##
**Periodic feedback for linear systems and optimal control of bilinear systems.**
*(English)*
Zbl 0945.93002

Berlin: Logos Verlag. xiv, 121 p. (1999).

This book considers \(T\)-periodic systems of the form
\[
\dot x= A(t)x+ B(t)u
\]
where \((A(t), B(t))\) is analytic in \(t\) and controllable on \([0,T]\). Given a linear feedback \(u= K(t)x\), the objective is to choose \(K\) so that
\[
\Phi(T,0)= \exp(MT)
\]
for any given matrix \(M\), i.e. to assign arbitrarily the monodromy matrix, where \(\Phi\) is the transition function. The gain \(K\) is determined by an associated bilinear quadratic optimal control problem solved via a Lax-type equation which parallels the Riccati equation solution in L-Q theory.

The first chapter considers the stabilization of the system \[ \dot x= f(x,\lambda)+ ug(x) \] which has a period-doubling bifurcation, by applying a linearisation and a centre manifold type reduction. The nonlinear period-doubling effect is cancelled by linear feedback obtained through the periodic L-Q theory by solving a periodic Riccati equation and conditions are given under which this is possible.

In chapter 2 the assignability of the monodromy matrix is considered. By using a Floquet transformation, it is shown that there are infinitely many stepwise constant periodic feedbacks which will achieve the assignment. Hence the restrictions which apply to Jordan block assignability in the linear, time-invariant case are no longer required in the periodic case.

The optimal choice of a feedback matrix for the system \[ \dot x= (A(t)+ B(t) K(t))x \] which assigns a particular monodromy matrix is considered in chapter 3, by solving an associated quadratic optimal control problem with bilinear constraint equation \[ \dot P= A(t)P- PM(t)+ B(t)KP. \] This equation derives from the fact that \(P(t)\) is a Lyapunov transformation and so puts the original system in the form \[ \dot y= M(t)y. \] Using nonlinear control theory and the maximum principle, a solution to the problem is found.

In chapter 4, the optimal control of bilinear systems on Lie groups is studied, by using the maximum principle and the Lax reduction of the Hamiltonian dynamics. It is shown that a simple bilinear system on a connected Lie group has completely integrable Hamiltonian dynamics for the optimal feedback system.

The final chapter compares the monodromy assignment technique presented earlier with the standard L-Q approach. The monodromy assignment problem via output feedback which holds under a natural observability assumption is solved.

This is an interesting monograph containing many ideas, some of which are new, and therefore should be useful to nonlinear system theorists.

The first chapter considers the stabilization of the system \[ \dot x= f(x,\lambda)+ ug(x) \] which has a period-doubling bifurcation, by applying a linearisation and a centre manifold type reduction. The nonlinear period-doubling effect is cancelled by linear feedback obtained through the periodic L-Q theory by solving a periodic Riccati equation and conditions are given under which this is possible.

In chapter 2 the assignability of the monodromy matrix is considered. By using a Floquet transformation, it is shown that there are infinitely many stepwise constant periodic feedbacks which will achieve the assignment. Hence the restrictions which apply to Jordan block assignability in the linear, time-invariant case are no longer required in the periodic case.

The optimal choice of a feedback matrix for the system \[ \dot x= (A(t)+ B(t) K(t))x \] which assigns a particular monodromy matrix is considered in chapter 3, by solving an associated quadratic optimal control problem with bilinear constraint equation \[ \dot P= A(t)P- PM(t)+ B(t)KP. \] This equation derives from the fact that \(P(t)\) is a Lyapunov transformation and so puts the original system in the form \[ \dot y= M(t)y. \] Using nonlinear control theory and the maximum principle, a solution to the problem is found.

In chapter 4, the optimal control of bilinear systems on Lie groups is studied, by using the maximum principle and the Lax reduction of the Hamiltonian dynamics. It is shown that a simple bilinear system on a connected Lie group has completely integrable Hamiltonian dynamics for the optimal feedback system.

The final chapter compares the monodromy assignment technique presented earlier with the standard L-Q approach. The monodromy assignment problem via output feedback which holds under a natural observability assumption is solved.

This is an interesting monograph containing many ideas, some of which are new, and therefore should be useful to nonlinear system theorists.

Reviewer: S.P.Banks (Sheffield)

### MSC:

93-02 | Research exposition (monographs, survey articles) pertaining to systems and control theory |

49N35 | Optimal feedback synthesis |

49N20 | Periodic optimal control problems |

93B52 | Feedback control |

93B29 | Differential-geometric methods in systems theory (MSC2000) |

93B17 | Transformations |

93D15 | Stabilization of systems by feedback |

93B55 | Pole and zero placement problems |

37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |

37G35 | Dynamical aspects of attractors and their bifurcations |

37B55 | Topological dynamics of nonautonomous systems |

34C15 | Nonlinear oscillations and coupled oscillators for ordinary differential equations |

34C20 | Transformation and reduction of ordinary differential equations and systems, normal forms |

34D08 | Characteristic and Lyapunov exponents of ordinary differential equations |