor of the system transfer function. The relation between nonminimum phase zeros and Hankel singular values of inner systems is studied and it is shown how the singular value structure of a suitably defined operator provides relevant insight about system invertibility and achievable tracking performance. The results are used to solve various tracking problems both on finite as well as on infinite time horizons. A typical receding horizon control scheme is considered and new conditions are derived to guarantee stabilizability of a receding horizon controller

Rough volatility and portfolio optimisation under small transaction costs

The first chapter of the thesis presents the study of the linear-quadratic ergodic control problem of fractional Brownian motion. Ergodic control problems arise naturally in the context of small cost asymptotic expansion of utility maximisation problems with frictions. The optimal solution to the ergodic control problem is derived through the use of an infinite dimensional Markovian representation of fractional Brownian motion as a superposition of Ornstein-Uhlenbeck processes. This solution then allows to compute explicit formulas for the minimised objective value through the variance of the stationary distribution of the Ornstein-Uhlenbeck processes. Building on the first chapter, the second chapter of the thesis presents the main result. This is motivated by the problem an agent faces when trying to minimise her utility loss in the presence of quadratic trading costs in a rough volatility model. Minimising the utility loss amounts to studying a tracking problem of a target that depends on the rough volatility process. This tracking problem is minimised at leading order by an asymptotically optimal strategy that is closely linked to the ergodic control problem of fractional Brownian motion. This asymptotically optimal strategy is explicitly derived. Moreover, the leading order of the small cost expansion is shown to depend only on the roughest part of the considered target. It therefore depends on the Hurst parameter. The third chapter is devoted to a numerical analysis of the utility loss studied in the second chapter. For this, we compare the utility loss in a rough volatility model to a semimartingale stochastic volatility model. The parameter values for both models are fitted to match frictionless utility for realistic values. By applying the result obtained in the second chapter of the thesis, the difference between leading order of utility loss can be explicitly compared

Nonlinear modulational stability of periodic traveling-wave solutions of the generalized Kuramoto-Sivashinsky equation

In this paper we consider the spectral and nonlinear stability of periodic traveling wave solutions of a generalized Kuramoto-Sivashinsky equation. In particular, we resolve the long-standing question of nonlinear modulational stability by demonstrating that spectrally stable waves are nonlinearly stable when subject to small localized (integrable) perturbations. Our analysis is based upon detailed estimates of the linearized solution operator, which are complicated by the fact that the (necessarily essential) spectrum of the associated linearization intersects the imaginary axis at the origin. We carry out a numerical Evans function study of the spectral problem and find bands of spectrally stable periodic traveling waves, in close agreement with previous numerical studies of Frisch-She-Thual, Bar-Nepomnyashchy, Chang-Demekhin-Kopelevich, and others carried out by other techniques. We also compare predictions of the associated Whitham modulation equations, which formally describe the dynamics of weak large scale perturbations of a periodic wave train, with numerical time evolution studies, demonstrating their effectiveness at a practical level. For the reader's convenience, we include in an appendix the corresponding treatment of the Swift-Hohenberg equation, a nonconservative counterpart of the generalized Kuramoto-Sivashinsky equation for which the nonlinear stability analysis is considerably simpler, together with numerical Evans function analyses extending spectral stability analyses of Mielke and Schneider.Comment: 78 pages, 11 figure