On the necessary optimality conditions for the fractional Cucker–Smale optimal control problem

doi:10.1016/j.cnsns.2020.105678

Communications in Nonlinear Science and Numerical Simulation

Volume 96, May 2021, 105678

https://doi.org/10.1016/j.cnsns.2020.105678 Get rights and content

Highlights

•
Study of sparse flocking control for the fractional Cucker–Smale multi-agent model.
•
Taking into consideration history or memory dependency in the selforganization of the group by applying the approach (for designing the Cucker–Smale model) with fractional derivative.
•
Proof of the Pontryagin Maximum Principle (PMP) for the Lagrange optimal control problem governed by a multi-order control system with the Caputo derivative.
•
Applying the PMP to the fractional Cucker–Smale optimal control problem. We provide necessary optimality conditions for external control with limited strength. This external controller by acting only on a few agents enforces flocking in the multi- agent system.
•
Analysis of some particular problems illustrated by numerical examples.

Abstract

This paper develops a sparse flocking control for the fractional Cucker–Smale multi-agent model. The Caputo fractional derivative, in the equations describing the dynamics of a consensus parameter, makes it possible to take into account in the self-organization of group its history and memory dependency. External control is designed based on necessary conditions for a local solution to the appropriate optimal control problem. Numerical simulations demonstrate the effectiveness of the control scheme.

Introduction

The Cucker–Smale (CS) flocking model, first analyzed in [11], is an elementary mathematical model describing a general tendency of multiple interacting agents to coordinate their behavior based on the dynamics of their neighbors. More precisely, it is focused on the alignment of agents’ velocities (in the general context called consensus parameters), when taking into account positions and momenta of each individual (see e.g., [3], [8], [24], [31]). Phenomenons that could be described by the CS model include flocking of birds, schooling of fish, or herding of mammals. Moreover, the question of motion synchronization appears when studying other professedly unconnected problems like goods distribution, spacecraft formation, sensor networks, and many others [10], [27], [28], [35].

In the references cited above, in order to describe the emergence of consensus, authors consider integer order derivatives. It is a well known fact that they are local operators which take into account only the values of a function in a neighborhood of a point. In recent years, however, it has been claimed that agreement could be influenced by the dynamics of each individual in previous times [2], [4], [25], [29], [30]. In order to acknowledge such memory effect some authors introduced fractional (real or complex) order derivatives to the CS model [12], [14], [23]. These operators are of non-local nature and because of that, they characterize time-dependent processes very efficiently [7], [33], [38]. Models with fractional operators play an important role in nonlinear sciences [13], [22], [26], [34], [36]. In the literature, we may find many definitions of fractional-order differential operators: Grunwald–Letnikov [18], Riemann–Liouville [32], Caputo [32], and their generalizations like Hilfer [13], [37] or derivative with respect to another function [1], [32]. The most common one is the Caputo fractional derivative because initial conditions for equations with this type of derivative are the same as the ones for integer-order differential equations, which is physically understandable and therefore more applicable to real-world problems. Therefore, in this paper, we replace the classical time derivative by the Caputo fractional derivative in the equations describing the dynamics of agents’ velocities. This strategy makes it possible to take into account history or memory dependency in the self-organization of the group. Additionally, the time-fractional order model may reduce errors resulting from the neglect of parameters in the model [33].

Despite the fact that self-organization is rather ubiquitous in natural bio-groups, in social or engineering multi-agent models it is legitimate to ask whether - in case of lost cohesion - additional forces acting on the agents of the system may restore stability and achieve consensus. One of the ways, that are proposed in the literature, to deal with this problem is by introducing an external controller [8], [9], [29], [39]. In order to promote self-organization in multi-agent systems, the authors of [8], [9] consider sparsity-promoting optimization of control. This is done by the design of a finite horizon optimal control problem with the cost functional that consists of two parts: flocking and sparsity. By the flocking part, the distance to a consensus is minimized and at the same time by a sparsity part, a control is penalized by means of the mixed $l_{1}^{N} - l_{2}^{d}$ norm. In other words, at any given time, instead of squandering resources on the entire group at once, the control acts on the least amount of agents, and there is a maximal amount of control resources that may be spent on interventions.

