## Abstract and Applied Analysis

### A Characterization of Semilinear Dense Range Operators and Applications

#### Abstract

We characterize a broad class of semilinear dense range operators ${G}_{H}:W\to Z$ given by the following formula, ${G}_{H}w=Gw+H\left(w\right),w\in W$, where $Z$, $W$ are Hilbert spaces, $G\in L\left(W,Z\right)$, and $H:W\to Z$ is a suitable nonlinear operator. First, we give a necessary and sufficient condition for the linear operator $G$ to have dense range. Second, under some condition on the nonlinear term $H$, we prove the following statement: If $\overline{\text{R}\text{a}\text{n}\text{g}\left(G\right)}=Z$, then $\overline{\text{R}\text{a}\text{n}\text{g}\left({G}_{H}\right)}=Z$ and for all $z\in Z$ there exists a sequence $\left\{{w}_{\alpha }\in Z:0<\alpha \le 1\right\}$ given by ${w}_{\alpha }={G}^{*}\left(\alpha I+G{G}^{*}{\right)}^{-1}\left(z-H\left({w}_{\alpha }\right)\right)$, such that $\mathrm{ }\text{l}\text{i}\text{m}\alpha \to {0}^{+}\left\{G{u}_{\alpha }+H\left({u}_{\alpha }\right)\right\}=z$. Finally, we apply this result to prove the approximate controllability of the following semilinear evolution equation: ${z}^{\prime }=Az+Bu\left(t\right)+F\left(t,z,u\left(t\right)\right),z\in Z,u\in U,t>0$, where $Z$, $U$ are Hilbert spaces, $A:D\left(A\right)\subset Z\to Z$ is the infinitesimal generator of strongly continuous compact semigroup $\left\{T\left(t\right){\right\}}_{t\ge 0}$ in $Z,B\in L\left(U,Z\right)$, the control function $u$ belongs to ${L}^{2}\left(0,\tau ;U\right)$, and $F:\left[0,\tau \right]×Z×U\to Z$ is a suitable function. As a particular case we consider the controlled semilinear heat equation.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 729093, 11 pages.

Dates
First available in Project Euclid: 27 February 2014

https://projecteuclid.org/euclid.aaa/1393511777

Digital Object Identifier
doi:10.1155/2013/729093

Mathematical Reviews number (MathSciNet)
MR3035310

Zentralblatt MATH identifier
1287.47054

#### Citation

Leiva, H.; Merentes, N.; Sanchez, J. A Characterization of Semilinear Dense Range Operators and Applications. Abstr. Appl. Anal. 2013 (2013), Article ID 729093, 11 pages. doi:10.1155/2013/729093. https://projecteuclid.org/euclid.aaa/1393511777

