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Abstract
The paper investigates the Lipschitz/Hölder stability with respect to perturbations of optimal control problems with linear dynamic and cost functional which is quadratic in the state and linear in the control variable. The optimal control is assumed to be of bang-bang type and the problem to enjoy certain convexity properties. Conditions for bi-metric regularity and (Hölder) metric sub-regularity are established, involving only the order of the zeros of the associated switching function and smoothness of the data. These results provide a basis for the investigation of various approximation methods. They are utilized in this paper for the convergence analysis of a Newton-type method applied to optimal control problems which are affine with respect to the control.
Highlights
Stability analysis of solutions is a crucial topic in optimization theory due, in particular, to its applications for obtaining error estimates of numerical approximations
Related investigations in optimal control theory accompany its development from its early stages, the systematic analysis of (Lipschitz) stability in the area started with the works of Dontchev, Hager and Malanowski
In the present paper we focus our attention on the following class of optimal control problems: minimize J (x, u) subject to x(t) = A(t)x(t) + B(t)u(t) + d(t), t ∈ [0, T ], u(t) ∈ U := [−1, 1]m, x(0) = x0, (P)
Summary
Stability analysis of solutions is a crucial topic in optimization theory due, in particular, to its applications for obtaining error estimates of numerical approximations. The general notion of strong bi-metric regularity was introduced in somewhat more restrictive form in [23], where applications to Mayer’s type problems for linear control systems were in the focus. As the strong metric regularity, it has the important property to be invariant with respect to small (in an appropriate sense) functional perturbations of F This property is often referred to as Lyusternik-Graves type theorem, see e.g. We prove strong bi-metric regularity of the mapping F associated with Problem (P), which extends the result in [23] concerning Mayer’s problems This extension is nontrivial, since, technically speaking, the integral cost introduces the state variable in the switching function, making this function nonsmooth. We prove the strong bi-metric regularity of the mapping F resulting from problem (P) and give a result about the invariance of this property under a class of non-linear perturbations.
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