Download General Pontryagin-Type Stochastic Maximum Principle and by Qi Lü, Xu Zhang (auth.) PDF

By Qi Lü, Xu Zhang (auth.)

The classical Pontryagin greatest precept (addressed to deterministic finite dimensional keep watch over structures) is likely one of the 3 milestones in glossy regulate thought. The corresponding thought is by means of now well-developed within the deterministic limitless dimensional atmosphere and for the stochastic differential equations. besides the fact that, little or no is understood in regards to the comparable challenge yet for managed stochastic (infinite dimensional) evolution equations while the diffusion time period comprises the regulate variables and the keep watch over domain names are allowed to be non-convex. certainly, it truly is one of many longstanding unsolved difficulties in stochastic regulate concept to set up the Pontryagin variety greatest precept for this sort of normal keep watch over platforms: this ebook goals to offer an answer to this challenge. This publication might be important for either rookies and specialists who're attracted to optimum keep an eye on thought for stochastic evolution equations.

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In the sequel, we denote by (w) − lim yn = y in Y n∀≥ when {yn }≥ n=1 weakly converges to y in Y ; and by (w*) − lim z n = z in Y → n∀≥ → → when {z n }≥ n=1 weakly converges to z in Y . Let us show first the following result (It seems for us that this is a known result. 1 Let X be a separable Banach space and let Y be a reflexive Banach space. Assume that {G n }≥ n=1 ∈ L (X, Y ) is a sequence of bounded linear operators is bounded for any given x ∞ X . Then, there exist a subsequence such that {G n x}≥ n=1 and a bounded linear operator G from X to Y such that {G n k }≥ k=1 (w) − lim G n k x = Gx in Y, ∗ x ∞ X, k∀≥ (w→) − lim G →n k y → = G → y → in X → , ∗ y → ∞ Y → , k∀≥ Q.

Proof Assume that (P(·), Q(·)) is another transposition solution to the Eq. 10). 2, for any t ≥ [0, T ], it follows that T 0 = E P(t) − P(t) ξ1 , ξ2 H +E P(s) − P(s) u1 (s), x2 (s) H ds t T T +E P(s)−P(s) x1(s), u2(s) ds+E t t P(s)−P(s) K(s)x1(s), v2(s) ds H H T +E P(s) − P(s) v1 (s), K(s)x2 (s) + v2 (s) t T +E ds T Q(s)−Q(s) v1 (s), x2 (s) t H H Q(s)−Q(s) x1 (s), v2 (s) ds+E H ds. 1) Q. Lü and X. 1007/978-3-319-06632-5_4, © The Author(s) 2014 38 4 Well-Posedness Result for the Operator-Valued BSEEs Choosing u1 = v1 = 0 and u2 = v2 = 0 in Eqs.

T1 , T ] × Ω. 11) depends on t, we also denote it by z t (·) whenever there exists a possible confusion. 11). 5), we obtain that 3 Well-Posedness of the Vector-Valued BSEEs 25 T E z (T ), yT t1 H −E z t1 (ω ), f (ω ) H dω t1 T =E T ρ(ω ), y (ω ) t1 H Δ (ω ), Y t1 (ω ) dω + E t1 H dω. 11). Clearly, z t2 (·) = 0, t ≥ [t2 , t1 ). 5), it follows that T E z (T ), yT t1 H −E z t1 (ω ), f (ω ) H dω t1 T =E T ρ(ω ), y (ω ) t2 H Δ (ω ), Y t2 (ω ) dω + E t1 H dω. 8), we obtain that for any ρ(·) ≥ L 1F (t1 , T ; L q (Ω; H )) and Δ (·) ≥ L 2F (t1 , T ; L q (Ω; H )), T E T ρ(ω ), y (ω ) − y (ω ) t1 t2 H dω + E t1 Δ (ω ), Y t1 (ω ) − Y t2 (ω ) H dω = 0.

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