Optimal Impulsive Control pp 1-18 | Cite as

# Linear Impulsive Control Problems

## Abstract

In this chapter, the simplest impulsive extension of a control problem which is feasible in the case of linear dynamical control systems is described. The chapter begins by considering several typical examples of linear control problems for which the appearance of discontinuities in admissible trajectories is natural, since it fits into their physical representation (under certain assumptions made from the point of view of the mathematical model). In particular, the well-known Lawden’s problem of the motion of a rocket is examined here and it is demonstrated how discontinuities of extremal trajectories inevitably arise. Next, we give a theorem on the existence of a solution to the extended problem and another theorem concerning necessary optimality conditions in the form of Pontryagins maximum principle, which, in the linear case, are expressed in a sufficiently simple and clear way. The chapter ends with 11 exercises.

## References

- 1.Arutyunov, A.V.: Optimality Conditions. Abnormal and degenerate problems. Mathematics and its applications. Kluwer Academic Publishers, Dordrecht (2000)zbMATHGoogle Scholar
- 2.Ioffe, A., Tikhomirov, V.: Studies in Mathematics and Its Applications, vol. 6. Elsevier Science, North-Holland, Amsterdam (1979)Google Scholar
- 3.Kolmogorov, A., Fomin, S.: Introductory Real Analysis (Dover Books on Mathematics). Dover Publications (1975)Google Scholar
- 4.Lawden, D.: Optimal Trajectories for Space Navigation (Butterworths mathematical texts) (1963)Google Scholar
- 5.Lawden, D.: Rocket trajectory optimization—1950–1963. J. Guid. Control Dyn.
**14**(4), 705–711 (1991)MathSciNetCrossRefGoogle Scholar - 6.Mordukhovich, B.: Maximum principle in the problem of time optimal response with nonsmooth constraints. J. Appl. Math. Mech.
**40**, 960–969 (1976)MathSciNetCrossRefGoogle Scholar - 7.Mordukhovich, B.: Variational Analysis and Generalized Differentiation I. Basic Theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 330. Springer, Berlin (2006)Google Scholar
- 8.Pontryagin, L., Boltyanskii, V., Gamkrelidze, R., Mishchenko, E.: Mathematical theory of optimal processes. Transl. from the Russian ed. by L.W. Neustadt, First Edition. Interscience Publishers, Wiley Inc. (1962)Google Scholar