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D. Properties of Continuous-Time Riccati Equations

D.4 Convergence of Solutions of the CTARE to the Stabilizing Solution of the CTARE

Finally, we show that, under reasonable conditions, the solution of the CTDRE will converge to the unique stabilizing solution of the CTARE. In the sequel, we will be particularly interested in the stabilizing solution to the CTARE.

Lemma D.5  Provided that $(A,B)$ is stabilizable and that $(\ensuremath{\boldsymbol{\Psi}} ^{\frac{1}{2}}\,, \ensuremath{\mathbf{A}} )$ has no unobservable poles on the imaginary axis and that $\ensuremath{\boldsymbol{\Psi}} _f>\ensuremath{\mathbf{P}} _{\infty}^{s}$, then

\begin{displaymath}\lim_{t_f\to \infty}\ensuremath{\mathbf{P}} (t)=\ensuremath{\mathbf{P}} _{\infty}^{s} \end{displaymath} (D.4.1)

Proof

We observe that the eigenvalues of \ensuremath{\mathbf{H}} can be grouped so that $\ensuremath{\boldsymbol{\Lambda}} _s $ contains only eigenvalues that lie in the left-half plane. We then have that


\begin{displaymath} \lim_{t_f\rightarrow \infty} e^{\ensuremath{\mathbf{H_s}} (... ...e^{- \ensuremath{\mathbf{H_u}} (t_f-t)}=\ensuremath{\mathbf{0}}\end{displaymath} (D.4.2)

given that $\ensuremath{\mathbf{H}} _s$ and $- \ensuremath{\mathbf{H}} _u$ are matrices with eigenvalues strictly inside the LHP.

The result then follows from (D.1.16) to (D.1.17).

Remark D.1  Actually, provided that $(\Psi^{\frac{1}{2}}\,, A)$ is detectable, then it suffices to have $\ensuremath{\boldsymbol{\Psi}} _f \geq \ensuremath{\mathbf{0}} $ in Lemma D.5

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