##### Department of Mathematics,

University of California San Diego

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### Math 288 C00 - Stochastic Systems Seminar (via Zoom)

## Miroslav Krstic

#### UCSD

## Stabilization of a Hyperbolic PDE of a Bioreactor With Distributed Age

##### Abstract:

For an advection-reaction PDE model of population, with a non-local boundary condition modeling ``birth", and with a multiplicative input whose nature is the ``harvesting rate", we design a feedback law that stabilizes a desired equilibrium profile (of population density vs. age). Without feedback the system has one eigenvalue at the origin and the remainder of its infinite spectrum has negative real parts, i.e., the systems is, as engineers call it, ``neutrally stable". Hence, a feedback is needed to move one eigenvalue to the left without making any of the other ones spill to the right of the imaginary axis. This control design objective is achieved by transforming the system into a control-theoretic canonical form consisting of a first-order ODE in which the input is present and whose eigenvalue needs to be made negative by feedback, and an infinite-dimensional input-free system called the ``zero dynamics", which we prove to be exponentially stable. The key feature of the overall PDE system and its feedback control law is the positivity of both the population density state and the harvesting rate input, which is a key element of the analysis, captured by a``control Lyapunov functional" which blows up when the population density or control approach zero.

### October 28, 2021

### 3:00 PM

Zoom info available by emailing Professor Williams.

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