Stability of Riccati's equation with random stationary coefficients (Q1304956)

The author considers the matrix Riccati equation \[ \dot P(t)= A(t)P(t)+ P(t)A^T(t) +B(t)B^T(t)- P(t)C^T(t) C(t)P(t) \tag{1} \] where \(P(t)\) is a nonnegative symmetric matrix and \((A(t),B(t), C(t))_{t\in\mathbb{R}}\) is a stationary ergodic process of matrices of appropriate dimension. Let \(H(t)\) be the fundamental matrix associated with \[ M(t)= \left[\begin{matrix} A(t) & B(t)B^T(t)\\ C^T(t)C(t) & -A^T(t) \end{matrix}\right].\tag{2} \] The author proves that the Lyapunov exponents of \(H(t)\) are nonzero when the process \((A(t)\), \(B(t),C(t))_{t\in\mathbb{R}}\) is weakly stabilizable and detectable. He also proves that weak observability or weak detectability implies exponential detectability. Similar results hold for the corresponding notions of stabilizability and weak controllability. He shows a construction of a stationary solution to equation (1) under weak conditions of stabilizability and detectability. It is also shown that this solution is attracting in the sense that every other solution converges to it exponentially fast at the rate given by the smallest positive Lyapunov exponent of \(H(t)\).