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On the observability and detectability of continuous-time Markov jump linear systems. (English) Zbl 1029.93007
The authors consider a class of linear stochastic uncertain systems with Markovian jump parameters described by ${dx(t)\over dt}=A\bigl(r (t)\bigr)x(t),\;x(0)=x_0,\;r(0)=r_0 \tag{1}$
$y(t)=C\bigl( r(t)\bigr)x(t)\tag{2}$ where $$\{r(t),t\in [0,+\infty)\}$$ is a homogeneous finite-state Markovian process with right continuous trajectories and taking values in a finite set $${\mathfrak I}=\{1,2, \dots,s\}$$, $$x\in\mathbb{R}^n$$ and $$y\in\mathbb{R}^q$$ are state and output vectors, respectively. $$A(r(t))$$ and $$C(r(t))$$ are known real constant matrices of appropriate dimensions.
The authors introduce a new concept of $$W$$-detectability and the set of observability matrices $$O$$ that is related to the concept of $$W$$-observability for system (1), (2). The obtained result provides a testable condition for $$W$$-detectability.

##### MSC:
 93B07 Observability 60J75 Jump processes (MSC2010) 93E03 Stochastic systems in control theory (general)
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