Evolution Equations and Control Theory最新文献

In this paper, we investigate the general decay rate of the solutions for a class of plate equations with memory term in the whole space begin{document}$ mathbb{R}^n $end{document}, begin{document}$ ngeq 1 $end{document}, given by

begin{document}$ begin{equation*} u_{tt}+Delta^2 u+ u+ int_0^t g(t-s)A u(s)ds = 0, end{equation*} $end{document}

with begin{document}$ A = Delta $end{document} or begin{document}$ A = -Id $end{document}. We use the energy method in the Fourier space to establish several general decay results which improve many recent results in the literature. We also present two illustrative examples by the end.

In this paper, we consider a one-dimensional linear Bresse system with only one infinite memory term acting in the third equation (longitudinal displacements). Under a general condition on the memory kernel (relaxation function), we establish a decay estimate of the energy of the system. Our decay result extends and improves some decay rates obtained in the literature such as the one in [27], [4], [33], [58] and [34]. The proof is based on the energy method together with convexity arguments. Numerical simulations are given to illustrate the theoretical decay result.

Let begin{document}$ f: {mathcal H} rightarrow mathbb{R} $end{document} be a convex differentiable function whose solution set begin{document}$ {{rm{argmin}}}; f $end{document} is nonempty. To attain a solution of the problem begin{document}$ min_{mathcal H}f $end{document}, we consider the second order dynamic system begin{document}$ ;ddot{x}(t) + alpha , dot{x}(t) + beta (t) nabla f(x(t)) + c x(t) = 0 $end{document}, where begin{document}$ beta $end{document} is a positive function such that begin{document}$ lim_{trightarrow +infty}beta(t) = +infty $end{document}. By imposing adequate hypothesis on first and second order derivatives of begin{document}$ beta $end{document}, we simultaneously prove that the value of the objective function in a generated trajectory converges in order begin{document}$ {mathcal O}big(frac{1}{beta(t)}big) $end{document} to the global minimum of the objective function, that the trajectory strongly converges to the minimum norm element of begin{document}$ {{rm{argmin}}}; f $end{document} and that begin{document}$ Vert dot{x}(t)Vert $end{document} converges to zero in order begin{document}$ mathcal{O} big( sqrt{frac{dot{beta}(t)}{beta (t)}}+ e^{-mu t} big) $end{document} where begin{document}$ mu. We then present two choices of begin{document}$ beta $end{document} to illustrate these results. On the basis of the Moreau regularization technique, we extend these results to non-smooth convex functions with extended real values.

The exact distributed controllability of the semilinear heat equation begin{document}$ partial_{t}y-Delta y + f(y) = v , 1_{omega} $end{document} posed over multi-dimensional and bounded domains, assuming that begin{document}$ f $end{document} is locally Lipschitz continuous and satisfies the growth condition begin{document}$ limsup_{| r|to infty} | f(r)| /(| r| ln^{3/2}| r|)leq beta $end{document} for some begin{document}$ beta $end{document} small enough has been obtained by Fernández-Cara and Zuazua in 2000. The proof based on a non constructive fixed point arguments makes use of precise estimates of the observability constant for a linearized heat equation. Under the same assumption, by introducing a different fixed point application, we present a different and somewhat simpler proof of the exact controllability, which is not based on the cost of observability of the heat equation with respect to potentials. Then, assuming that begin{document}$ f $end{document} is locally Lipschitz continuous and satisfies the growth condition begin{document}$ limsup_{| r|to infty} | f^prime(r)|/ln^{3/2}| r|leq beta $end{document} for some begin{document}$ beta $end{document} small enough, we show that the above fixed point application is contracting yielding a constructive method to compute the controls for the semilinear equation. Numerical experiments illustrate the results.

In this paper, we address exponential stabilization of transmission problem of the wave equation with linear dynamical feedback control. Using the classical energy method and multiplier technique, we prove that the energy of system exponentially decays.