A Note about the Linear Time-Optimal Control Problem

Maxim V. Balashov

doi:10.68381/jca34003

Abstract

In non-autonomous time-optimal control problems the question of the sufficiency of the maximum principle is considered. We also briefly discuss the question of uniqueness of optimal control. New conditions are obtained that guarantee the sufficiency of the maximum principle in terms of convex analysis, properties of matrices in the linear system, the geometry of the control set and the nonempty interior of some set-valued integrals. Examples that demonstrate the unimprovability of the obtained results are considered

Author information

Contact details are reproduced from the original publication and may be historical.

Maxim V. Balashov

V. A. Trapeznikov Institute of Control Sciences, Moscow, Russia

balashov73@mail.ru

Keywords

Maximum principle
support function
set-valued integral
nonsmooth analysis

Mathematics Subject Classification

49J52, 49J53, 49N05; 93C05

References

J.-P. Aubin, H. Frankowska: Set-Valued Analysis, Birkhäuser, Boston (2009).https://doi.org/10.1007/978-0-8176-4848-0 Google Scholar
G. Birkhoff, G.-C. Rota: Ordinary Differential Equations, 4th ed., John Wiley and Sons, New York (1989).Google Scholar
V. G. Boltyanskii: Mathematical Methods of Optimal Control, Holt, Rinehart and Winston, New York (1971).https://doi.org/10.1115/1.3426517 Google Scholar
P. Cannarsa, C. Sinestrari: Convexity properties of the minimum time function, Calc. Var. 3 (1995) 273–298.https://doi.org/10.1007/BF01189393 Google Scholar
L. C. Evans, M. R. Janaes: The Hamilton-Jacobi-Bellman equation for time optimal control, SIAM J. Control Optim. 27 (1989) 1477–1489.https://doi.org/10.1137/0327076 Google Scholar
A. F. Filippov: On some questions of the theory of optimal control, Moscow Univ. Phys. Bull. 2 (1959) 25–32.Google Scholar
C. A. Harvey, E. B. Lee: On the uniqueness of time-optimal control for linear processes, J. Math. Anal. Appl. 5 (1962) 258–268.https://doi.org/10.1016/S0022-247X(62)80009-2 Google Scholar
H. Hermes, J. P. LaSalle: Functional Analysis and Time Optimal Control, Academic Press, New York (1969).Google Scholar
A. D. Ioffe, V. M. Tikhomirov: Theory of Extremal Problems, Elsevier North-Holland, Amsterdam (1979).Google Scholar
S. G. Krantz, H. R. Parks: A Primer of Real Analytic Functions, 2nd ed., Birkhäuser, Boston (2002).https://doi.org/10.1007/978-0-8176-8134-0 Google Scholar
E. B. Lee, L. Markus: Foundations of Optimal Control Theory, John Wiley and Sons, New York (1967).Google Scholar
N. N. Petrov: Controllability of autonomous systems, Diff. Equations 4 (1968) 311–317.Google Scholar
N. N. Petrov: On the Bellman function for the time-optimal process problem, J. Appl. Math. Mech. 34 (1970) 785–791.https://doi.org/10.1016/0021-8928(70)90060-2 Google Scholar
L. S. Pontryagin, V. G. Boltyanskij, R. V. Gamkrelidze, E. F. Mishchenko: The Mathematical Theory of Optimal Processes, Interscience Publishers, John Wiley and Sons, New York (1962).Google Scholar
P. Soravia: Holder continuity of the minimum-time function for C 1 -manifold targets, J. Optim. Theory Appl. 75 (1992) 401–421.https://doi.org/10.1007/BF00941476 Google Scholar

Suggested citation

[1] J.-P. Aubin, H. Frankowska: Set-Valued Analysis, Birkhäuser, Boston (2009).https://doi.org/10.1007/978-0-8176-4848-0 Google Scholar

[2] G. Birkhoff, G.-C. Rota: Ordinary Differential Equations, 4th ed., John Wiley and Sons, New York (1989).Google Scholar

[3] V. G. Boltyanskii: Mathematical Methods of Optimal Control, Holt, Rinehart and Winston, New York (1971).https://doi.org/10.1115/1.3426517 Google Scholar

[4] P. Cannarsa, C. Sinestrari: Convexity properties of the minimum time function, Calc. Var. 3 (1995) 273–298.https://doi.org/10.1007/BF01189393 Google Scholar

[5] L. C. Evans, M. R. Janaes: The Hamilton-Jacobi-Bellman equation for time optimal control, SIAM J. Control Optim. 27 (1989) 1477–1489.https://doi.org/10.1137/0327076 Google Scholar

[6] A. F. Filippov: On some questions of the theory of optimal control, Moscow Univ. Phys. Bull. 2 (1959) 25–32.Google Scholar

[7] C. A. Harvey, E. B. Lee: On the uniqueness of time-optimal control for linear processes, J. Math. Anal. Appl. 5 (1962) 258–268.https://doi.org/10.1016/S0022-247X(62)80009-2 Google Scholar

[8] H. Hermes, J. P. LaSalle: Functional Analysis and Time Optimal Control, Academic Press, New York (1969).Google Scholar

[9] A. D. Ioffe, V. M. Tikhomirov: Theory of Extremal Problems, Elsevier North-Holland, Amsterdam (1979).Google Scholar

[10] S. G. Krantz, H. R. Parks: A Primer of Real Analytic Functions, 2nd ed., Birkhäuser, Boston (2002).https://doi.org/10.1007/978-0-8176-8134-0 Google Scholar

[11] E. B. Lee, L. Markus: Foundations of Optimal Control Theory, John Wiley and Sons, New York (1967).Google Scholar

[12] N. N. Petrov: Controllability of autonomous systems, Diff. Equations 4 (1968) 311–317.Google Scholar

[13] N. N. Petrov: On the Bellman function for the time-optimal process problem, J. Appl. Math. Mech. 34 (1970) 785–791.https://doi.org/10.1016/0021-8928(70)90060-2 Google Scholar

[14] L. S. Pontryagin, V. G. Boltyanskij, R. V. Gamkrelidze, E. F. Mishchenko: The Mathematical Theory of Optimal Processes, Interscience Publishers, John Wiley and Sons, New York (1962).Google Scholar

[15] P. Soravia: Holder continuity of the minimum-time function for C 1 -manifold targets, J. Optim. Theory Appl. 75 (1992) 401–421.https://doi.org/10.1007/BF00941476 Google Scholar