Hamiltonian Pontryagin's principles for control problems governed by semilinear parabolic equations
The system is \[ {\partial y \over \partial t} + Ay + f(t, x, y, u) = 0, \] where \(A\) is a second order elliptic operator in an \(m\)-dimensional domain \(\Omega\) with boundary \(\Gamma\) and \(0 \leq t \leq T\). Besides the distributed control \(u,\) there are controls \(v, w\) acting on the initial and the boundary condition, \[ y(0) = w \quad (x \in \Omega), \qquad {\partial y \over \partial n_A} + g(t, x, y, v) = 0 \quad(x \in \Gamma). \] Here \(u\) (resp. \(v, w)\) belongs to \(L^q((0, T) \times \Omega)\) (resp. \(L^r((0, T) \times \Gamma)\), \(L^\infty(\Omega))\). The controls are subject to the pointwise constraints \[ u(t, x) \in K_U, \quad v(t, x) \in K_V, \quad w(x) \in K_W. \] The cost functional undergoing minimization is \[ J(y, u, v, w)= \int_{(0, T) \times \Omega} F(t, x, y, u)dx dt+ \int_{(0, T) \times \Gamma} G(t, x, y, v) d\sigma dt + \int_\Omega \ell(x, y(T), w) dx. \] The authors define three Hamiltonians (resp. \textit{distributed}, \textit{boundary} and \textit{initial}) by \[ \begin{aligned} H_d(t, x, y, u, p) &= F(t, x, y, u) - pf(t, x, y, u),\\ H_b(t, x, y, v, p) &= G(t, x, y, v) - pg(t, x, y, v),\\ H_i(x, y, w, p) &= \ell(x, y, w) + pw,\end{aligned} \] and obtain necessary conditions for optimal controls \(\bar u(t, x),\) \(\bar v(t, x),\) \(\bar w(x)\) and optimal solutions \(\bar y(t, x)\) in the form of three Pontryagin's principles. These minimum principles are \textit{decoupled} in the sense that each one involves only one of the controls. Since both the distributed and the boundary control may be unbounded, the proof is technically complicated and needs new regularity results for linear and nonlinear parabolic equations.
scientific article; zbMATH DE number 1271357