The aim of this paper is to use the aforementioned approach to the fractional CS model which is obtained by replacing the usual time derivative by the Caputo fractional time derivative in the equation describing the dynamics of a consensus parameter. To this end, we state and prove the Pontryagin Maximum Principle for a Lagrange problem described by a multi-order control system with Caputo derivative. This result allows us to derive the necessary optimality conditions for the fractional Cucker–Smale optimal control problem. In this problem, we seek for a sparse control for which the associated solution to the fractional system of the CS type tends to consensus. By numerical simulations, we demonstrate the effectiveness of the proposed control scheme. Summing up, the contributions of this paper are as follows:

•
The approach for designing the Cucker–Smale model with fractional derivative takes into consideration history or memory dependency in the self-organization of the group.
•
We prove Pontryagin Maximum Principle (PMP) for the Lagrange optimal control problem governed by a multi-order control system with the Caputo derivative.
•
The PMP is applied for the fractional Cucker–Smale optimal control problem. We provide necessary optimality conditions for external control with limited strength. This external controller by acting only on a few agents enforces flocking in the multi-agent system.

The remainder of the paper is organized as follows. In Section 2, we provide necessary definitions and recall or extend some results from fractional calculus. Section 3 is devoted to the Lagrange problem described by a multi-order control system with Caputo derivatives. The Pontryagin Maximum Principle for this problem is formulated there. For detailed proof of PMP, which is a new result by itself, we refer the reader to Appendix A. This result allows us to obtain the necessary optimality conditions for the optimal control problem of the system governed by the modified CS system proposed in [12]. Namely, the usual time derivative is replaced by the Caputo fractional time derivative in the second equation of the classical CS system while in the first equation the usual time derivative is kept. Therefore, the first equation could be still treated as a position of an agent and simultaneously, by the second equation, it is possible to incorporate the memory factor into the consensus process. Section 4 discusses the fractional CS flocking model with sparse control. Our main result is stated and proved: PMP for the finite time optimal control problem of the fractional Cucker–Smale system with the minimization criterion that is a combination of the distance from consensus and the $l_{1}^{N} - l_{2}^{d}$ norm of the control. In Section 5, the theoretical analysis is supported by numerical examples showing the effectiveness of the proposed sparsity-promoting optimization of control. Section 6 draws conclusions.

Section snippets

Preliminaries

In this section, the fundamental concepts of fractional operators are recalled (see [21], [32]). We also prove preliminary results that will be used in the following sections.

Let $[a, b] \subset R$ be any bounded interval, $α > 0,$ and $f (\cdot) \in L^{1} ([a, b], R^{n})$ . By the left–sided and the right-sided Riemann–Liouville integrals of function $f$ of order $α$ we mean functions $\begin{matrix} (I_{a +}^{α} f) (t) : = \frac{1}{Γ (α)} \int_{a}^{t} \frac{f (τ)}{(t - τ)^{1 - α}} d τ, t \in [a, b] a . e . \\ and \\ (I_{b -}^{α} f) (t) : = \frac{1}{Γ (α)} \int_{t}^{b} \frac{f (τ)}{(τ - t)^{1 - α}} d τ, t \in [a, b] a . e ., \end{matrix}$ respectively.

In view of convergence (cf. [32,

Pontryagin maximum principle for a Lagrange problem governed by a multi-order control system with Caputo derivatives

Let us consider the following Lagrange problem: $minimize H (x (\cdot), u (\cdot)) = \int_{0}^{T} f_{0} (t, x (t), u (t)) d t,$ subject to $\begin{matrix} {\begin{matrix} (^{C} D_{0 +}^{α_{1}} x_{1}) (t) = f_{1} (t, x (t), u (t)), \\ ⋮ & t \in [0, T] a . e ., \\ (^{C} D_{0 +}^{α_{n}} x_{n}) (t) = f_{n} (t, x (t), u (t)), \\ x (0) = x_{0}, \\ u (t) \in M \subset R^{m}, \end{matrix} \end{matrix}$ where $x_{i} : [0, T] \to R^{r},$ $x = (x_{1}, \dots, x_{n}),$ $x_{0} = (x_{10}, \dots, x_{n 0}) \in R^{n r},$ $α_{i} \in (0, 1],$ $f_{i} : [0, T] \times R^{n r} \times M \to R^{r},$ $i = 1, \dots, n,$ $f_{0} : [0, T] \times R^{n r} \times M \to R$ .

