Anti-periodic oscillations of some second-order differential equations and optimal control problems
The authors study the existence, uniqueness and continuous dependence (on \(f)\) of the mild solutions \(y\) of the following perturbed wave equation \[ y_{tt} (t,x) + \alpha y_t (t,x) - \Delta_x y(t,x) = f(t,x) \quad \text{in} \quad (0,T) \times \Omega = Q \tag{1} \] with the following Dirichlet boundary condition in \(x \in \mathbb{R}^n\) \[ y(t,x) = 0\text{ on }(0,T) \times \partial \Omega \tag{2} \] and anti-periodic conditions in \(t\) \[ y(0,x) + y(T,x) = 0,\quad y_t (0,x) + y_t (T,x) = 0\text{ in }\Omega \tag{3} \] where \(f \in L^2 (Q)\). This study is carried out via the \(C_0\)-semigroup \(S_\alpha(t)\), \(t \geq 0\), in the Hilbert space \(H = H (\Omega) = H_0' (\Omega) \times L^2 (\Omega)\). The main optimal control problem which is studied is the following: Minimize \[ L(y,u) = \int^T_0 \int_\Omega \biggl( g \bigl( y(t,x) \bigr) + h \bigl( u(t,x) \bigr) \biggr) dx dt \tag{4} \] on all pairs \((y,u)\) with \(u \in L^2 (Q)\) and \(y\) the mild solution of (1)--(3) with \(f(t,x) + u(t,x)\) in place of \(f \in L^2 (Q)\).
scientific article; zbMATH DE number 711755