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Let Θ1 = (A1 , B1 , C1 , D) and Θ2 = (A2 , B2 , C2 , D) be controllable and observable realizations of Σ with state space dimensions n1 and n2 , respectively. Let G(·) be the impulse response matrix of Σ, then j−1 G(j) = C1 Aj−1 1 B1 = C2 A2 B2 , j ≥ 1. 10) Let n be the largest of the two numbers n1 and n2 , and let Hn be the block Hankel matrix deﬁned by ⎛ ⎞ G(1) G(2) ... G(n) ⎜ G(2) G(3) . . G(n + 1) ⎟ ⎜ ⎟ Hn = ⎜ . ⎟. .. ⎝ .. ⎠ . G(n) G(n + 1) . . 10) that Hn = Γ1 (n)Λ1 (n) = Γ2 (n)Λ2 (n), where ⎛ ⎞ Ci ⎜ Ci Ai ⎟ ⎜ ⎟ Γi (n) = ⎜ Bi , i = 1, 2.

Let Θ = (A, B, C, D) be a realization of the system Σ with state space dimension n. Then the following statements are equivalent: (i) Θ is observable, ⎛ ⎞ C ⎜ CA ⎟ ⎜ ⎟ (ii) rank ⎜ . ⎟ = n, . ⎝ . ⎠ CAn−1 (iii) the matrix n−1 T j T j j=0 (A ) C CA is non-singular. Proof. (i) ⇔ (ii). By the Cayley–Hamilton theorem, the matrix At (t ≥ n) is a linear combination of the matrices I, A, . . , An−1 . This means that Nt (Θ) = Nn (Θ) for all t ≥ n. 5), the realization is observable if and only if {0} = ∩k≥1 Nk (Θ), which in turn is equivalent to {0} = Nn (Θ).

See for instance [6] for an excellent book on dynamic programming. 1). 1 works just as well and probably even more transparent in the more general case discussed here. 3. 1. It is assumed that the price trajectory {p(t) | t = 0, . . , N } is known to the trader. Before giving a formal 58 Chapter 5. Optimal Control analysis, common sense suggests the following strategy. On day t, the trader has to decide whether to buy or to sell the good. This decision has no eﬀect on his future options, as the only restriction is that |u(t)| ≤ K on all days.