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In this article, we report the results we obtained when investigating the numerical solution of some nonlinear eigenvalue problems for the Monge-Ampère operator v → det D2v. The methodology we employ relies on the following ingredients: (i) a divergence formulation of the eigenvalue problems under consideration. (ii) The time discretization by operator-splitting of an initial value problem (a kind of gradient flow) associated with each eigenvalue problem. (iii) A finite element approximation relying on spaces of continuous piecewise affine functions. To validate the above methodology, we applied it to the solution of problems with known exact solutions: The results we obtained suggest convergence to the exact solution when the space discretization step h → 0. We considered also test problems with no known exact solutions. [ABSTRACT FROM AUTHOR]

This paper presents a unified mathematical theory of swarms where the dynamics of social behaviors interacts with the mechanical dynamics of self-propelled particles. The term behavioral swarms is introduced to characterize the specific object of the theory which is subsequently followed by applications. As concrete examples for our unified approach, we show that several Cucker-Smale type models with internal variables fall down to our framework. The second part of the paper shows how the modeling can be developed, beyond the Cucker-Smale approach. This will be illustrated with the aid of numerical simulations in swarms whose movement strategy is sensitive to individual social behaviors. Finally, the presentation looks ahead to research perspectives. [ABSTRACT FROM AUTHOR]

We show that under Kalman's rank condition on the coupling matrices, the uniqueness of solution to a complex system of elliptic operators can be reduced to the observability of a scalar problem. Based on this result, we establish the asymptotic stability and the asymptotic synchronization for a large class of linear dissipative systems. [ABSTRACT FROM AUTHOR]

We study the existence of sign-changing solutions to the nonlinear heat equation ∂tu = Δu + |u|αu on ℝN, N ≥ 3, with 2/N−2 <α<α0, where α0=4/N−4+2√N−1 ∈ (2/N−2,4/N−2), which are singular at x = 0 on an interval of time. In particular, for certain μ > 0 that can be arbitrarily large, we prove that for any u0 ∈ Lloc∞(ℝN\{0}) which is bounded at infinity and equals μ|x|−2/α in a neighborhood of 0, there exists a local (in time) solution u of the nonlinear heat equation with initial value u0, which is sign-changing, bounded at infinity and has the singularity β|x|−2/α at the origin in the sense that for t > 0, |x|2/αu(t,x) → β as |x|→ 0, where β=2/α(N−2−2/α). These solutions in general are neither stationary nor self-similar. [ABSTRACT FROM AUTHOR]

This paper studies the stability of a 1-dim system which comprises a wave equation and a degenerate heat equation in two connected bounded intervals. The coupling between these two different components occurs at the interface with certain transmission conditions. We find an explicit polynomial decay rate for solutions of this system. This rate depends on the degree of the degeneration for the diffusion coefficient near the interface. Besides, the well-posedness of this degenerate coupled system is proved by the semigroup theory. [ABSTRACT FROM AUTHOR]

We study a linear quadratic optimal control problem for mean-field stochastic evolution equation with the assumption that all the coefficients concerned in the problem are deterministic. We show that the existence of optimal feedback operators is equivalent to that of regular solution to the system which is coupled by two Riccati equations and an explicit formula of the optimal feedback control operator is given via the regular solution. We also show that the mentioned Riccati equations admit a unique strongly regular solution when the cost functional is uniformly convex. [ABSTRACT FROM AUTHOR]

We consider the finite-time stabilization of homogeneous quasilinear hyperbolic systems with one side controls and with nonlinear boundary condition at the other side. We present time-independent feedbacks leading to the finite-time stabilization in any time larger than the optimal time for the null controllability of the linearized system if the initial condition is sufficiently small. One of the key technical points is to establish the local well-posedness of quasilinear hyperbolic systems with nonlinear, non-local boundary conditions. [ABSTRACT FROM AUTHOR]