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On the stability of some relay-type regulation system. (English) Zbl 1161.34026

This note is concerned with the stability problem for solutions of a system which arises in the mathematical modeling of a temperature regulation device. It is well-known that, for autonomous systems, a periodic regulation process can be asymptotically stable only if it is stationary. In this paper, the case is considered where the “heating” equation is of the form \(\dot x=f(t,x)\) and the cooling equation reads \(\dot x=-h(x)\), where f and h are continuous and satisfy some boundedness conditions. In this case it is shown that, under suitable hypotheses, the solution is asymptotically stable, and estimates for the approximation of solutions are also given.

MSC:

34D20 Stability of solutions to ordinary differential equations
34A36 Discontinuous ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
93D09 Robust stability
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34C60 Qualitative investigation and simulation of ordinary differential equation models
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References:

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