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MSc Mathematics Pre-thesis Seminar Date: Friday, 10 November 2023 Time: 2:00 - 3:00 PM Venue: Lecture Hall 202 Asymptotic Stability in a variety of semilinear PDEs Bhavesh Devdatta Padwal Chennai Mathematical Institute. 10-11-23 Abstract The observer (state observer) was first proposed and developed by David.G.Luenberger in 1964. In control theory, a state observer is a system that provides an estimate of the internal state of the given dynamical system D, from the available observations/ measurements of the input and output of D. We are interested in the convergence of the estimated state to the `true state' as given by the solution of the original dynamical system as time goes to infinity. We rephrase convergence of the estimate as the asymptotic stability of the zero-solution for the `state-estimate error equations'. This motivates us to study the stability theory for semilinear PDEs. In the talk, as an example of the general strategy, we will first analyse the stability theory of nonlinear diffusion equations using $L^1 - L^{\infty}$ norm estimates. Then we will see some toy problems which motivate us to derive some results about the asymptotic stability of a variety of nonlinear PDEs. We will discuss the implications of the result which states that in a given nonlinear PDE, if the linear semigroup satisfies a suitable upper-bound, and the nonlinearity satisfies a growth-bound (specified in terms of a bound on some suitable integral), then the solution converges to the zero-solution asymptotically in time. Later, if time permits we will also discuss some observations and results which are based on the same.
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