#### References

• E. Iturriaga and H. Leiva, “A necessary and sufficient condition for the controllability of linear systems in Hilbert spaces and applications,” IMA Journal of Mathematical Control and Information, vol. 25, no. 3, pp. 269–280, 2008.
• H. Leiva, “Exact controllability of the suspension bridge model proposed by Lazer and McKenna,” Journal of Mathematical Analysis and Applications, vol. 309, no. 2, pp. 404–419, 2005.
• H. Leiva, “Exact controllability of a non-linear generalized damped wave equation: application to the sine-Gordon equation,” in Proceedings of the Electronic Journal of Differential Equations, vol. 13, pp. 75–88, 2005.
• H. Leiva, “Exact controllability of semilinear evolution equation and applications,” International Journal of Communication Systems, vol. 1, no. 1, 2008.
• H. Leiva and J. Uzcategui, “Exact controllability for semilinear difference equation and application,” Journal of Difference Equations and Applications, vol. 14, no. 7, pp. 671–679, 2008.
• H. Leiva, “Appxoximate controllability of semilinear cascade systems in $H={L}^{2}(\Omega )$,” International Mathematical Forum, vol. 7, no. 57, pp. 2797–2813, 2012.
• H. Leiva, N. Merentes, and J. L. Sanchez, “Interior controllability of the nD semilinear heat equation,” African Diaspora Journal of Mathematics, vol. 12, no. 2, pp. 1–12, 2011.
• H. Leiva, N. Merentes, and J. L. Sánchez, “Approximate controllability of semilinear reaction diffusion equations,” Mathematical Control and Related Fields, vol. 2, no. 2, pp. 171–182, 2012.
• X. Zhang, “A remark on null exact controllability of the heat equation,” SIAM Journal on Control and Optimization, vol. 40, no. 1, pp. 39–53, 2001.
• H. Leiva and Y. Quintana, “Interior controllability of a broad class of reaction diffusion equations,” Mathematical Problems in Engineering, vol. 2009, Article ID 708516, 8 pages, 2009.
• J. I. Díaz, J. Henry, and A. M. Ramos, “On the approximate controllability of some semilinear parabolic boundary-value problems,” Applied Mathematics and Optimization, vol. 37, no. 1, pp. 71–97, 1998.
• E. Fernandez-Cara, “Remark on approximate and null controllability of semilinear parabolic equations,” in Proceedings of the Controle et Equations AUX Derivees Partielles, ESAIM, vol. 4, pp. 73–81, 1998.
• E. Fernández-Cara and E. Zuazua, “Controllability for blowing up semilinear parabolic equations,” Comptes Rendus de l'Académie des Sciences I, vol. 330, no. 3, pp. 199–204, 2000.
• D. Bárcenas, H. Leiva, and W. Urbina, “Controllability of the Ornstein-Uhlenbeck equation,” IMA Journal of Mathematical Control and Information, vol. 23, no. 1, pp. 1–9, 2006.
• D. Barcenas, H. Leiva, Y. Quintana, and W. Urbina, “Controllability of Laguerre and Jacobi equations,” International Journal of Control, vol. 80, no. 8, pp. 1307–1315, 2007.
• R. F. Curtain and A. J. Pritchard, Infinite Dimensional Linear Systems Theory, vol. 8 of Lecture Notes in Control and Information Sciences, Springer, Berlin, Germany, 1978.
• R. F. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, vol. 21 of Texts in Applied Mathematics, Springer, New York, NY, USA, 1995.
• K. Balachandran, J. Y. Park, and J. J. Trujillo, “Controllability of nonlinear fractional dynamical systems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 75, no. 4, pp. 1919–1926, 2012.
• A. E. Bashirov and N. I. Mahmudov, “On Concepts of controllability for deterministic and stochastic systems,” SIAM Journal on Control and Optimization, vol. 37, no. 6, pp. 1808–1821, 1999.
• J. P. Dauer and N. I. Mahmudov, “Approximate controllability of semilinear functional equations in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 273, no. 2, pp. 310–327, 2002.
• J. P. Dauer and N. I. Mahmudov, “Controllability of some nonlinear systems in Hilbert spaces,” Journal of Optimization Theory and Applications, vol. 123, no. 2, pp. 319–329, 2004.
• N. I. Mahmudov, “Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces,” SIAM Journal on Control and Optimization, vol. 42, no. 5, pp. 1604–1622, 2003.
• L. de Teresa, “Approximate controllability of a semilinear heat equation in ${\mathbb{R}}^{N}$,” SIAM Journal on Control and Optimization, vol. 36, no. 6, pp. 2128–2147, 1998.
• L. de Teresa and E. Zuazua, “Approximate controllability of a semilinear heat equation in unbounded domains,” Nonlinear Analysis: Theory, Methods & Applications, vol. 37, no. 8, pp. 1059–1090, 1999.
• K. Naito, “Controllability of semilinear control systems dominated by the linear part,” SIAM Journal on Control and Optimization, vol. 25, no. 3, pp. 715–722, 1987.
• K. Naito, “Approximate controllability for trajectories of semilinear control systems,” Journal of Optimization Theory and Applications, vol. 60, no. 1, pp. 57–65, 1989.
• D. Barcenas, H. Leiva, and Z. Sívoli, “A broad class of evolution equations are approximately controllable, but never exactly controllable,” IMA Journal of Mathematical Control and Information, vol. 22, no. 3, pp. 310–320, 2005.
• M. H. Protter, “Unique continuation for elliptic equations,” Transactions of the American Mathematical Society, vol. 95, pp. 81–91, 1960.