By Panos M. Pardalos, Vitaliy A. Yatsenko
Covers advancements in bilinear platforms concept
Focuses at the keep an eye on of open actual strategies 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 an eye on of actual procedures, biophysics, superconducting magnetism, and actual info science
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Extra resources for Optimization and control of bilinear systems: Theory, algorithms, and applications
Sample text
Q, y˙ ji (t) = yji+1 (t) + (Li−1 Qj (x(t)), G(x(t))u(t)), y˙ 0i (t) = y0i+1 (t) + (Li−1 h(x(t))x G(x(t))u(t)). 11) has a finite-dimensional sensor orbit for j = 0, . . , q, q Mq y˙ jMj = m q Mk A(j, k, i)yki (t) k=0 i=1 Bj (i, Mj , k, l)ykl (t)ui (t) i=1 k=0 l=1 and, for r = 1, . . , Mj − 1, j = 0, . . , q, m q Mk y˙ jr = yjr+1 (t)dt + Bj (i, r, k, l)ykl (t)ui (t), i=1 k=0 l=1 16 Optimization and Control of Bilinear Systems where A(j, k, i) and Bj (i, r, k, l) are constant matrices. Let y˙ = [y01 , .
Vr ), respectively. 1. The nonlinear system, m x(t) ˙ = Gi ui (t) x(t), F+ i=1 p z(t) = q ϕr (x(t), . . 18) i=1 where A, B, C, Di are some constant matrices of appropriate sizes and, for some positive integers Mi , i = 0, . . 13) holds. 1. System-Theoretical Description of Open Physical Processes 3. 19 Identification of Bilinear Control Systems Due to the widespread use of bilinear models, there is strong motivation to develop identification algorithms for such systems given noisy observations (Fnaiech, Ljung, and Fliess, 1987; Dang Van Mien and Norman-Cyrot, 1984; Krishnaprasad and Marcus, 1982).
Bm (x) by A, B1 , . . , Bm , respectively, where A and B are elements in G(G), the Lie algebra of G. Then we have the following lemma. 3. Assume h ∈ H. Then Ad (r)h ∈ H for all r ∈ {G(e)}G . 35) for all r ∈ G(e). 36) Proof: First, we claim that Ad (r)h ∈ H Because R(e) is a semigroup, for any r˜ ∈ R(e) we have r˜r ∈ R(e). Thus, (˜ rr)h(˜ rr)−1 = r˜(rhr−1 )1 r˜−1 ∈ C for all r ∈ R(e). It follows that rhr−1 ∈ H. From its defining properties, it is clear that {R(e)}G = exp(ts Xs ) | ti ∈ R, s ∈ Z + , m Xi ∈ uj Bj uj ∈ R , i = 1, .
