Abstract
The paper is devoted to studying the impulse optimal control problem with inequality-type state constraints and geometric control constraints defined by a measurable multivalued mapping. The author obtains necessary optimality conditions in the form of the Pontryagin maximum principle and nondegeneracy conditions for the latter.
Similar content being viewed by others
References
A. V. Arutyunov, Necessary Extremum Conditions [in Russian], Faktorial, Moscow (1997).
A. V. Arutyunov, “Relaxations and perturbations of optimal control problems,” Tr. Mat. Inst. Ross. Akad. Nauk, 220, 27–34 (1998).
A. V. Arutyunov, “Perturbations of constrained extremal problems and necessary optimality conditions,” In: Progress in Science and Technology, Series on Mathematical Analysis [in Russian], Vol. 27, All-Union Institute for Scientific and Technical Information (VINITI), Akad. Nauk SSSR, Moscow (1989), pp. 147–235.
A. V. Arutyunov and N. T. Tynyanskii, “On the maximum principle in the state-constrained problem,” Izv. Akad. Nauk SSSR, Ser. Tekh. Kibern., 4, 60–68 (1984).
A. V. Arutyunov, “On necessary optimality conditions in the state constrained problem,” Dokl. Akad. Nauk SSSR, 280, No. 5, 1033–1037 (1985).
A. V. Arutyunov, “First-order necessary conditions in the state constrained optimal control problem,” Tr. Inst. Prikl. Mat. Tbilis. Univ., 27, 46–59 (1988).
A. V. Arutyunov, “Maximum principle and second-order necessary optimality conditions in the optimal control problem with delays,” Soobshch. Akad. Nauk Grus.SSR, 122, No. 2, 265–268 (1986).
A. V. Arutyunov, S. M. Aseev, and V. I. Blagodatskikh, “First-order necessary conditions in the optimal control problem for a differential inclusion with state constraints,” Mat. Sb., 184, No. 6, 3–32 (1993).
A. V. Arutyunov and S. M. Aseev, “Investigation of the degeneracy phenomenon of the maximum principle for optimal control problems with state constraints,” SIAM J. Control Optim., 35, No. 3, 930–952 (1997).
A. V. Arutyunov, V. Jacimovic, and F. L. Pereira, “Second order necessary conditions of optimality for impulsive control problems,” Int. J. Dyn. Contr. Syst., 9, No. 1, 131–153 (2003).
A. V. Arutyunov, V. A. Dykhta, and F. L. Pereira, “Necessary conditions for impulsive nonlinear optimal control problems without a priori normality assumptions,” J. Optim. Theory Appl. (in press).
V. I. Blagodatskikh and A. F. Filippov, “Differential inclusions and optimal control,” Tr. Mat. Inst. Akad. Nauk SSSR, 169, 194–252 (1985).
A. Bressan and F. Rampazzo, “Impulsive control systems with commutative vector fields,” J. Optim. Theory Appl. 71, 67–83 (1991).
F. Clarke, Optimization and Nonsmooth Analysis [Russian translation], Nauka, Moscow (1988).
V. F. Dem’yanov, Minimax: Differentiability in Directions [in Russian], LGU, Leningrad (1974).
A. Ya. Dubovitskii and V. A. Dubovitskii, “Necessary conditions for the strong minimum in optimal control problems with degeneration of endpoint and state constraints,” Usp. Mat. Nauk, 40, No. 2, 175–176 (1985).
A. Ya. Dubovitskii and A. A. Milyutin, “Extremum problems under constraints,” Dokl. Akad. Nauk SSSR, 149, No. 4, 759–762 (1963); Zh. Vychisl. Mat. Mat. Fiz., 5, No. 3, 395–453 (1965).
V. A. Dykhta and O. N. Samsonyuk, Optimal Impulse Control with Applications [in Russian], Fizmatlit, Moscow (2000).
A. F. Filippov, “On certain problems of optimal regulation,” Vestn. MGU, Mat. Mekh., No. 2, 25–38 (1959).
A. F. Filippov, “Differential equations with discontinuous right-hand side,” Mat. Sb., 51, No. 2 100–128 (1966).
R. V. Gamkrelidze, Foundations of Optimal Control [in Russian], Tbilis. Univ., Tbilisi (1977).
M. I. Gusev, “On optimal control of generalized processes under nonconvex state constraints,” in: Differential Games and Control Problems [in Russian], UNTs Akad. Nauk SSSR, Sverdlovsk (1975).
