Control of Wave and Beam PDEs pp 1-25 | Cite as

# Preliminaries

- 269 Downloads

## Abstract

This chapter serves as a simple introduction on the necessary mathematics basis for later chapters. It first introduces some basic results in functional analysis. This is the minimal knowledge in understanding the system control theory of partial differential equations, where the state spaces are infinite dimensional, contrast to the finite-dimensional systems where the state space is the Euclidean space \(\mathbb {R}^n\) or \(\mathbb {C}^n\). This chapter introduces briefly some basic facts on \(C_0\)-semigroup theory without proofs. To study systems described by the partial differential equations, the theory of the Sobolev spaces is also necessary in the sense that the solution to a partial differential equation is usually the weak solution. This is in sharp contrast to finite-dimensional systems, where the derivative is always the classical derivative. This chapter only lists some very basic results of the Sobolev space for the convenience of citations in later chapters.

## References

- Adams RA (1975) Sobolev spaces. Academic Press, BostonzbMATHGoogle Scholar
- Arendt W, Grabosch A, Greiner G, Groh U, Lotz HP, Moustakas U, Nagel R, Neubrander F, Schlotterbeck U (1999) One-parameter semigroups of positive operators. Lecture notes in mathematics, vol 1184. Springer, BerlinGoogle Scholar
- Engel KJ, Nagel R (1999) One-parameter semigroups for linear evolution equations. Spinger, BerlinzbMATHGoogle Scholar
- Kaashoek MA, Verduyn Lunel SM (1994) An integrability condition on the resolvent for hyperbolicity of the semigroup. J Differ Eqs 112:374–406Google Scholar
- Luo ZH, Guo BZ, Mörgül O (1999) Stability and stabilization of infinite dimensional system with applications. Springer, LondonCrossRefGoogle Scholar
- Pazy A (1983) Semigroups of linear operators and applications to partial differential equations. Springer, New YorkCrossRefGoogle Scholar
- Taylor AE, Lay D (1980) An introduction to functional analysis, 2nd edn. Wiley, New YorkzbMATHGoogle Scholar