Abstract
In this chapter, the complexity of the dynamical control system in the optimal control problem under extension increases. Herein, it is not linear w.r.t. x and u but is still linear w.r.t. the impulsive control variable. Moreover, the matrix-multiplier for the impulsive control depends on the conventional control \(u(\cdot )\) given by Borel functions. The right-hand side of the dynamical system is assumed to be Borel w.r.t. u. The results of the first chapter are derived for this more general formulation. The concept of extension itself does not change so far, as the space of Borel measures yet suffices to describe all feasible trajectories. The chapter ends with seven exercises.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
That is, measurable w.r.t. the Lebesgue measure \(\ell \) generated by length and w.r.t. the Lebesgue–Stieltjes measure \(\mu \), which is the unique Lebesgue completion of the corresponding Borel measure.
- 2.
While the first integral by ds is understood in the conventional Lebesgue sense, the second integral by the vector-valued Borel measure \(d\mu \) is understood in the sense of the Lebesgue–Stieltjes integral of the vector-valued Borel function G(u(s), s)w(s) by the Borel measure \(|\mu |\), where function \(w(\cdot )\) means the Radon–Nikodym derivative of \(\mu \) w.r.t. \(|\mu |\).
- 3.
- 4.
Weak-* convergence in \(\mathrm{BV}(T;\mathbb {R}^n)\).
- 5.
Note that (2.26) is a free right endpoint problem, and therefore, its extremals are normal.
References
Clarke, F.: Optimization and Nonsmooth Analysis. Wiley-Interscience, New York (1983)
Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)
Filippov, A.: On certain problems of optimal regulation. Bull. Moscow State Univ. Ser. Math. Mech. (2), 25–38 (1959)
Mordukhovich, B.: Variational analysis and generalized differentiation I. Basic theory. In: Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 330. Springer, Berlin (2006)
Mordukhovich, B.: Variational analysis and generalized differentiation II. Applications. In: Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 331. Springer, Berlin (2006)
Rockafellar, R., Wets, R.B.: Variational Analysis. Grundlehren der Math. Wissenschaften. Springer (1998)
Vinter, R.: Optimal Control. Birkhäuser, Boston (2000)
Vinter, R., Pereira, F.: Maximum principle for optimal processes with discontinuous trajectories. SIAM J. Control Optim. 26(1), 205–229 (1988)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Arutyunov, A., Karamzin, D., Lobo Pereira, F. (2019). Impulsive Control Problems Under Borel Measurability. In: Optimal Impulsive Control. Lecture Notes in Control and Information Sciences, vol 477. Springer, Cham. https://doi.org/10.1007/978-3-030-02260-0_2
Download citation
DOI: https://doi.org/10.1007/978-3-030-02260-0_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-02259-4
Online ISBN: 978-3-030-02260-0
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)