Mathematical Problems in Engineering

Volume 2015, Article ID 850124, 14 pages

http://dx.doi.org/10.1155/2015/850124

## Uncertainty Quantification in Control Problems for Flocking Models

^{1}Fakultät für Mathematik, Technische Universität München, Boltzmannstraße 3, Garching, 85748 Munich, Germany^{2}Department of Mathematics and Computer Science, University of Ferrara, Via N. Machiavelli 35, 44121 Ferrara, Italy

Received 19 January 2015; Revised 27 April 2015; Accepted 28 April 2015

Academic Editor: Angel Sánchez

Copyright © 2015 Giacomo Albi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The optimal control of flocking models with random inputs is investigated from a numerical point of view. The effect of uncertainty in the interaction parameters is studied for a Cucker-Smale type model using a generalized polynomial chaos (gPC) approach. Numerical evidence of threshold effects in the alignment dynamic due to the random parameters is given. The use of a selective model predictive control permits steering of the system towards the desired state even in unstable regimes.

#### 1. Introduction

The aggregate motion of a multiagent system is frequently seen in the real world. Common examples are schools of fishes, swarms of bees and herds of sheep, natural phenomena that inspired important applications in many fields such as biology, engineering and economy [1]. As a consequence, the significance of new mathematical models, for understanding and predicting these complex dynamics, is widely recognized. Several heuristic rules of flocking have been introduced as alignment, separation, and cohesion [2, 3]. Nowadays these mathematical problems, and their constrained versions, are deeply studied from both the microscopic viewpoint [4–7] and their kinetic and mean-field approximations [8–13]. We refer to [1] for a recent introduction on the subject.

In an applicative framework a fundamental step for the study of such models is represented by the introduction of stochastic parameters reflecting the uncertainty due to wide range of phenomena, such as the weathers influence during an experiment, temperature variations, or even human errors. Therefore quantifying the influence of uncertainties on the main dynamics is of paramount importance to build more realistic models and to give better predictions of their behavior. In the modeling of self-organized system, different ways to include random sources have been studied and analyzed; see, for example, [3, 14–17]. In this paper we focus on the case where the uncertainty acts directly in the parameter characterizing the interaction dynamic between the agents.

We present a numerical approach having roots in the numerical techniques for uncertainty quantification (UQ) and model predictive control (MPC). Among the most popular methods for UQ, the generalized polynomial chaos (gPC) has recently received deepest attentions [18]. Jointly with Stochastic Galerkin (SG) this class of numerical methods is usually applied in physical and engineering problems, for which fast convergence is needed. Applications of gPC-Galerkin schemes to flocking dynamics, and their controlled versions, are almost unexplored in the actual state of art.

We give numerical evidence of threshold effects in the alignment dynamic due to the random parameters. In particular the presence of a negative tail in the distribution of the random inputs leads to the divergence of the expected values for the system velocities. The use of a selective model predictive control permits steering of the system towards the desired state even in such unstable regimes.

The rest of the paper is organized as follows. In Section 2 we introduce briefly a Cucker-Smale dynamic with interaction function depending on stochastic parameters and analyze the system behavior in the case of uniform interactions. The gPC approach is then summarized in Section 3. Subsequently, in Section 4 we consider the gPC scheme in a constrained setting and derive a selective model predictive approximation of the system. Next, in Section 5 we report several numerical experiments which illustrate the different features of the numerical method. Extensions of the present approach are finally discussed in Section 6.

#### 2. Cucker-Smale Dynamic with Random Inputs

We introduce a Cucker-Smale type [10] differential system depending on a random variable with a given distribution . Let , , evolving as follows: where is a time-dependent random function characterizing the uncertainty in the interaction rates and is a symmetric function describing the dependence of the alignment dynamic from the agents positions. A classical choice of space-dependent interaction function is related to the distance between two agents where and are given parameters. Mathematical results concerning the system behavior in the deterministic case () can be found in [10]. In particular unconditional alignment emerges for . Let us observe that, even for the model with random inputs (1), the mean velocity of the system is conserved in time since the symmetry of implies Therefore for each we have .

##### 2.1. The Uniform Interaction Case

To better understand the leading dynamic let us consider the simpler uniform interaction case when , leading to the following equation for the velocities:

The differential equation (5) admits an exact solution depending on the random input . More precisely if the initial velocities are deterministically known we have that where is the mean velocity of the system. In what follows we analyze the evolution of (6) for different choices of and of the distribution of the random variable .

*Example 1. *Let us consider a random scattering rate written in terms of the following decomposition: where is a nonnegative function depending on . The expected velocity of the th agent is defined by Then each agent evolves its expected velocity according to For example, let us choose , where the random variable is normally distributed; that is, . Then, for each , we need to evaluate the following integral:

The explicit form is easily found through standard techniques and yields

From (11) we observe a threshold effect in the asymptotic convergence of the mean velocity of each agent toward . It is immediately seen that if it follows that, for , the expected velocity diverges. In particular, if we have that the solution starts to diverge as soon as . Note that this threshold effect is essential due to the negative tail of the normal distribution. In fact, if we now consider a random variable taking only nonnegative values, for example, exponentially distributed for some positive parameter , from (9) we obtain which corresponds to and therefore as . Then independently from the choice of the rate we obtain for each agent convergences toward the average initial velocity of the system. Finally, in case of a uniform random variable we obtain that is,which implies the divergence of the system in time as soon as assumes negative values.

*Example 2. *Next we consider a random scattering rate with time-dependent distribution function; that is, with . As an example we investigate the case of a normally distributed random parameter with given mean and time-dependent variance: . It is straightforward to rewrite as a translation of a standard normal-distributed variable ; that is, where . The expected velocities read which correspond toSimilarly to the case of a time-independent normal variable a threshold effect occurs for large times; that is, the following condition on the variance of the distribution implies the divergence of system (5). As a consequence instability can be avoided by assuming a variance decreasing sufficiently fast in time. The simplest choice is represented by for some . Condition (21) becomes For example, if the previous condition implies that the system diverges for each .

#### 3. A gPC Based Numerical Approach

In this section we approximate the Cucker-Smale model with random inputs using a generalized polynomial chaos approach. For the sake of clarity we first recall some basic facts concerning gPC approximations.

##### 3.1. Preliminaries on gPC Approximations

Let be a probability space, that is, an ordered triple with any set, a -algebra, and a probability measure on , where we define a random variable with the Borel set of . Moreover let us consider , , and certain spatial and temporal subsets. For the sake of simplicity we focus on real-valued functions depending on a single random input

In any case it is possible to extend the setup of the problem to a -dimensional vector of random variables ; see [19]. Let us consider now the linear space of polynomials of of degree up to , namely, . From classical results in approximation theory it is possible to represent the distribution of random functions with orthogonal polynomials , that is, an orthogonal basis of : with as the Kronecker delta function. Assuming that the probability law , involved in the definition of the introduced function , has finite second-order moment, then the complete polynomial chaos expansion of is given by

According to the Cameron-Martin theorem and to the Askey scheme, results that pave a connection between random variables and orthogonal polynomials [18, 20, 21], we chose a set of polynomials which constitutes the optimal basis with respect to the distribution of the introduced random variable in agreement with Table 1.