About the Project
The project concerns various stability notions associated with certain classes of nonlinear evolution equations, which may be subject to exogenous forcing. As is well-known, even scalar nonlinear differential equations which have stable equilibria in the absence of forcing can exhibit pathological behaviour in the presence of arbitrarily small, but persistent, forcing - behaviour not seen in the stable linear case. Exogenous disturbances, or forcing, is a natural phenomenon in many physically meaningful settings, however; capturing unmodelled dynamics, for instance. Consequently, there are a range of stability notions, and an associated theory, which encompass the role of forcing in evolution equations. The nonlinear evolution equations presently of interest comprise both a linear part (such as a Laplacian, or elliptic operator), and a nonlinear component, and the interplay between the linear and nonlinear components is crucial. The aim of the project is to investigate and establish sufficient conditions for the above mentioned related stability notions for certain classes of nonlinear evolution equations using (readily checkable) properties of a certain operator-valued holomorphic function associated with the linear part of the evolution equation. These properties include boundedness on the right-half complex plane, or an accretive-type property there.
The technical crux is anticipated to be deriving state-space properties (particularly which give rise to candidate Lyapunov-type functions) from frequency domain properties of the above mentioned holomorphic function. These equivalences are known to hold in a finite-dimensional setting, but the infinite-dimensional case is more involved and subtle. These equivalences shall be used in the current project as a means to ensuring stability, but are of independent interest as well. It is anticipated that the results will be applied to a number of nonlinear partial- and delay-differential equations. The research may also extend to infinite-dimensional nonlinear difference equations (again with a certain structure), and analogous questions may be posed and solved in discrete-time.
The project will suit someone with a background, and enthusiasm, in analysis and the study of evolution equations. The lead supervisor, Chris Guiver, is part of the mathematical systems and control theory research group, within the analysis research group in the Department of Mathematical Sciences.
Please feel free to get in contact if you would like more information or to discuss the project ([Email Address Removed]).
Candidate:
Applicants should hold, or expect to receive, a First Class or high Upper Second Class UK Honours degree (or the equivalent qualification gained outside the UK) in a relevant subject. A master’s level qualification would also be advantageous.
Applications:
Formal applications should be made via the University of Bath’s online application form:
https://samis.bath.ac.uk/urd/sits.urd/run/siw_ipp_lgn.login?process=siw_ipp_app&code1=RDUMA-FP03&code2=0013
Please ensure that you quote the supervisor’s name and project title in the ‘Your research interests’ section.
More information about applying for a PhD at Bath may be found here:
http://www.bath.ac.uk/guides/how-to-apply-for-doctoral-study/
Anticipated start date: 30 September 2019.
Funding Notes
Candidates may be considered for a University Research Studentship which will cover UK/EU tuition fees, a training support fee of £1,000 per annum and a tax-free maintenance allowance at the UKRI Doctoral Stipend rate (£14,777 in 2018-19) for a period of up to 3.5 years.
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