# Polynomial Stability of Abstract Wave Equations

##### Kosonen, Jasper (2023)

Kosonen, Jasper

2023

Teknis-luonnontieteellinen DI-ohjelma - Master's Programme in Science and Engineering

Tekniikan ja luonnontieteiden tiedekunta - Faculty of Engineering and Natural Sciences

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##### Hyväksymispäivämäärä

2023-09-18**Julkaisun pysyvä osoite on**

https://urn.fi/URN:NBN:fi:tuni-202309138166

##### Tiivistelmä

Abstract wave equations model a large class of linear dynamical systems, i.e., systems that evolve linearly over time. For example, this vast class of abstract wave equations encloses many important second-order partial differential equations that are frequent in applications. As with any partial differential equation, an important question arises whether a solution to an abstract wave equation exists for all possible initial conditions. By a solution we simply mean a function that satisfies both the abstract wave equation and the existing boundary and initial conditions. Answering this question leads us to the widely known criteria for well-posed problems by Hadamard. It turns out that all abstract wave equations are well-posed, which is implied by the theory of so-called strongly continuous semigroups.

Having established the existence of solutions for abstract wave equations, it is natural to ask how the solutions behave over time. In particular, we are interested in the limiting, that is, asymptotic behaviour of the solutions as time elapses. If the solutions corresponding to all initial conditions eventually converge to some equilibria, then we call the associated abstract wave equation asymptotically stable. In case the rate of convergence is also uniform for all solutions, the solutions actually converge to their equilibria at an exponential rate, yielding exponential stability. In general, a solution to an asymptotically stable abstract wave equation can approach its equilibrium arbitrarily slowly and thus preclude any uniform rate of convergence. However, with certain assumptions we obtain results for strongly continuous semigroups that guarantee both asymptotic stability and a uniform rate of convergence for a particular subset of solutions called classical solutions.

In this thesis we examine the polynomial stability of abstract wave equations. Put simply, all classical solutions to an abstract wave equation should converge to their equilibria at a polynomial rate. A polynomially stable system is always asymptotically stable but not necessarily exponentially stable. Although polynomial stability is a special case of a more general semi-uniform stability, for the time being counterparts to important results implying exponential stability only exist for polynomial stability. The key idea in these results is to investigate how the norm of a resolvent associated with the abstract wave equation grows on the imaginary axis. The slower this norm grows, the faster the classical solutions converge. At the end of this thesis we analyze a system from the literature and its two variants in great detail. We recast these systems as abstract wave equations and study their stability with the theory and tools we obtain along the way.

Having established the existence of solutions for abstract wave equations, it is natural to ask how the solutions behave over time. In particular, we are interested in the limiting, that is, asymptotic behaviour of the solutions as time elapses. If the solutions corresponding to all initial conditions eventually converge to some equilibria, then we call the associated abstract wave equation asymptotically stable. In case the rate of convergence is also uniform for all solutions, the solutions actually converge to their equilibria at an exponential rate, yielding exponential stability. In general, a solution to an asymptotically stable abstract wave equation can approach its equilibrium arbitrarily slowly and thus preclude any uniform rate of convergence. However, with certain assumptions we obtain results for strongly continuous semigroups that guarantee both asymptotic stability and a uniform rate of convergence for a particular subset of solutions called classical solutions.

In this thesis we examine the polynomial stability of abstract wave equations. Put simply, all classical solutions to an abstract wave equation should converge to their equilibria at a polynomial rate. A polynomially stable system is always asymptotically stable but not necessarily exponentially stable. Although polynomial stability is a special case of a more general semi-uniform stability, for the time being counterparts to important results implying exponential stability only exist for polynomial stability. The key idea in these results is to investigate how the norm of a resolvent associated with the abstract wave equation grows on the imaginary axis. The slower this norm grows, the faster the classical solutions converge. At the end of this thesis we analyze a system from the literature and its two variants in great detail. We recast these systems as abstract wave equations and study their stability with the theory and tools we obtain along the way.