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My graduation thesis was about stability theorems for $C_0$-semigroups (see the Wikipedia article for the definitions: http://en.wikipedia.org/wiki/C0-semigroup). I would like to know if there is some applicability of the stability theorems I know in this field. The only applications I found for my thesis were about the Hille-Yosida theorem and some of its applications to existence and uniqueness of solutions of partial differential equations.

I will not put any names to my theorems, since maybe they are not known to the world as my teachers name them. Here are some of them:

The $C_0$-semigroup $\{T(t)\}_{t \geq 0}$ is exponentially stable if and only if there exists $p \geq 1$ such that $\int_0^\infty \|T(t)\|^pdt <\infty$.

The $C_0$-semigroup $\{T(t)\}_{t \geq 0}$ is exponentially stable if and only if it satisfies the following condition: For any $f \in \mathcal{C}$ it follows that $x_f \in \mathcal{C}$ where $x_f: \Bbb{R}_+ \to X,\ x_f(t)=\int_0^t T(t-s)f(s)ds$, and $\mathcal{C} = \{ f : \Bbb{R}_+ \to X,\ f \text{ continuous and bounded } \}$.

The last theorem can be formulated and proved in some cases for $(L^p,L^q)$ spaces with $(p,q) \neq (1,\infty)$. A more general concept, dichotomy can be formulated (the space splits into two spaces, on one of them there is stability, and on the other one there is instability.

All these sound very nice, and have quite beautiful proofs, but are they applicable to some branches of applied math, such as ordinary or partial differential equations, or they are just pure math, and thats it?

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  • $\begingroup$ I don't know if I have the right tags. $\endgroup$
    – Beni Bogosel
    Commented Jun 1, 2011 at 12:16

1 Answer 1

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The first theorem (proved by Richard Datko in the Hilbert space case and by Pazy in the Banach space case) is very useful to establish controllability for systems governed by hyperbolic differential equations. Examples include the boundary control of plates, rods, and other elastic structures. There is a book by Jack Lagnese and Jacques-Louis Lions on this general topic.

There, I have done it and mentioned the names of two of my former colleagues

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  • $\begingroup$ All these results can be proved by other methods (and were in fact proved first by other methods, right?). At present, I know no result say in differential equations that was first proved using $C_0$ semigroups. On the other hand, the theory is indeed very appealing, my favorite being Pazy's book and Engel-Nagel's book (I recommend also the section by Schnaubelt and the outstanding exposition on the exponential function). $\endgroup$
    – John B
    Commented Dec 19, 2015 at 21:14

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