Delay systems and optimal control (Q1568228)

The authors of this paper generalize the Gronwall lemma to the case with time lags and at the same time use it to study controlled delay systems. It is considered the controlled delay systems associated with \(C_0\)-semigroup and analytic semigroup, \[ \dot x(t)=Ax(t)+f(t,x_t,x(t))+B(t)u(t)\quad (0\leq t\leq T),\qquad x(t)=\varphi (t)\quad (-r\leq t\leq 0). \] Here \(\varphi\in C\), \(x_t\in C([-r,a],X)\), \(A:D(A)\to X\) is the infinitesimal generator of a \(C_0-\)semigroup \(T(t)\) (\(t\geq 0\)) satisfying \(|T(t)|_{L}\leq Me^{\omega t}\) (\(M\geq 1\), \(\omega >0\)), \(f\) is Hölder continuous with respect to \(t\) and locally Lipschitz continuous with respect to other arguments. The main result is that there exist mild solutions and optimal control. It is shown a uniqueness result as well. An example, which illustrates the main idea, is discussed.