Relaxed dissipativity assumptions and a simplified algorithm for multiobjective MPC

We consider nonlinear model predictive control (MPC) with multiple competing cost functions. In each step of the scheme, a multiobjective optimal control problem with a nonlinear system and terminal conditions is solved. We propose an algorithm and give performance guarantees for the resulting MPC closed loop system. Thereby, we significantly simplify the assumptions made in the literature so far by assuming strict dissipativity and the existence of a compatible terminal cost for one of the competing objective functions only. We give conditions which ensure asymptotic stability of the closed loop and, what is more, obtain performance estimates for all cost criteria. Numerical simulations on various instances illustrate our findings. The proposed algorithm requires the selection of an efficient solution in each iteration, thus we examine several selection rules and their impact on the results

On the relationship between stochastic turnpike and dissipativity notions

In this paper, we introduce and study different dissipativity notions and different turnpike properties for discrete-time stochastic nonlinear optimal control problems. The proposed stochastic dissipativity notions extend the classic notion of Jan C. Willems to $L^r$ random variables and to probability measures. Our stochastic turnpike properties range from a formulation for random variables via turnpike phenomena in probability and in probability measures to the turnpike property for the moments. Moreover, we investigate how different metrics (such as Wasserstein or L\'evy-Prokhorov) can be leveraged in the analysis. Our results are built upon stationarity concepts in distribution and in random variables and on the formulation of the stochastic optimal control problem as a finite-horizon Markov decision process. We investigate how the proposed dissipativity notions connect to the various stochastic turnpike properties and we work out the link between these two different forms of dissipativity

A Polynomial Chaos Approach to Stochastic LQ Optimal Control: Error Bounds and Infinite-Horizon Results

The stochastic linear-quadratic regulator problem subject to Gaussian disturbances is well known and usually addressed via a moment-based reformulation. Here, we leverage polynomial chaos expansions, which model random variables via series expansions in a suitable $\mathcal{L}^2$ probability space, to tackle the non-Gaussian case. We present the optimal solutions for finite and infinite horizons. Moreover, we quantify the truncation error and we analyze the infinite-horizon asymptotics. We show that the limit of the optimal trajectory is the unique solution to a stationary optimization problem in the sense of probability measures. A numerical example illustrates our findings

Pathwise turnpike and dissipativity results for discrete-time stochastic linear-quadratic optimal control problems

We investigate pathwise turnpike behavior of discrete-time stochastic linear-quadratic optimal control problems. Our analysis is based on a novel strict dissipativity notion for such problems, in which a stationary stochastic process replaces the optimal steady state of the deterministic setting. The analytical findings are illustrated by a numerical example

Turnpike and dissipativity in generalized discrete-time stochastic linear-quadratic optimal control

We investigate different turnpike phenomena of generalized discrete-time stochastic linear-quadratic optimal control problems. Our analysis is based on a novel strict dissipativity notion for such problems, in which a stationary stochastic process replaces the optimal steady state of the deterministic setting. We show that from this time-varying dissipativity notion, we can conclude turnpike behaviors concerning different objects like distributions, moments, or sample paths of the stochastic system and that the distributions of the stationary pair can be characterized by a stationary optimization problem. The analytical findings are illustrated by numerical simulations

Microhabitat use, population densities, and size distributions of sulfur cave-dwelling Poecilia mexicana

The Cueva del Azufre in Tabasco, Mexico, is a nutrient-rich cave and its inhabitants need to cope with high levels of dissolved hydrogen sulfide and extreme hypoxia. One of the successful colonizers of this cave is the poeciliid fish Poecilia mexicana, which has received considerable attention as a model organism to examine evolutionary adaptations to extreme environmental conditions. Nonetheless, basic ecological data on the endemic cave molly population are still missing; here we aim to provide data on population densities, size class compositions and use of different microhabitats. We found high overall densities in the cave and highest densities at the middle part of the cave with more than 200 individuals per square meter. These sites have lower H2S concentrations compared to the inner parts where most large sulfide sources are located, but they are annually exposed to a religious harvesting ceremony of local Zoque people called La Pesca. We found a marked shift in size/age compositions towards an overabundance of smaller, juvenile fish at those sites. We discuss these findings in relation to several environmental gradients within the cave (i.e., differences in toxicity and lighting conditions), but we also tentatively argue that the annual fish harvest during a religious ceremony (La Pesca) locally diminishes competition (and possibly, cannibalism by large adults), which is followed by a phase of overcompensation of fish densities