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Optimal control of semilinear elliptic equation with state constraint : maximum principle for minimizing sequence, regularity, normality, sensitivity

Sumin M.

Control and Cybernetics

2000

Vol. 29, no 2

449-472

This article deals with state constrained optimal control problem for semilinear elliptic equation in a domain Omega. The state constraint is lumped on the compactum X contained in/implied by Omega n and contains a functional parameter q in 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 normality in the so-called "linear-convex" problem, and the normality of the problem for some fixed value of the parameter q in C(X ) implies the Lipschitz continuity of its value function in a neighborhood of q. The paper contains illustrative examples.

Lipschitz stability of solutions to parametric optimal control for elliptic equations

Malanowski K., Troeltzsch F.

Control and Cybernetics

2000

Vol. 29, no 1

237-256

: A family of parameter dependent elliptic optimal control problems with nonlinear boundary control is considered. The control function is subject to amplitude constraints. A characterization of conditions is given under which solutions to the problems exist, are locally unique and Lipschitz continuous in a neighborhood of the reference value of the parameter.

Application of the classical implicit function theorem in sensitivity analysis of parametric optimal control

Malanowski K.

Control and Cybernetics

1998

Vol. 27, no 3

335-352

A family {(O[sub h])} of parametric optimal control problems for nonlinear ODEs is considered. The problems are subject to pointwise inequality type state constraints. It is assumed that the reference solution is regular. The original problems (O[sub h]) are substituted by problems [...] subject to equality type constraints with the sets of activity depending on the parameter. Using the classical implicit function theorem, conditions are derived under which stationary points of [...] are Frechet differentiable functions of the parameter. It is shown that, under additional conditions, the stationary points of [...] correspond to the solutions and Lagrange multipliers of (O[sub h]9).