2010
DOI: 10.4236/am.2010.14033
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A Characterization of Semilinear Surjective Operators and Applications to Control Problems

Abstract: In this paper we characterize a broad class of semilinear surjective operatorsgiven by the following formula ( ) H

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Cited by 9 publications
(6 citation statements)
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“…Lemma 16 The following statements are equivalent. Ranfalse(frakturCfalse)=n$$ \mathrm{Ran}\left(\mathfrak{C}\right)={\mathbb{R}}^n $$. kerfalse(frakturCfalse)=false{0false}$$ \ker \left({\mathfrak{C}}^{\ast}\right)=\left\{0\right\} $$. γ>0:3.0235ptfrakturCfrakturCz,z>γfalse‖zfalse‖2,0.1emz0$$ \exists \gamma >0:\kern3.0235pt \left\langle \mathfrak{C}{\mathfrak{C}}^{\ast }z,z\right\rangle >\gamma {\left\Vert z\right\Vert}^2,z\ne 0 $$ in n$$ {\mathbb{R}}^n $$. frakturW1Lfalse(nfalse)$$ \exists {\mathfrak{W}}^{-1}\in L\left({\mathbb{R}}^n\right) $$ ( frakturW1$$ {\mathfrak{W}}^{-1} $$ is bounded). Bfalse(sfalse)Efalse(τ,sfalse)z=0,0.30emsfalse[0,τfalse]z=0$$ {B}^{\ast }(s){E}^{\ast}\left(\tau, s\right)z=0,\kern0.30em \forall s\in \left[0,\tau \right]\Rightarrow z=0 $$. …”
Section: Preliminariesmentioning
confidence: 98%
“…Lemma 16 The following statements are equivalent. Ranfalse(frakturCfalse)=n$$ \mathrm{Ran}\left(\mathfrak{C}\right)={\mathbb{R}}^n $$. kerfalse(frakturCfalse)=false{0false}$$ \ker \left({\mathfrak{C}}^{\ast}\right)=\left\{0\right\} $$. γ>0:3.0235ptfrakturCfrakturCz,z>γfalse‖zfalse‖2,0.1emz0$$ \exists \gamma >0:\kern3.0235pt \left\langle \mathfrak{C}{\mathfrak{C}}^{\ast }z,z\right\rangle >\gamma {\left\Vert z\right\Vert}^2,z\ne 0 $$ in n$$ {\mathbb{R}}^n $$. frakturW1Lfalse(nfalse)$$ \exists {\mathfrak{W}}^{-1}\in L\left({\mathbb{R}}^n\right) $$ ( frakturW1$$ {\mathfrak{W}}^{-1} $$ is bounded). Bfalse(sfalse)Efalse(τ,sfalse)z=0,0.30emsfalse[0,τfalse]z=0$$ {B}^{\ast }(s){E}^{\ast}\left(\tau, s\right)z=0,\kern0.30em \forall s\in \left[0,\tau \right]\Rightarrow z=0 $$. …”
Section: Preliminariesmentioning
confidence: 98%
“…is called the steering operator and it is a right inverse of G, in the sense that 17) such that 18) and a control steering the system (1.6) from initial state z to a nal state z at time τ > is given by…”
Section: Controllability Of Linear Systemsmentioning
confidence: 99%
“…Lemma 3.7. (Iturriaga and Leiva [12]) Let W and Z be Hilbert spaces, S ∈ L(W, Z) and S * ∈ L(Z, W ) be the adjoined operator of S, and dim(Z) < +∞, then the following statements are equivalent…”
Section: System Descriptionmentioning
confidence: 99%
“…steering the system (13) from the initial complete state z(0) = (0, 0) to a final state x = x(2), given by the formula (12).…”
Section: Examplesmentioning
confidence: 99%