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Stability analysis of variational inequalities for bang-singular-bang controls

Felgenhauer U.

Control and Cybernetics

2013

Vol. 42, no. 3

557--592

The paper is related to parameter dependent optimal control problems for control-affine systems. The case of scalar reference control with bang-singular-bang structure is considered. The analysis starts from a variational inequality (VI) formulation of Pontryagin’s Maximum Principle. In a first step, under appropriate higher-order sufficient optimality conditions, the existence of solutions for the linearized problem (LVI) is proven. In a second step, for a certain class of right-hand side perturbation, it is show that the controls from LVI have bang-singular-bang structure and, in L1 topology, depend Lipschitz continuously on the data. Applying finally a common fixed-point approach to VI, the results are brought together to obtain existence and structural stability results for extremals of the original control problem under parameter perturbation.