Abstract:
We show that the optimal feedback control $\widehat u$ in the
standard nonhomogeneous LQG-problem with infinite horizon has
the following property. There is a constant $b_*$ such that,
whatever $b> b_*$ is, the deficiency process of optimal control
with respect to any possible control $u$, i.e., the difference
$J_T(\widehat u\hspace*{0.2pt})- J_T(u)$ between the optimal cost process
$J_T(\widehat u\hspace*{0.2pt})$ and the cost process corresponding to control
$u$, is majorated at infinity by a deterministic function $b\log
T$. In other words, $b\log T$ is an upper function for any
deficiency process. This result, combined with an example of an
LQG-regulator where,
for certain $b>0$, the function $b\log T$ is
not an upper function for certain deficiency processes, gives an
answer to the long-standing open problem about the best possible
rate function for sensitive probabilistic criteria. Our setting
covers the optimal tracking problem.
Keywords:linear-quadratic regulator, optimality almost surely, observability, controllability, Riccati equation, martingale law of large numbers, upper functions, Ornstein–Uhlenbeck process.
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