The analog of the frequency theorem for nonstationary systems (Q1899679)

The author considers a nonlinear control system with nonstationary linear part \[ \dot x= A(t) x(t)+ b(t)\xi(t),\tag{1} \] where \(A\), \(b\) are \(n\times n\) and \(n\times m\) matrices respectively. \(x\) represents the input, \(\xi\) the output. A quadratic relationship is imposed \[ \exists t_0,\;t_k\to \infty,\;\exists \gamma>0: \int^{t_k}_{t_0} F(t, x, \xi)dt\geq -\gamma.\tag{2} \] \(F\) is a fixed quadratic form: \[ F(t, x, \xi)= \textstyle{{1\over 2}} [x^T G(t) x+ 2 x^T g(t)\xi+ \xi^T \Gamma(t)\xi]. \] The problem of absolute stability of this system is investigated with the help of a Lyapunov function \(V= x^T R(t)x\), where \(R\) is symmetric and \(|R|\leq \text{const}\) \(\forall t\geq 0\). The following definitions are given: The system (1), (2) is called minimally stable if there exists a matrix \(\mu(t)\) such that \(|\mu|\leq \text{const}\) \(\forall t\geq 0\) and all solutions with feedback \(\xi(t)= \mu(t) x(t)\) are in \(L_2[0, \infty)\) while \(F(t, x, \mu(t)x)\geq 0\) \(\forall t\) and all \(x\). Matrix \(B(\cdot)\) is called \(L_2\) stable if all solutions of the vector equation \(\dot x= Bx\) belong to \(L_2[0, \infty)\). The following theorems are proved: Suppose that (3) \(\exists\varepsilon> 0\): \(\Gamma(t)\leq -\varepsilon I_m\) (\(m\times m\) unit matrix). Let \[ J= \Biggl[\begin{matrix} 0 & - I_n\\ I_n & 0\end{matrix}\Biggr],\quad H= \Biggl[\begin{matrix} G- g\Gamma^{-1} g^T & A- b\Gamma^{- 1} g^T\\ A- b\Gamma^{-1} g^T & -b\Gamma^{- 1} b^T\end{matrix}\Biggr]. \] Suppose that the system (1), (2) is minimally stable and the corresponding Hamiltonian system \(J\dot z= Hz\) is 0- nonoscillatory. Then the (1)--(2) system is absolutely stable, an appropriate Lyapunov function exists and a matrix \(R(t)\) exists and satisfies a matrix Riccati differential equation.