Optimal control of semilinear elliptic equation with state constraint: Maximum principle for minimizing sequence, regularity, normality, sensitivity
The paper considers standard optimal control problem for the semilinear elliptic equation NEWLINE\[NEWLINE\text{div}[A(x)\nabla z(x)]+ f(x,z(x), u(x))= 0\quad\text{in }\Omega,\quad z\in H^1_0(\Omega),\tag{1}NEWLINE\]NEWLINE with additional pointwise constraints on the state NEWLINE\[NEWLINEG(x, z(x))\leq q(x),\quad x\in\overline D,\quad\overline D\subset\Omega.NEWLINE\]NEWLINE The author gives necessary conditions which must satisfy minimizing approximate sequences of controls \(\{u_s\}\). These conditions are of the type NEWLINE\[NEWLINE\sup_{u\in U} \int_\Omega H(x, z_s(x), \phi_s(x), u(x)) dx- \int_\Omega H(x, z_s(x), \phi_s(x), u_s(x)) dx= \gamma_s\to 0\text{ as }s\to\infty,NEWLINE\]NEWLINE where \(U\) is the set of admissible controls, \(z_s\) the solution of (1) with \(u= u_s\), \(H\) is the standard Pontryagin's function and \(\phi_s\) is the solution of the adjacent equation corresponding to \((u_s, z_s)\). The adjoint equation involves measure supported on the set where \(|G(x, z_s(x))- q(x)|\leq \gamma_s\). After that the author investigates the dependence of the cost of the problem from the functional parameter \(q\).
scientific article; zbMATH DE number 1625993