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Register a new userSingular asymptotic expansion of the exact control for a linear model of the Rayleigh beam
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The Petrowsky type equation $y_{tt}^eps+eps y_{xxxx}^eps - y_{xx}^eps=0$, $eps>0$ encountered in linear beams theory is null controllable through Neumann boundary controls. Due to the boundary layer of size of order $sqrt{eps}$ occurring at the extremities, these boundary controls get singular as $eps$ goes to $0$. Using the matched asymptotic method, we describe the boundary layer of the solution $y^eps$ then derive a rigorous second order asymptotic expansion of the control of minimal $L^2-$norm, with respect to the parameter $eps$. In particular, we recover that the leading term of the expansion is a null Dirichlet control for the limit hyperbolic wave equation, in agreement with earlier results due to J-.L. Lions in the eighties. Numerical experiments support the analysis.
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We develop an asymptotical control theory for one of the simplest distributed oscillating systems, namely, for a closed string under a bounded load applied to a single distinguished point. We find exact classes of string states that admit complete damping and an asymptotically exact value of the required time. By using approximate reachable sets instead of exact ones, we design a dry-friction like feedback control, which turns out to be asymptotically optimal. We prove the existence of motion under the control using a rather explicit solution of a nonlinear wave equation. Remarkably, the solution is determined via purely algebraic operations. The main result is a proof of asymptotic optimality of the control thus constructed.
In a separable Hilbert space $X$, we study the linear evolution equation begin{equation*} u(t)+Au(t)+p(t)Bu(t)=0, end{equation*} where $A$ is an accretive self-adjoint linear operator, $B$ is a bounded linear operator on $X$, and $pin L^2_{loc}(0,+infty)$ is a bilinear control. We give sufficient conditions in order for the above control system to be locally controllable to the ground state solution, that is, the solution of the free equation ($pequiv0$) starting from the ground state of $A$. We also derive global controllability results in large time and discuss applications to parabolic equations in low space dimension.
We give an alternate proof of the existence of the asymptotic expansion of the Bergman kernel associated to the $k$-th tensor powers of a positive line bundle $L$ in a $frac{1}{sqrt{k}}$-neighborhood of the diagonal using elementary methods. We use the observation that after rescaling the Kahler potential $kvarphi$ in a $frac{1}{sqrt{k}}$-neighborhood of a given point, the potential becomes an asymptotic perturbation of the Bargmann-Fock metric. We then prove that the Bergman kernel is also an asymptotic perturbation of the Bargmann-Fock Bergman kernel.
The long-time average behaviour of the value function in the calculus of variations, where both the Lagrangian and Hamiltonian are Tonelli, is known to be connected to the existence of the limit of the corresponding Abel means as the discount factor goes to zero. Still in the Tonelli case, such a limit is in turn related to the existence of solutions of the critical (or, ergodic) Hamilton-Jacobi equation. The goal of this paper is to address similar issues when the Hamiltonian fails to be Tonelli: in particular, for control systems that can be associated with a family of vector fields which satisfies the Lie Algebra rank condition. First, following a dynamical approach we characterise the unique constant for which the ergodic equation admits solutions. Then, we construct a critical solution which coincides with its Lax-Oleinik evolution.
We address the design of decentralized feedback control laws inducing consensus and prescribed spatial patterns over a singular interacting particle system of Cucker-Smale type. The control design consists of a feedback term regulating the distance between each agent and pre-assigned subset of neighbours. Such a design represents a multidimensional extension of existing control laws for 1d platoon formation control. For the proposed controller we study consensus emergence, collision-avoidance and formation control features in terms of energy estimates for the closed-loop system. Numerical experiments in 1, 2 and 3 dimensions assess the different features of the proposed design.
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