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1

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

Felgenhauer U.

Control and Cybernetics

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2013

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Vol. 42, no. 3

557--592

EN

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.

2

Optimality and stability result for bang-bang optimal controls with simple and double switch behaviour

Felgenhauer U., Poggiolini L., Stefani G.

Control and Cybernetics

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2009

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Vol. 38, no 4B

1305-1325

EN

The paper considers parametric optimal control problems with bang-bang control vector function. For this problem we give regularity and second-order optimality conditions at the nominal solution which are sufficient to: (i) existence and local uniqueness of extremals, (ii) local structure stability, (iii) strong local optimality, under parameter perturbations. Here "local" means in a L∞ neighbourhood of the nominal trajectory, regardless of the control values. Stability results were obtained by the first author using the shooting approach, while optimality results were obtained by the other authors, using the Hamiltonian approach. The paper, combining both approaches, allows to unify the assumptions and to close some gaps between optimality and stability results.

3

The shooting approach in analyzing bang-bang extremals with simultaneous control switches

Felgenhauer U.

Control and Cybernetics

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2008

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Vol. 37, no 2

307-327

EN

The paper is devoted to stability investigation of optimal structure and switching points position for parametric bang-bang control problem with special focus on simultaneous switches of two control components. In contrast to problems where only simple switches occur, the switching points in general are no longer differentiable functions of input parameters. Conditions for Lipschitz stability are found which generalize known sufficient optimality conditions to nonsmooth situation. The analysis makes use of backward shooting representation of extremals, and of generalized implicit function theorems. The Lipschitz properties are illustrated for an example by constructing backward parameterized family of extremals and providing first-order switching points prediction.

4

Optimality properties of controls with bang-bang components in problems with semilinear state equation

Felgenhauer U.

Control and Cybernetics

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2005

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Vol. 34, no 3

763-785

EN

In this paper we study optimal control problems with bang-bang solution behavior for a special class of semilinear dynamics. Generalizing a former result for linear systems, optimlity conditions are derived by a duality based approach. The results apply for scalar as well as for vector control functions and, in particular, for the case of the so-called multiple switches, too. Further, an iterative procedure for determining switching points is proposed, and convergence results are provided.

5

Optimality and sensitivity for semilinear bang-bang type optimal control problems

Felgenhauer U.

International Journal of Applied Mathematics and Computer Science

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2004

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Vol. 14, no 4

447-454

EN

In optimal control problems with quadratic terminal cost functionals and systems dynamics linear with respect to control, the solution often has a bang-bang character. Our aim is to investigate structural solution stability when the problem data are subject to perturbations. Throughout the paper, we assume that the problem has a (possibly local) optimum such that the control is piecewise constant and almost everywhere takes extremal values. The points of discontinuity are the switching points. In particular, we will exclude the so-called singular control arcs, see Assumptions 1 and 2, Section 2. It is known from the results by Agrachev et al. (2002) stating that regularity assumptions, together with a certain strict second-order condition for the optimization problem formulated in switching points, are sufficient for strong local optimality of a state-control solution pair. This finite-dimensional problem is analyzed in Section 3 and optimality conditions are formulated (Lemma 2). Using well-known results concerning solution sensitivity for mathematical programs in Rn (Fiacco, 1983) one may further conclude that, under parameter changes in the problem data, the switching points will change Lipschitz continuously. The last section completes these qualitative statements by calculating sensitivity differentials (Theorem 2, Lemma 6). The method requires a simultaneous solution of certain linearized multipoint boundary value problems.