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We assume, for a distributed parameter control system, that a linear stabilizing is available. We then seek a stabilizing, necessarily nonlinear, subject to an a priori bound on the control.
Suggested by conversations in 1991 (Mark Krasnosel'skiĭ and Aleksei Pokrovskiĭ with TIS), this paper generalizes earlier work  of theirs by defining a setting of hybrid systems with isotone switching rules for a partially ordered set of modes and then obtaining a periodicity result in that context. An application is given to a partial differential equation modeling calcium release and diffusion in cardiac cells.
The convergence is established for a sequence of operator semigroups, where the limiting object is the transition semigroup for a reflected stable processes. For semilinear equations involving the generators of these transition semigroups, an approximation method is developed as well. This makes it possible to derive an a priori bound for solutions to these equations, and therefore prove global existence of solutions. An application to epidemiology is also given.
We consider an optimal control problem involving the use of bacteria for pollution removal where the model assumes the bacteria switch instantaneously between active and dormant states, determined by threshold sensitivity to the local concentration $v$ of a diffusing critical nutrient; compare , ,  in which nutrient transport is convective. It is shown that the direct problem has a solution for each boundary control $ψ = ∂v/∂n$ and that optimal controls exist, minimizing a combination of residual pollutant and aggregated cost of the nutrient.
For a chemical reaction/diffusion system, a very fast reaction $A+B\to C$ implies non-coexistence of $A,B$ with resulting interfaces. We try to understand how these interfaces evolve in time. In particular, we seek a characterizing system of equations and conditions for the sharp interface limit: when this fast reaction is taken as infinitely fast.
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