Optimal control of semilinear elliptic equation with state constraint : maximum principle for minimizing sequence, regularity, normality, sensitivity
Mikhail Iosifovich Sumin, Nizhnii Novgorod · Control and Cybernetics · 2000
This article deals with state constrained optimal con trol problem for semilinear elliptic equation in a domain n. The state constraint is lumped on the compactum X c n and contains a functional parameter q E C(X). It is shown that any minimizing approximate solution (m.a.s. ) in the sense of J. Warga satisfies the pointwise maximum principle (the maximum principle for m.a.s.) if the problem is meaningful, i.e., the value of the problem is finite. It is also shown that a condition of Slater's type is sufficient for the nor mality in the so-called linear-convex problem, and the normality of the problem for some fixed value of the parameter q E C(X) im plies the Lipschitz continuity of its value function in a neighborhood of q. The paper contains illustrative examples.