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By Panos M. Pardalos, Vitaliy A. Yatsenko

Covers advancements in bilinear structures conception makes a speciality of the keep an eye on of open actual techniques functioning in a non-equilibrium mode Emphasis is on 3 basic disciplines: sleek differential geometry, keep an eye on of dynamical structures, and optimization idea contains purposes to the fields of quantum and molecular computing, regulate of actual approaches, biophysics, superconducting magnetism, and actual details technological know-how

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4). A control belongs to a class of real piecewise analytical functions on (0, ∞). 8 Optimization and Control of Bilinear Systems Bilinear systems. 5) i=1 where x ∈ Rn0 = Rn − {0}; C is a constant q × n matrix; ui (t) is a scalar function of time; and A, Bi are constant n × n dimensional matrices. 5) is isomorphic to the matrix algebra Lie of a minimal dimension, which contains the matrices {A, Bi }. We can introduce a control semigroup S(U ) for BS, which assumes some representation in the form of the matrix semigroup.

6 Optimization and Control of Bilinear Systems For any X ∈ L let adX denote the endomorphism of L ad X : Y → [X, Y ] (Y ∈ L). An endomorphism adX is also a derivation of L and X → adX is a representation of L in L, called the adjoint representation. Let G be a topological group and at the same time a differentiable manifold. G is a Lie group if the mapping (x, y) → xy : G × G → G and the mapping x → x−1 : G → G are both C ∞ mappings. Given a Lie group G, there is an essentially unique way to define its Lie algebra.

16) i=1 Consider the jth column, say CAi Bj y(t), of (Li h(x(t)))k G(x(t)). 13), the kth column of Bj (A )i C can be expressed as q M0 l−1 (A ) M0 (A )l−1 Dr B0 (j, i, r, l, k) C B0 (j, i, 0, l, k) + r=1 l=1 l=1 for some constant p-vectors B0 (j, i, r, l, k). Hence q M0 Bj (A )i C = M0 (A )l−1 C B0 (j, i, 0, l) + (A )l−1 Dr B0 (j, i, r, l), r=1 l=1 l=1 where B0 (j, l, r, i) = [B0 (j, i, r, l, 1), . . , B0 (j, i, r, l, p)] and M0 i B0 (j, i, 0, l)Ll−1 h(x(t)) CA Bj y(t) = l=1 q Mr B0 (j, i, r, l)Ll−1 Qr (x(t)).

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