A. D. Ioffe and V. M. Tikhomirov, “Several remarks on variational principles,” Mat. Zametki, 61, No. 2, 305–311 (1997).
A. D. Ioffe and V. M. Tikhomirov, Theory of Extremal Problems [in Russian], Nauka, Moscow (1974).
A. P. Kartashev and B. L. Rozhdestvenskii, Mathematical Analysis [in Russian], Moscow (1984).
A. A. Kirillov and A. D. Gvishiani, Theorems and Problems of Functional Analysis [in Russian], Nauka, Moscow (1979).
A. N. Kolmogorov and S. V. Fomin, Elements of Function Theory and Functional Analysis [in Russian], Nauka, Moscow (1968).
N. N. Krasovskii, Theory of Control Motion [in Russian], Nauka, Moscow (1968).
A. B. Kurzhanskii, “Optimal systems with impulse controls,” in: Differential Games and Control Problems [in Russian], UNTs, Akad. Nauk SSSR, Sverdlovsk (1975), pp. 131–156.
A. B. Kurzhanskii and Yu. S. Osipov, “On the control of a linear system by generalized forcings,” Differents. Uravn., 5, No. 8, 1360–1370 (1969).
A. S. Matveev, “On necessary extremum conditions in an optimal control problem,” Differents. Uravn., 23, No. 4, 629–639 (1987).
B. M. Miller, “Optimality conditions in generalized control problems,” Avtomat. Telemekh., No. 5, 50–58 (1992).
B. M. Miller, “Generalized solutions in nonlinear optimization problems with impulse control, I. Existence problem of a solution. II. Representation of a solution via a differential equation with a measure,” Avtomat. Telemekh., No. 4, 62–76 (1995); No. 5, 56–70 (1995).
I. P. Natanson, Theory of Functions of a Real Variable [in Russian], Nauka, Moscow (1974).
F. L. Pereira, “A maximum principle for impulsive control problems with state constraints,” Comput. Appl. Math. 19, No. 2, 137–155 (2000).
F. L. Pereira and G. N. Silva, “Necessary conditions of optimality for vector-valued impulsive control problems,” Systems Control Lett., 40, 205–215 (2000).
F. L. Pereira and G. N. Silva, “Stability for impulsive control systems,” Dyn. Syst., 17, 421–434 (2002).
F. L. Pereira, A. Coimbra de Matos, and G. N. Silva, “Hamilton-Jacobi conditions for an impulsive control problem,” in: Nonlinear Control Systems, Fevereiro (2002), pp. 1297–1302.
F. L. Pereira and G. N. Silva, “Necessary conditions of optimality for vector-valued impulsive control problems,” Syst. Control Lett., 40, 205–215 (2000).
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, Mathematical Theory of Optimal Processes [in Russian], Nauka, Moscow (1976).
W. Rudin, Functional Analysis [Russian translation], Mir, Moscow (1975).
G. N. Silva and R. B. Vinter, “Measure differential inclusions,” J. Math. Anal. Appl., 202, 727–746 (1996).
G. N. Silva and R. B. Vinter, “Necessary conditions for optimal impulsive control problems,” SIAM J. Control Optim., 35, 1829–1846 (1997).
G. N. Silva, I. S. Litvinchev, M. Rojas-Medar, and A. J. V. Brandao, “State constraints in optimal impulsive controls, Comput. Appl. Math. 19, No. 2, 179–206 (2000).
F. P. Vasil’ev, Numerical Methods for Solving Extremal Problems [in Russian], Moscow (1980).
R. B. Vinter and F. L. Pereira, “Necessary conditions for optimal control problems with discontinuous trajectories,” J. Econom. Dyn. Contr., 10, 115–118 (1986).
R. B. Vinter and F. L. Pereira, “A maximum principle for optimal processes with discontinuous trajectories,” SIAM J. Control Optim., 26, 205–229 (1988).
R. B. Vinter and M. M. A. Ferreira, “When is the maximum principle for state constrained problems nondegenerate?” J. Math. Anal. Appl., 187, 438–467 (1994).
J. Warga, Optimal Control of Differential and Functional Equations[Russian translation], Nauka, Moscow (1977).
S. T. Zavalishin and A. N. Sesekin, Impulse Processes: Models and Applications [in Russian], Nauka, Moscow (1991).
Author information
Authors and Affiliations
Additional information
__________
Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 24, Dynamical Systems and Optimization, 2005.
Rights and permissions
About this article
Cite this article
Karamzin, D.Y. Necessary conditions of the minimum in an impulse optimal control problem. J Math Sci 139, 7087–7150 (2006). https://doi.org/10.1007/s10958-006-0408-z
Issue Date:
DOI: https://doi.org/10.1007/s10958-006-0408-z