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Register a new userSparse control of Hegselmann-Krause models: Black hole and declustering
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This paper elaborates control strategies to prevent clustering effects in opinion formation models. This is the exact opposite of numerous situations encountered in the literature where, on the contrary, one seeks controls promoting consensus. In order to promote declustering, instead of using the classical variance that does not capture well the phenomenon of dispersion, we introduce an entropy-type functional that is adapted to measuring pairwise distances between agents. We then focus on a Hegselmann-Krause-type system and design declustering sparse controls both in finite-dimensional and kinetic models. We provide general conditions characterizing whether clustering can be avoided as function of the initial data. Such results include the description of black holes (where complete collapse to consensus is not avoidable), safety zones (where the control can keep the system far from clustering), basins of attraction (attractive zones around the clustering set) and collapse prevention (when convergence to the clustering set can be avoided).
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The original Hegselmann-Krause (HK) model is composed of a finite number of agents characterized by their opinion, a number in $[0,1]$. An agent updates its opinion via taking the average opinion of its neighbors whose opinion differs by at most $epsilon$ for $epsilon>0$ a confidence bound. An agent is absolutely stubborn if it does not change its opinion while update, and absolutely open-minded if its update is the average opinion of its neighbors. There are two types of HK models--the synchronous HK model and the asynchronous HK model. The paper is about a variant of the HK dynamics, called the mixed model, where each agent can choose its degree of stubbornness and mix its opinion with the average opinion of its neighbors at all times. The mixed model reduces to the synchronous HK model if all agents are absolutely open-minded all the time, and the asynchronous HK model if only one uniformly randomly selected agent is absolutely open-minded and the others are absolutely stubborn at all times. In cite{mhk}, we discuss the mixed model deterministically. Point out some properties of the synchronous HK model, such as finite-time convergence, do not hold for the mixed model. In this topic, we study the mixed model nondeterministically. List some properties of the asynchronous model which do not hold for the mixed model. Then, study circumstances under which the asymptotic stability holds.
The original Hegselmann-Krause (HK) model consists of a set of~$n$ agents that are characterized by their opinion, a number in~$[0, 1]$. Each agent, say agent~$i$, updates its opinion~$x_i$ by taking the average opinion of all its neighbors, the agents whose opinion differs from~$x_i$ by at most~$epsilon$. There are two types of~HK models: the synchronous~HK model and the asynchronous~HK model. For the synchronous model, all the agents update their opinion simultaneously at each time step, whereas for the asynchronous~HK model, only one agent chosen uniformly at random updates its opinion at each time step. This paper is concerned with a variant of the~HK opinion dynamics, called the mixed~HK model, where each agent can choose its degree of stubbornness and mix its opinion with the average opinion of its neighbors at each update. The degree of the stubbornness of agents can be different and/or vary over time. An agent is not stubborn or absolutely open-minded if its new opinion at each update is the average opinion of its neighbors, and absolutely stubborn if its opinion does not change at the time of the update. The particular case where, at each time step, all the agents are absolutely open-minded is the synchronous~HK model. In contrast, the asynchronous model corresponds to the particular case where, at each time step, all the agents are absolutely stubborn except for one agent chosen uniformly at random who is absolutely open-minded. We first show that some of the common properties of the synchronous~HK model, such as finite-time convergence, do not hold for the mixed model. We then investigate conditions under which the asymptotic stability holds, or a consensus can be achieved for the mixed model.
A new class of cost functionals for optimal control of quantum systems which produces controls which are sparse in frequency and smooth in time is proposed. This is achieved by penalizing a suitable time-frequency representation of the control field, rather than the control field itself, and by employing norms which are of $L^1$ or measure form with respect to frequency but smooth with respect to time. We prove existence of optimal controls for the resulting nonsmooth optimization problem, derive necessary optimality conditions, and rigorously establish the frequency-sparsity of the optimizers. More precisely, we show that the time-frequency representation of the control field, which a priori admits a continuum of frequencies, is supported on only textit{ finitely many} frequencies. These results cover important systems of physical interest, including (infinite-dimensional) Schrodinger dynamics on multiple potential energy surfaces as arising in laser control of chemical reactions. Numerical simulations confirm that the optimal controls, unlike those obtained with the usual $L^2$ costs, concentrate on just a few frequencies, even in the infinite-dimensional case of laser-controlled chemical reactions.
In this paper, we investigate a sparse optimal control of continuous-time stochastic systems. We adopt the dynamic programming approach and analyze the optimal control via the value function. Due to the non-smoothness of the $L^0$ cost functional, in general, the value function is not differentiable in the domain. Then, we characterize the value function as a viscosity solution to the associated Hamilton-Jacobi-Bellman (HJB) equation. Based on the result, we derive a necessary and sufficient condition for the $L^0$ optimality, which immediately gives the optimal feedback map. Especially for control-affine systems, we consider the relationship with $L^1$ optimal control problem and show an equivalence theorem.
We consider nonlinear transport equations with non-local velocity, describing the time-evolution of a measure, which in practice may represent the density of a crowd. Such equations often appear by taking the mean-field limit of finite-dimensional systems modelling collective dynamics. We first give a sense to dissipativity of these mean-field equations in terms of Lie derivatives of a Lyapunov function depending on the measure. Then, we address the problem of controlling such equations by means of a time-varying bounded control action localized on a time-varying control subset with bounded Lebesgue measure (sparsity space constraint). Finite-dimension
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