Linear Control Systems on Unbounded Time Intervals and Invariant Measures of Ornstein--Uhlenbeck Processes in Hilbert Spaces
M Fuhrman, AM Paganoni - SIAM journal on control and optimization, 2003 - SIAM
M Fuhrman, AM Paganoni
SIAM journal on control and optimization, 2003•SIAMWe consider linear control systems in a Hilbert space over an unbounded time interval of the
form y_ α'(t)=(A-α I) y_ α (t)+ Bu (t),\qquad t ∈ (-∞, T, with bounded control operator B, under
appropriate stability assumptions on the operator A. We study how the space of states
reachable at time T depends on the parameter α≧0. We apply the results to study the
dependence on α of the Cameron--Martin spaces of the invariant measures of the Ornstein--
Uhlenbeck processes X_α defined by the equation driven by the Wiener process W: dX_ α …
form y_ α'(t)=(A-α I) y_ α (t)+ Bu (t),\qquad t ∈ (-∞, T, with bounded control operator B, under
appropriate stability assumptions on the operator A. We study how the space of states
reachable at time T depends on the parameter α≧0. We apply the results to study the
dependence on α of the Cameron--Martin spaces of the invariant measures of the Ornstein--
Uhlenbeck processes X_α defined by the equation driven by the Wiener process W: dX_ α …
We consider linear control systems in a Hilbert space over an unbounded time interval of the form $$ y_\alpha'(t)=(A-\alpha I)y_\alpha(t)+Bu(t), \qquad t\in (-\infty, T], $$ with bounded control operator B, under appropriate stability assumptions on the operator A. We study how the space of states reachable at time T depends on the parameter . We apply the results to study the dependence on of the Cameron--Martin spaces of the invariant measures of the Ornstein--Uhlenbeck processes defined by the equation driven by the Wiener process W: $$ dX_\alpha(t) = (A-\alpha I) X_\alpha(t)\; dt + B\; dW(t),\qquad t\geq 0.
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