Differential stability of solutions to constrained optimization problems. (English) Zbl 0572.49012
Results concerning directional (one-sided) differentiability of the mapping of projection onto a closed, convex set in a Hilbert space are quoted and several examples are provided. These results are used for sensitivity analysis of constrained optimal control problems. An abstract convex optimal control problem subject to control constraints is considered. The cost functional and the state equation depend on a real parameter. Using the results of directional differentiability of the projection mapping it is shown that the right-derivative of the optimal solution exists and it is given by the solution of an auxiliary quadratic optimal control problem.
The abstract result is illustrated by an example of an optimal control problem for a parabolic equation. The problem of second order right- differentiability is discussed.
The abstract result is illustrated by an example of an optimal control problem for a parabolic equation. The problem of second order right- differentiability is discussed.
Reviewer: K.Malanowski
MSC:
| 49K40 | Sensitivity, stability, well-posedness |
| 49J50 | Fréchet and Gateaux differentiability in optimization |
| 49K20 | Optimality conditions for problems involving partial differential equations |
| 46C99 | Inner product spaces and their generalizations, Hilbert spaces |
| 52A07 | Convex sets in topological vector spaces (aspects of convex geometry) |
| 46G05 | Derivatives of functions in infinite-dimensional spaces |
| 93B35 | Sensitivity (robustness) |
| 93C20 | Control/observation systems governed by partial differential equations |
| 93C05 | Linear systems in control theory |
Keywords:
directional (one-sided) differentiability; projection onto a closed, convex set; Hilbert space; sensitivity analysis; constrained optimal control problemsReferences:
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