TY - JOUR A1 - Biccari, Umberto A1 - Zuazua, Enrique T1 - A Stochastic Approach to the Synchronization of Coupled Oscillators N2 - This paper deals with an optimal control problem associated with the Kuramoto model describing the dynamical behavior of a network of coupled oscillators. Our aim is to design a suitable control function allowing us to steer the system to a synchronized configuration in which all the oscillators are aligned on the same phase. This control is computed via the minimization of a given cost functional associated with the dynamics considered. For this minimization, we propose a novel approach based on the combination of a standard Gradient Descent (GD) methodology with the recently-developed Random Batch Method (RBM) for the efficient numerical approximation of collective dynamics. Our simulations show that the employment of RBM improves the performances of the GD algorithm, reducing the computational complexity of the minimization process and allowing for a more efficient control calculation. KW - coupled oscillators KW - gradient descent KW - Kuramoto model KW - optimal control KW - random batch method Y1 - 2020 U6 - https://doi.org/10.3389/fenrg.2020.00115 VL - 8 ER - TY - JOUR A1 - Heiland, Jan A1 - Zuazua, Enrique T1 - Classical system theory revisited for Turnpike in standard state space systems and impulse controllable descriptor systems N2 - The concept of turnpike connects the solution of long but finite time horizon optimal control problems with steady state optimal controls. A key ingredient of the analysis of the turnpike is the linear quadratic regulator problem and the convergence of the solution of the associated differential Riccati equation as the terminal time approaches infinity. This convergence has been investigated in linear systems theory in the 1980s. We extend classical system theoretic results for the investigation of turnpike properties of standard state space systems and descriptor systems. We present conditions for turnpike in the nondetectable case and for impulse controllable descriptor systems. For the latter, in line with the theory for standard linear systems, we establish existence and convergence of solutions to a generalized differential Riccati equation. KW - descriptor systems KW - linear systems KW - long time behavior KW - optimal control KW - Riccati equations Y1 - 2020 ER - TY - JOUR A1 - Esteve, Carlos A1 - Geshkovski, Borjan A1 - Pighin, Dario A1 - Zuazua, Enrique T1 - Large-time asymptotics in deep learning N2 - It is by now well-known that practical deep supervised learning may roughly be cast as an optimal control problem for a specific discrete-time, nonlinear dynamical system called an artificial neural network. In this work, we consider the continuous-time formulation of the deep supervised learning problem, and study the latter’s behavior when the final time horizon increases, a fact that can be interpreted as increasing the number of layers in the neural network setting. When considering the classical regularized empirical risk minimization problem, we show that, in long time, the optimal states converge to zero training error, namely approach the zero training error regime, whilst the optimal control parameters approach, on an appropriate scale, minimal norm parameters with corresponding states precisely in the zero training error regime. This result provides an alternative theoretical underpinning to the notion that neural networks learn best in the overparametrized regime, when seen from the large layer perspective. We also propose a learning problem consisting of minimizing a cost with a state tracking term, and establish the well-known turnpike property, which indicates that the solutions of the learning problem in long time intervals consist of three pieces, the first and the last of which being transient short-time arcs, and the middle piece being a long-time arc staying exponentially close to the optimal solution of an associated static learning problem. This property in fact stipulates a quantitative estimate for the number of layers required to reach the zero training error regime. Both of the aforementioned asymptotic regimes are addressed in the context of continuous-time and continuous space-time neural networks, the latter taking the form of nonlinear, integro-differential equations, hence covering residual neural networks with both fixed and possibly variable depths. KW - deep learning KW - Neural ODEs KW - optimal control KW - Residual Neural Networks KW - Supervised Learning Y1 - 2020 ER - TY - JOUR A1 - Adelhütte, Dennis A1 - Aßmann, Denis A1 - Gonzàlez Grandòn, Tatiana A1 - Gugat, Martin A1 - Heitsch, Holger A1 - Liers, Frauke A1 - Henrion, René A1 - Nitsche, Sabrina A1 - Schultz, Rüdiger A1 - Stingl, Michael A1 - Wintergerst, David T1 - Joint model of probabilistic/robust (probust) constraints applied to gas network optimization N2 - Optimization tasks under uncertain conditions abound in many real-life applications. Whereas solution approaches for probabilistic constraints are often developed in case the uncertainties can be assumed to follow a certain probability distribution, robust approaches are usually used in case solutions are sought that are feasible for all realizations of uncertainties within some pre-defined uncertainty set. As many applications contain different types of uncertainties that require robust as well as probabilistic treatments, we deal with a class of joint probabilistic/robust constraints as its appears in optimization problems under uncertainty. Focusing on complex uncertain gas network optimization problems, we show the relevance of this class of problems for the task of maximizing free booked capacities in an algebraic model for a stationary gas network. We furthermore present approaches for their solution. Finally, we study the problem of controlling a transient system that is governed by the wave equation. The task consists in determining controls such that a certain robustness measure remains below some given upper bound, with high probability. KW - robust optimization KW - chance constraints KW - optimal control KW - spheric-radial decomposition Y1 - 2017 U6 - https://doi.org/10.1007/s10013-020-00434-y ER - TY - JOUR A1 - Hante, Falk A1 - Leugering, Günter A1 - Martin, Alexander A1 - Schewe, Lars A1 - Schmidt, Martin T1 - Challenges in optimal control problems for gas and fluid flow in networks of pipes and canals: From modeling to industrial applications N2 - We consider optimal control problems for the flow of gas or fresh water in pipe networks as well as drainage or sewer systems in open canals. The equations of motion are taken to be represented by the nonlinear isothermal Euler gas equations, the water hammer equations, or the St.~Venant equations for flow. We formulate model hierarchies and derive an abstract model for such network flow problems including pipes, junctions, and controllable elements such as valves, weirs, pumps, as well as compressors. We use the abstract model to give an overview of the known results and challenges concerning equilibria, well-posedness, controllability, and optimal control. A major challenge concerning the optimization is to deal with switching on-off states that are inherent to controllable devices in such applications combined with continuous simulation and optimization of the gas flow. We formulate the corresponding mixed-integer nonlinear optimal control problems and outline a decomposition approach as a solution technique. KW - Networks KW - pipes KW - optimal control KW - Euler and St. Venant equations KW - hierarchy of models Y1 - 2016 ER -