Stabilization and random linear regulator problem for random linear control processes
The authors consider the feedback stabilization problem for linear time varying control systems by interpreting the system matrices as being defined by a dynamical system: Let \(C\) and \(D\) be spaces of matrix functions with appropriate topologies, and define the shift \(\tau_t : C \times D \to C \times D\) by \(\tau_t (a,b) (s) = (a (t + s), b(t + s))\) for \((a, b) \in C \times D\). Let \(\Omega \subset C \times D\) be a compact, shift-invariant set and let \(A : \Omega \to M(n,n)\), \(B : \Omega \to M (n,m)\) be matrix valued functions. Then a (family of) linear, varying control systems is given by \[ \dot x = A \bigl( \tau_t (w) \bigr) x + B \bigl( \tau_t (w) \bigr) u, \quad w \in \Omega. \] The linear quadratic regulator problem for these systems is solved in the following way: Assume the (suitable modified) controllability assumptions for the system and its adjoint, construct an exponential dichotomy over \(\Omega\) via properties of the rotation numbers of the associated Hamiltonian system, interpret the stable bundle of the dichotomy as the solution of a Riccati equation, and define the optimal feedback in the usual way, using this solution. The advantage of this approach is that the resulting feedback inherits smoothness, robustness, and invariance properties (w.r.t. te shift \(\tau)\) from the stable bundle, uniformly, in \(\Omega\). Choosing the \(Q\)-matrix of the LQR problem as \(\delta \cdot\) identity, the authors show that the Lyapunov exponents of the system lie to the left of \(- \delta \cdot\) constant, uniformly in \(w \in Q\). This result can be considered as a time varying version of the pole placement theorem.
scientific article; zbMATH DE number 902462