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

International Journal of Applied Mathematics and Computer Science (2004)

- Volume: 14, Issue: 4, page 447-454
- ISSN: 1641-876X

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topFelgenhauer, Ursula. "Optimality and sensitivity for semilinear bang-bang type optimal control problems." International Journal of Applied Mathematics and Computer Science 14.4 (2004): 447-454. <http://eudml.org/doc/207709>.

@article{Felgenhauer2004,

abstract = {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 R^n (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.},

author = {Felgenhauer, Ursula},

journal = {International Journal of Applied Mathematics and Computer Science},

keywords = {strong local optima; optimality conditions; sensitivity differentials; bang-bang control; stability in optimal control; solution structure},

language = {eng},

number = {4},

pages = {447-454},

title = {Optimality and sensitivity for semilinear bang-bang type optimal control problems},

url = {http://eudml.org/doc/207709},

volume = {14},

year = {2004},

}

TY - JOUR

AU - Felgenhauer, Ursula

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

JO - International Journal of Applied Mathematics and Computer Science

PY - 2004

VL - 14

IS - 4

SP - 447

EP - 454

AB - 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 R^n (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.

LA - eng

KW - strong local optima; optimality conditions; sensitivity differentials; bang-bang control; stability in optimal control; solution structure

UR - http://eudml.org/doc/207709

ER -

## References

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- Dontchev A. and Malanowski K. (2000): A characterization of Lipschitzian stability in optimal control, In: Calculus of Variations and Optimal Control (A. Ioffe et al., Eds.). - Boca Raton, FL: Chapman and Hall CRC, Vol. 411, pp. 62-76. Zbl0970.49021
- Felgenhauer U. (2003a): On stability of bang-bang type controls. - SIAM J. Contr. Optim., Vol. 41, No. 6, pp. 1843-1867. Zbl1031.49026
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- Felgenhauer U. (2003c): On sensitivity results for bang-bang type controls of linear systems.- Preprint M-01/2003, Technical University Cottbus, available at http://www.math.tu-cottbus.de/INSTITUT/lsopt/publication/preprints.html. Zbl1031.49026
- Felgenhauer U. (2003d): Optimality and sensitivity properties of bang-bang controls for linear systems. - Proc. 21-st IFIP Conf. System Modeling and Optimization, Sophia Antipolis, France, Dordrecht, The Netherlands: Kluwer, (submitted). Zbl1031.49026
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