Controllability and observability of time-invariant linear dynamic systems. (English) Zbl 1265.34334

The paper is concerned with linear dynamic time-invariant control systems of the forms \[ x^{\Delta }(t)=-Ax^{\sigma }(t)+Bu(t),\;\;\;y(t)=Cx^{\sigma }(t),\tag{1} \] where \(x\), \(y\), \(u\) denote the state, input, and output of the system, respectively. It is assumed that \(u\) is an rd-continuous function, and \(A\) is a regressive matrix.
The authors derive necessary and sufficient conditions for the controllability and reachability of system (1), which generalize the usual ones which are well known in the classical control theory.
The final result is a duality theorem, which says that the controllability of (1) is equivalent to the observability of \[ x^{\Delta }(t)=A^T x(t)+C^T u(t),\;\;\;y(t)=B^T x(t). \]


34N05 Dynamic equations on time scales or measure chains
93B05 Controllability
93B07 Observability
34H05 Control problems involving ordinary differential equations
34A30 Linear ordinary differential equations and systems
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