Stability of Lurie-type evolution equations with multiple nonlinearities in Hilbert spaces (Q2714171)

The system \(\dot x=Ax+\sum_{i=1}^m b_i\mu_i\), \(\dot \mu_i=\phi_i(\sigma_i)\), \(\sigma_i=(c_i,x)-\rho_i\mu_i\), \(i\in\{1,2,\ldots,m\}\), is studied, where \(A\) generates a \(C_0\) group on a real Hilbert space \(X\) with inner product \((\cdot,\cdot)\), \(b_i,c_i\in X\), \(\rho_i\in \mathbb{R}\), \(\phi:\mathbb{R}\to \mathbb{R}\) is Lipschitz continuous. Under certain conditions (e.g., the origin is the only stationary point, and \(\dot x=Ax\) is exponentially stable) the global uniform asymptotic stability of the zero solution is proved. A similar result is given for the nonlinear version of the above system, where \(Ax\) is replaced by a monotonic function \(f:X\to X\).