Let $1 \leq p < \infty,$ $0 < \frac{1}{p} < α_{i} \leq 1,$ $i = 1, \dots, n,$ and $α = (α_{1}, \dots, α_{n})$ . We define the set of controls $U_{M}$ and the space ${AC}_{0 +}^{α, p}$ in the following way: $U_{M} : = {u (\cdot) : [0, T] \to R^{m} - m e a s u r a b l e o n [0, T] a n d u (t) \in M, t \in [0, T] a . e .},$ ${AC}_{0 +}^{α, p} : =_{C} A C_{0 +}^{α_{1}, p} ($

Optimal control of the fractional Cucker–Smale model

In the seminal paper [11], Cucker and Smale introduced a mathematical model of flocking, currently known as the Cucker–Smale flocking model. Its dynamics is based solely on the currently available information of the network, namely is based on the assumption that the velocity of each individual is determined in terms of position and velocities of other members of the group. On the other hand, it is quite obvious that each individual’s motion (behavior) is also based on its past observation

Numerical examples

In this section, we analyze two particular problems to demonstrate the effectiveness of the proposed control strategy. In both examples we consider the following communication weight function $η (s) = \frac{10}{(1 + s)^{\frac{1}{2}}} .$

Example 1

Let us consider problem (10)–(11) with parameters: $d = 1,$ $N = 3,$ $γ = 1,$ $μ = 0.9,$ $K = 1$ . Namely, the problem reads: minimize the functional $H ((z (\cdot), v (\cdot)), u (\cdot)) = \int_{0}^{20} (\sum_{i = 1}^{3} (v_{i} (t) - \frac{1}{3} \sum_{j = 1}^{3} v_{j} (t))^{2} + \sum_{i = 1}^{3} | u_{i} (t) |) d t,$ subject to $\begin{matrix} {\begin{matrix} {\dot{z}}_{i} (t) = v_{i} (t), \\ ^{C} D_{0^{+}}^{0.9} v_{i} (t) = \frac{1}{3} \sum_{j = 1}^{3} η ((z_{j} (t) - z_{i} (t))^{2}) (v_{j} (t) - v_{i} (t)) + u_{i} (t), t \in [0, 20] a . e ., \\ (z_{i} ( \end{matrix} \end{matrix}$

Conclusions

In this work, we have considered Cucker–Smale sparse optimal control problem, where the dynamics of the consensus parameter is expressed via Caputo fractional derivative. This approach is novel and more accurate since fractional derivatives are non-local operators and allow us to take into account the dynamics of each individual in previous times. The main result of this work refers to the PMP type theorem, which provides necessary optimality conditions for a solution to the considered

CRediT authorship contribution statement

Ricardo Almeida: Formal analysis, Writing - review & editing. Rafał Kamocki: Investigation, Resources, Conceptualization, Writing - review & editing. Agnieszka B. Malinowska: Supervision, Conceptualization, Investigation, Writing - review & editing. Tatiana Odzijewicz: Investigation, Conceptualization, Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgment

R. Almeida is supported by Portuguese funds through the CIDMA - Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (FCT-Fundação para a Ciência e a Tecnologia), within project UIDB/04106/2020. A. B. Malinowska is supported by the Bialystok University of Technology Grant and T. Odzijewicz by the SGH Warsaw School of Economics Grant, both financed from a subsidy provided by the Minister of Science and Higher Education in

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Research paperOn the necessary optimality conditions for the fractional Cucker–Smale optimal control problem

Highlights

Abstract

Introduction

Section snippets

Preliminaries

Pontryagin maximum principle for a Lagrange problem governed by a multi-order control system with Caputo derivatives

Optimal control of the fractional Cucker–Smale model

Numerical examples

Conclusions

CRediT authorship contribution statement

Declaration of Competing Interest

Acknowledgment

Commun Nonlinear Sci Numer Simul

IFAC- PapersOnLine

J Math Anal Appl

IEEE Control Syst

Nonlinear Dyn

Control of multi-agent systems with applications to distributed frequency control of power systems

Licentiate Thesis, Stockholm, Sweden

Ring patterns and their bifurcations in a nonlocal model of biological swarms

Commun Math Sci

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ESAIM Contr Optim Ca

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IEEE Trans Syst Man Cybern Syst, Part B