Optimization and control of bilinear systems: Theory, by Panos M. Pardalos, Vitaliy A. Yatsenko

By Panos M. Pardalos, Vitaliy A. Yatsenko

Covers advancements in bilinear platforms thought

Focuses at the keep watch over of open actual methods functioning in a non-equilibrium mode

Emphasis is on 3 fundamental disciplines: smooth differential geometry, keep an eye on of dynamical structures, and optimization concept

Includes purposes to the fields of quantum and molecular computing, keep watch over of actual methods, biophysics, superconducting magnetism, and actual info science

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Extra resources for Optimization and control of bilinear systems: Theory, algorithms, and applications

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A(i, j, M ) ⎡ ⎤ B0 (i, 0) B0 (i, 1) . . B0 (i, q) ⎢B1 (i, 0) B1 (i, 1) . . B1 (i, q)⎥ ⎢ ⎥ Bi = ⎢ .. ⎥ , ⎣ . ⎦ ⎡ Bq (i, 0) Bj (i, 1, k, 1) Bj (i, 2, k, 1) .. ⎢ ⎢ Bj (i, k) = ⎢ ⎣ Bj (i, Mk , k, 1) ...... Bq (i, q) Bj (i, 1, k, 2) . . ... ... Bj (i, 1, k, Mi ) Bj (i, 2, k, Mi ) .. ⎤ ⎥ ⎥ ⎥. 12) holds, where C = [I 0 . . 0], D = [0 ( ... i−1 j=0 0 I 0 ... 0]. 13) is true. This completes the proof of sufficiency. Necessity. As the bilinear and nonlinear systems are dynamically equivalent, we have, for a control function v, p z(t) = h(x(t)) + Q(x(t))v(t) = Di vi (t) y(t).

7. 11) is said to have a finite-dimensional sensor orbit, if there exist integers Mi , i = 0, . . , q, such that for k = 1, . . , q, and all state trajectories x(t), t ∈ T , M0 −1 M0 L A(0, 0, i + 1)Li h(x(t)) h(x(t)) = i=0 q Mq −1 A(0, j, i + 1)Li Qj (x(t)), + j=1 i=0 M0 −1 Mk L A(k, 0, i + 1)Li h(x(t)) Qk (x(t)) = i=0 q Mq −1 A(k, j, i + 1)Li Qj (x(t)), + j=1 i=0 where A(i, j, k) are constant p × p matrices, and every column of (Li h(x(t)))x G(x(t)) and (Li Qj (x(t)))x G(x(t)), i = 0, . . , (Mj − 1), j = 1, .

5) is that Gl(m, R) is a Lie group and each Aj defines a right-invariant vector field Aj X on this group, hence a member of the associated Lie algebra gl(m, R) of all m × m real matrices. This algebra is finite-dimensional over the field R and the multiplication is defined by the Lie bracket [Ai , Aj ] = Aj Ai − Ai Aj . This is a noncommutative and nonassociative operation that satisfies the skew-symmetry and Jacobi relations, [Ai , Aj ] = −[Aj , Ai ], and [Ai , [Aj , Ak ]] = [[Ai , Aj ], Ak ] + [Aj , [Ai , Ak ]].

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