Controllability of Bilinear Systems

The systems we study are homogeneous bilinear, single-input: $$\dot{x}=(A+{{u}_{t}}B)x$$ (1.1) $${{x}_{k+1}}=(A+{{u}_{k}}B){{x}_{k}}$$ (1.2) where the state space is R 0 n = Rn - {0} (since the origin is an isolated equilibrium point), and the controls u are scalar. We are concerned with conditions, sufficient or necessary, for state controllability of these systems for bounded or unbounded controls.

[1]  R. R. Mohler,et al.  Completely Controllable Bilinear Systems , 1968 .

[2]  William M. Boothby,et al.  A transitivity problem from control theory , 1975 .

[3]  H. Sussmann,et al.  Control systems on Lie groups , 1972 .

[4]  D. Elliott.  A consequence of controllability , 1971 .

[5]  C. Bruni,et al.  Bilinear systems: An appealing class of "nearly linear" systems in theory and applications , 1974 .

[6]  Wei-Liang Chow.  Über Systeme von linearen partiellen Differential-gleichungen erster Ordnung , 1941 .

[7]  J. Zaborszky,et al.  On the controllability of a class of discrete bilinear systems , 1973 .

[8]  A. Krener.  A Generalization of Chow’s Theorem and the Bang-Bang Theorem to Nonlinear Control Problems , 1974 .

[9]  D. Elliott,et al.  Controllability of discrete bilinear systems with bounded control , 1973 .

[10]  L. Bittner.  S. H. Lehnigk, Stability Theorems For Linear Motions. (International Series in Applied Mathematics.) XI + 251 S. m. Fig. Englewood Cliffs. N. J. 1966. Prentice‐Hall, Inc. Preis geb. 96.– s. net , 1971 .

[11]  H. Sussmann,et al.  Controllability of nonlinear systems , 1972 .

[12]  Jan Kucera,et al.  Title: Solution in large of control problem $\dot x=(A(1-u)+Bu)x$ , 1966 .

[13]  R. Brockett.  System Theory on Group Manifolds and Coset Spaces , 1972 .

[14]  C. Desoer,et al.  The minimal time regulator problem for linear sampled-data systems: General theory☆ , 1961 .

[15]  D. Elliott,et al.  Controllability and Observability for Bilinear Systems , 1971 .

[16]  Wei-Liang Chow.  Über Systeme von liearren partiellen Differentialgleichungen erster Ordnung , 1940 .

[17]  R. Hermann.  On the Accessibility Problem in Control Theory , 1963 .

[18]  Tzyh Jong Tarn.  Singular Control of Bilinear Discrete Systems , 1972, Inf. Control..