zbMATH — the first resource for mathematics

Hamiltonian Pontryagin’s principles for control problems governed by semilinear parabolic equations. (English) Zbl 0922.49013
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. distributed, boundary and 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 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.

49K20 Optimality conditions for problems involving partial differential equations
93C20 Control/observation systems governed by partial differential equations
35K15 Initial value problems for second-order parabolic equations
35K20 Initial-boundary value problems for second-order parabolic equations
Full Text: DOI