Let us consider a single input T-periodic system
where x(0)=x0 and x in Rn. We assume that it is controllable on [0,T] and that the system matrices depend analytically on time. The main result of the thesis is to show that it is possible to find a T-periodic feedback of the form
such that any matrix M in Mn(R) which we call the monodromy matrix, can be chosen such that the monodromy operator Phi(T,0) of the closed-loop system satisfies
In this way, we perform monodromy matrix assignment which generalizes to periodic systems the notion of pole placement for linear autonomous systems. We give a constructive way of synthesizing the periodic feedback via an auxiliary optimal control problem with fixed end point conditions.
The research is structured in the following way:
We provide first a motivating example for studying such a problem. The stabilization of a periodic orbit which undergoes possibly a period doubling bifurcation is studied. After suitable coordinate transformations, the linearization around the periodic solution leads to a periodic linear control system which will be generically controllable. A classical stabilization approach via the periodic Riccati equation is proposed. In case the mode corresponding to the period doubling is linearly uncontrollable, the bifurcation cannot be eliminated by linear feedback but a center manifold approach shows that the stability of the citical orbit can be guaranteed in general by nonlinear feedback.
Next, we show that it is possible to construct a periodic feedback which answers the problem of monodromy matrix assignment. A consequence is that the limitations for Jordan block assignment in the autonomous case disappear in a periodic context.
Then, we show how to construct the periodic gain via an optimal control problem. We have existence of a solution, normality and appearance of two analogues to the Riccati equation which arises in the L-Q optimization problem. These are two equations in the Lax form. They simplify the analytical treatment and help characterizing the boundary conditions of the optimal control problem.
This leads to the next step. We provide a theoretical justification for the two Lax equations obtained in the previous chapter. They describe the dynamics on coadjoint orbits for two group actions which appear naturally when studying affine control systems made of right-invariant and left-invariant vector fields on Lie groups. The analoque to the solution to the Riccati equation can be identified with the momentum mappings associated to the corresponding hamiltonian actions. This helps studying complete integrability of the hamiltonian dynamics corresponding to the optimal control problem under study. Such a dynamics is completely integrable in general in a suitable sense.
Finally, we present a framework for comparing our optimal control problem with the standard L-Q problem. The advantage is that we place the poles whereas the L-Q method stabilizes only. We finally show that the limitations concerning ouput pole placement disappear in a periodic context: with a SISO controllable and observable system, we can generically place any monodromy matrix.