Hamzi - Eurandom

Review of Some Concepts for Linear Stochastic Differential
Equations
• The mean satisfies d
dt E[x] = AE[x] and thus E[x(t)] = eAt E[x(0)]. If A
is Hurwitz, limt→∞ E[x(t)] = 0.
• The covariance satisfies d
dt E[xxT ] = AE[xx T ] + E[xx T ]A + BB T .
• Hence, Q = limt→∞ E[xx ⊤ ] satisfies the Lyapunov system
AQ + QA ⊤ = −BB ⊤ . So, Q = Wc = ∫ ∞
0 eAt BB ⊤ e A⊤ t dt, where Wc is the
controllability Gramian, which is positive iff. the pair (A, B) is
controllable.
. . . . . .
Boumediene Hamzi (Imperial College) On Control and RDS in RKHS June 4th, 2012 42 / 55