Pontryagin's principle for time-optimal problems
The paper considers time-optimal control problems for the semilinear parabolic equation \[ y_t- \sum^n_{i,j= 1}{\partial\over\partial x_j} a_{ij}(x) y_{x_i}+ f(x, t,y,u)= 0\quad\text{in }\Omega\times (0,T), \] \[ {\partial y\over\partial v}= 0\quad\text{on }\partial\Omega\times(0, T),\quad y(x,0)= y_0(x)\quad\text{in }\Omega, \] with constraints on the state \(y\) at the terminal time \(T\); the terminal time \(T\) and \[ u\in U= \{u\in L^s_{\text{loc}}((0, \infty); L^s(\Omega))|u(\cdot, t)\in K(x)\text{ a.e. }x\in\Omega\} \] being the controls. Here \(\Omega\subset \mathbb{R}^n\) is a bounded domain with a smooth boundary \(\partial\Omega\) and \(K\) is a measurable multimapping with nonempty closed values. The target functional involves integrals over \(\Omega\times(0, T)\) and over \(\Omega\) at the terminal time \(T\). It is proved that the optimal triple \((y_*, u_*, T_*)\) satisfies Pontryagin's maximum principle. The proof uses Ekeland's variational principle and a change of variables (to pass to a problem with a fixed terminal time).
scientific article; zbMATH DE number 1300020