Cheap Control versus Controllability with Vanishing Energy

Luciano Pandolfi* and Enrico Priola

Cheap control is a version of Tikonov regularization for control systems, and addresses the following problem. We consider the system \begin{equation} \tag{$\star$} \dot x=Ax+Bu\,,\qquad y=C x \end{equation} (finite dimensional, distributed, with boundary controls\dots) and a reference signal $ r(t) $ (quite often $ r(t)=Ca^{At}x_0$ or, more in general, r(t) is the output of a second system). Let $ \Lambda $ be the input output operator of system $(\star)$. It is required to solve \[ \Lambda u =r\,. \] Of course this problem is not solvable, and it is replaced by its Tikonov regularization (here $ \alpha>0 $) \[ \min _{u\in L^2(0,+\infty)}\int_0^{+\infty} \left \{ |Cx(t)-r(t)|^2+\alpha |u(t)|^2 \right \} {\rm d}\, t=I_\alpha (r)\,. \] The system is $\textit{cheap}$ when \[ \min I_{\alpha }(r)=0\qquad \forall r\in L^2(0,+\infty)\,. \] \newline Null controllability with vanishing energy (NCVE) is the following problem ($ x_0 $ is the initial condition) \[ \inf \left \{ \| u\|_{L^2(0,T)}\,, \quad \exists T \quad {\rm for \ which}\quad x(T;x_0,u)=0\right \} =0 \] for every $ x_0 $. \newline In this talk we contrast the properties of the two problems, which turn out to be oppose one to the other in the sense that cheap control depends on the location of the \textit{zeros} of the transfer function (to be interpreted as the presence/absence of outer factors in the case of distributed systems) while NCVE depends on the \textit{poles} (or singularities) of the transfer function. \newline Roughly speaking we shall prove that, for a large class of systems, \emph{the system is cheap if the transfer function does not have unstable zeros} while \emph{NCVE holds if the system does not have unstable poles.} In both the cases, zeros or poles on the imaginary axis are admitted.

Mathematics Subject Classification: 49N05 35Q93

Keywords: Optimization; Tikonov regularization; null controllability

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