Optimal control of multiscale systems using reduced-order models

We study optimal control of diffusions with slow and fast variables and address a question raised by practitioners: is it possible to first eliminate the fast variables before solving the optimal control problem and then use the optimal control computed from the reduced-order model to control the original, high-dimensional system? The strategy "first reduce, then optimize"--rather than "first optimize, then reduce"--is motivated by the fact that solving optimal control problems for high-dimensional multiscale systems is numerically challenging and often computationally prohibitive. We state sufficient and necessary conditions, under which the "first reduce, then control" strategy can be employed and discuss when it should be avoided. We further give numerical examples that illustrate the "first reduce, then optmize" approach and discuss possible pitfalls

Stabilizability in optimal control

We extend the well known concepts of sampling and Euler solutions for control systems associated to discontinuous feedbacks by considering also associated costs; in particular, we introduce the notions of Sample and Euler stabilizability to a closed target set C with W-regulated cost, which roughly means that we require the existence of a stabilizing feedback such that all the corresponding sampling and Euler solutions have finite costs, bounded above by a continuous, state-dependent function W, divided by some positive constant c. We prove that the existence of a special Control Lyapunov Function W, called c-Minimum Restraint function, c-MRF, implies Sample and Euler stabilizability to C with W-regulated cost, so extending [Motta, Rampazzo 2013], [Lai, Motta, Rampazzo, 2016], where the existence of a c-MRF was only shown to yield global asymptotic controllability to C with W-regulated cost

A singular stochastic control problem with interconnected dynamics

In this paper we study a Markovian two-dimensional bounded-variation stochastic control problem whose state process consists of a difusive mean-reverting component and of a purely controlled one. The main problem's characteristic lies in the interaction of the two components of the state process: the mean-reversion level of the difusive component is an afne function of the current value of the purely controlled one. By relying on a combination of techniques from viscosity theory and free-boundary analysis, we provide the structure of the value function and we show that it satisfes a second-order smooth-ft principle. Such a regularity is then exploited in order to determine a system of functional equations solved by the two monotone continuous curves (free boundaries) that split the control problem's state space into three connected regions. Further properties of the free boundaries are also obtained

International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more