Nonautonomous delay differential equations in Hilbert spaces and Lyapunov exponents
Let \(m\) be a positive integer, \(\mathbb{C}\) the set of complex numbers and \(\tau >0\). Let \(X=\mathbb{C}^m \times L^2([-\tau ,0]; \mathbb{C}^{m})\) be a Hilbert space with the inner product \(\left \langle x,y\right \rangle =v^Hu + \int _{-\tau }^0 \psi ^{H}(t)\varphi (t)\, dt\) for \(x=(u,\varphi )\), \(y =(v,\psi ) \in X\). For given \(s\geq 0\), the author gives conditions on \(A,B\:[s,\infty ) \to \mathbb{C}^{m \times m}\) and \(D(t,\cdot ) \in L^2([-\tau ,0]; \mathbb{C}^{m \times m})\) (\(t \geq s\)) which guarantee that, for each \(T>s\) and each \((u,\varphi ) \in X\), the initial value problem \[ x'(t)=A(t)x(t) + B(t)x(t-\tau ) + \int _{-\tau }^0 D(t,\xi )x(t+\xi )\, d \xi,\;t \geq s, \] \(x(s)=u\), \(x(s+\xi )=\varphi (\xi )\) for a.e. \(\xi \in [-\tau ,0)\) has a unique solution \(x(\cdot )=x(\cdot ;s,u,\varphi )\) on \([s - \tau ,T]\) such that \(x \in W^{1,2}[s,T]\). Using this result, for \(t \geq s\) an operator \(U(t,s)\: X \to X\) is defined by the formula \(U(t,s)(u,\varphi )=(x(t),x_{t})\), where \(x\) is the unique solution of the above problem and \(x_{t}(\xi )=x(t+\xi )\) for \(\xi \in [-\tau ,0]\). It is proved that \(\{U(t,s)\}_{t \geq s}\) is an evolution family and that \(U(t,s)\) is a compact operator for all \(t \geq s+\tau \). Further, an operator \(F(t)\: \operatorname {dom}(F(t)) \subset X \to X\) is introduced having the property that the abstract Cauchy problem \(z'(t)=F(t)z(t)\), \(t \geq s\), \(z(s)=w \in X\) has the unique solution \(z(t)=U(t,s)w\) as soon as \(w \in \operatorname {dom}(F(s))\). Finally, the existence of infinitely many Lyapunov exponents for the associated evolution is proved and their meaning is discussed.
nonautonomous delay differential equation
scientific article; zbMATH DE number 5944978