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The exact distributed controllability of the semilinear wave equation $y_{tt}-y_{xx} + g(y)=f ,1_{omega}$, assuming that $g$ satisfies the growth condition $vert g(s)vert /(vert svert log^{2}(vert svert))rightarrow 0$ as $vert svert rightarrow infty$ and that $g^primein L^infty_{loc}(mathbb{R})$ has been obtained by Zuazua in the nineties. The proof based on a Leray-Schauder fixed point argument makes use of precise estimates of the observability constant for a linearized wave equation. It does not provide however an explicit construction of a null control. Assuming that $g^primein L^infty_{loc}(mathbb{R})$, that $sup_{a,bin mathbb{R},a eq b} vert g^prime(a)-g^{prime}(b)vert/vert a-bvert^r<infty $ for some $rin (0,1]$ and that $g^prime$ satisfies the growth condition $vert g^prime(s)vert/log^{2}(vert svert)rightarrow 0$ as $vert svert rightarrow infty$, we construct an explicit sequence converging strongly to a null control for the solution of the semilinear equation. The method, based on a least-squares approach guarantees the convergence whatever the initial element of the sequence may be. In particular, after a finite number of iterations, the convergence is super linear with rate $1+r$. This general method provides a constructive proof of the exact controllability for the semilinear wave equation.
The null distributed controllability of the semilinear heat equation $y_t-Delta y + g(y)=f ,1_{omega}$, assuming that $g$ satisfies the growth condition $g(s)/(vert svert log^{3/2}(1+vert svert))rightarrow 0$ as $vert svert rightarrow infty$ and that
It has been proved by Zuazua in the nineties that the internally controlled semilinear 1D wave equation $partial_{tt}y-partial_{xx}y + g(y)=f 1_{omega}$, with Dirichlet boundary conditions, is exactly controllable in $H^1_0(0,1)cap L^2(0,1)$ with con
The exact distributed controllability of the semilinear wave equation $partial_{tt}y-Delta y + g(y)=f ,1_{omega}$ posed over multi-dimensional and bounded domains, assuming that $gin C^1(mathbb{R})$ satisfies the growth condition $limsup_{rto infty}
The epsilon alternating least squares ($epsilon$-ALS) is developed and analyzed for canonical polyadic decomposition (approximation) of a higher-order tensor where one or more of the factor matrices are assumed to be columnwisely orthonormal. It is s
We consider the problem of efficiently solving large-scale linear least squares problems that have one or more linear constraints that must be satisfied exactly. Whilst some classical approaches are theoretically well founded, they can face difficult