By Andrej A. Agračev
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Additional resources for Mathematical control theory, Summer School, ICTP, Trieste, Italy, 3-28 September 2001
Zabczyk, Appl. Math. Optimization 3, 383 (1976).
6) is called Bellman’s equation. 1 that, under appropriate conditions, W (T, x) is the minimal value of the functional JT (x, ·). 2). 6) is attained. The function vˆ(·, ·) from part (ii) of the theorem is a selector of the multivalued function U (·, ·) in the sense that vˆ(t, x) ∈ U (t, x), (t, x) ∈ [0, T ] × E. 10) should have a well defined, absolutely continuous, solution. Remark A similar result holds for a more general cost functional T JT (x, u( · )) = e−αt g(y(t), u(t)) dt + e¯αT G(y(T )).
0 0 . . 0 γk 0 0 ... 0 0 Setting ω0 = ω(A) we get r r ω0 t k=1 wk (t) ≤ e xk , q(t) t ≥ 0, k=1 where q is a polynomial of order at most max(jk − 1), k = 1, . . , r. If ω > ω0 and M0 = sup q(t)e(ω0 −ω)t , t ≥ 0 , then M0 < +∞ and r r k=1 wk (t) ≤ M0 eωt xk , k=1 t ≥ 0. Therefore for a new constant M1 w(t) ≤ M1 eωt x , t ≥ 0. Finally z x (t) = P w(t)P −1 ≤ M1 eωt P P −1 x , t ≥ 0, 32 J. Zabczyk and this is enough to define M = M1 P P −1 . Proof of the theorem. Assume ω0 ≥ 0.
Mathematical control theory, Summer School, ICTP, Trieste, Italy, 3-28 September 2001 by Andrej A. Agračev